From news3.cis.umn.edu!umn.edu!sander Wed Mar 31 12:00:07 CST 1993 Article: 63 of geometry.forum Newsgroups: geometry.forum Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Greetings Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: nielsen.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Tue, 30 Mar 1993 21:55:04 GMT Lines: 24 Status: OR Hello, my name is Evelyn Sander, and I have just started as the Minnesota Geometry Center correspondent to the Geometry Forum news groups. Let me tell you a bit about myself: I am a third year graduate student at the University of Minnesota, specializing in Dynamical Systems. A native Midwesterner, being originally from Ann Arbor, Michigan, I attended college at Northwestern University in Evanston, Illinois. It is quite exciting to have the opportunity to write about mathematics. You will soon see a regular flow of articles regarding the people, programs, research, and miscellaneous goings on at the Geometry Center. If you have any questions, comments, or requests, feel free to send me email, and I will respond as much as I can. Look forward to getting to know more about the readers, contributors, and of course about geometry! Evelyn Sander sander@geom.umn.edu (612)626-8316 From news3.cis.umn.edu!umn.edu!sander Wed Mar 31 17:29:52 CST 1993 Article: 15 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Rick Wicklin, 4/1, University of Minnesota Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 31 Mar 1993 23:18:48 GMT Lines: 10 Status: OR University of Minnesota Dynamics and Mechanics Seminar Thursday, April 1, 1993, 11:15 a.m. Murphy Hall 130 (Minneapolis Campus) Rick Wicklin from the Center for Applied Mathematics, Cornell University will speak on: "Dynamics Near Multiple Resonances of Multi-frequency Systems" Rick Wicklin will be a postdoc at the Geometry Center next year. From news3.cis.umn.edu!umn.edu!sander Wed Mar 31 18:55:49 CST 1993 Article: 28 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Interview with Bob Devaney, Part 1: Communication Among All Educators Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Thu, 1 Apr 1993 00:51:22 GMT Lines: 59 Status: OR Boston University professor Bob Devaney is in Minnesota this week, as part of the Minnesota Math Mobilization program. He is usually in Boston doing research in Complex Dynamics, but for his spring break, he has chosen to travel around the state giving talks to audiences of high school math teachers as well as some college students. I had a chance to talk to him this morning immediately before he left for Duluth. Professor Devaney is concerned about the lack of communication between high school math teachers and college math teachers and researchers. "It's ridiculous to have two groups of mathematicians so divided; for the most part, they do not talk to one another. Most mathematicians at the college level don't even know what NCTM stands for. And yet high school teachers are the ones responsible for getting students interested in mathematics." What exactly is turning students away from math? "Our society currently views it as acceptable to dislike math. High school students decide math is boring so they're going to stop taking it; their parents don't object. They say, 'I got by with only eighth grade math, so my kids will be able to do the same.' "Most high school teachers are afraid of changing the curiculum to incorporate technology and modern mathematics. The lack of communication with researchers makes it very difficult for them. "These attitudes are part of why our country is in the state it's in today. We need to make an effort to get students interested again. I think it's time for mathematicians to do what everyone else has been doing for a long time; namely, popularize and advertize." How can we popularize mathematics? Professor Devaney has been active in a program affiliated with Boston University which works with Boston inner city schools: "The purpose of the program was twofold; first, we wanted to help the teachers learn to use the technology effectively. The schools all had computers, but we had to find a way to incorporate them into the curiculum. "The second goal of the program was to add modern mathematics into the courses. The teachers really resisted this, so in order to convivce them, we formed an after school Chaos Club. Once a week we would have some activity, usually on the computer. It was always something fun but also always having to do with math. The club was very popular with students. The teachers were amazed at all these students staying after school to do math. It was quite persuasive of the importance of this kind of material. Now the teachers have taken over the club." For the specific material that Professor Devaney feels is appropriate for high school students, as well as a desription of the talks he is giving in Minnesota this week, please see the part two of this article. From news3.cis.umn.edu!umn.edu!sander Thu Apr 1 18:04:30 CST 1993 Article: 29 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: World Construction Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: klein.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Thu, 1 Apr 1993 23:59:49 GMT Lines: 61 Status: OR At the beginning of May, the University of Minnesota's Institute of Technology will have their annual festival, known as I.T. Week. On May 4, the organizing committee has invited students grades 4-9 to come help celebrate by building a 42 foot 1:1,000,000 scale model of the earth in Northrup Mall, the central area of campus. Starting this month, I.T. alumni will go to elementary and junior high schools all over Minnesota for the first part of the construction project. The students will cut and paint 1620 triangular panels, each approximately one yard across, which will eventually make up the globe. One May 4, the students will come to the university put all the panels together to build the earth. The construction will involve a bicycle powered hydrolic lift, enabling the students to be responsible for every part of the project. This new world project is a quite interdisiplinary effort, calling for interaction between engineers, such as organizer Bryan Beaulieu, elementary and junior high teachers, students, I.T. alumni, and, as you might expect by the newsgroup choice, geometers. Stuart Levy of the Geometry Center got involved with the project because in order to use flat material to build a sphere, someone needs to calculate the geodesic structure. I spoke to Mr. Levy this afternoon about the project. He gave me some of the particulars of working on this applied project. It is quite intriguing, especially since it is somewhat rare for engineers and geometers to work together, so concerns such as strength of construction and size of panel are easy to forget. The material used is a thin plastic which comes in one yard sheets. In order to maximize the stability of the globe, it was best for the panels to be triangles. Under these conditions, Mr. Levy came up with the following triangulation scheme: starting with an iscosohedron, split the faces nine-fold, slightly raising each of the new triangles to give a closer approximation of the sphere. Use two successive applications of this process to get the proper size of panels. After designing the triangulation and underlying structure of the earth, the next step was projecting maps onto each of the panels. The students will use these maps to paint the panels with rivers, other geography, as well as political boundaries. This involved choosing the best projection to use, which turns out to be the stereographic projection. In order to make a map which is larger than one panel, a better choice of projection is one which unrolls the surface onto a tangent plane while preserving arclength. On a larger map, the stereographic projection distorts size in a distracting way. This program is an exciting way to bring people of all ages and many diciplines together for a common purpose. If you are in the area on May 4, come to campus to watch the final construction of the world. From news3.cis.umn.edu!umn.edu!sander Fri Apr 9 16:30:02 CDT 1993 Article: 38 of geometry.pre-college Newsgroups: geometry.pre-college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Some creative teaching techniques Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: klein.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 9 Apr 1993 21:07:25 GMT Lines: 109 Status: OR "The world comes to you in a mess, and out of the mess you have to discover something mathematical." This is the opinion of Arnie Cutler, Education Consultant at the Minnesota Geometry Center. These words seem to serve as a guide to him; in addition to his position at the center, he has instructed in the math education department at the University of Minnesota, in three weeks will begin his term as President of the Minnesota Council of Teachers of Mathematics, and a math teacher at a local high school. In this article, rather than talk about organizations and committees, I will describe Cutler's actions and opinions as a teacher. "I make a big distinction between problems and exercises," he tells me. "Exercises are canned versions of what has already been done in the book. They have an easy answer, and the student doesn't learn much by doing them. Problems are more difficult. They are often taken from real situations. The students have to figure out what method to use to start them. "In my classes I try to have a lot of problems and very few exercises. There are plenty of exercises in the books if they want to work them. In class we need to do something more challenging. Sometimes students complain, saying that they don't know what to do to solve a problem. I tell them that there would be no point giving them a problem that they already knew how to solve." In order to solve a problem, Cutler breaks the class into small groups and let them work on it for a while. Solving may take days, or even weeks. He encourages them to think for themselves, always saying, "The real world doesn't have an answer book." After the groups have worked for some time, Cutler chooses someone in the class to go to the board and present what their group has done. The students quickly learn that there is more than one method to do a problem. Even if someone gives a correct answer to a problem, he asks other to show their solutions. In past years, Cutler has assigned a variety of problems to his classes. For example, last year his calculus class considered the path and length of path taken by a man standing on a ladder as the ladder slips down a wall. The work groups each designed a model for the problem, the most successful being a lego construction with a pen attached; the pen traced the path on a wall beside the ladder. Another problem from earlier this year was to calculate center of mass of the textbook example of a meter stick with weights at the ends. The students realized that they were going to have to use integration now that the meter stick itself had nonzero mass. Most recently, Cutler's calculus class spent several weeks working on the problem of designing a food drop for Bosnia. They spent the first week doing background research. For example, they had to read the paper and decide which cities most needed the supplies. Then they had to find the coordinates of these cities and the coordinates of the air bases. They considered which airplane to use for the drop, evaluating the specifications on each plane. After some work, the class decided that the Defense Department was right; the C-130 is the best plane for the job. It turned out to be quite difficult to find a weather service with information on the current prevailing winds near the cities chosen for the drop. The next step was to come up with a mathematical model. At first the class got bogged down in small details. They worried about whether there would be a small hill or someone standing right where they were trying to make the drop. They soon realized the need for simplifying assumptions in mathematical modelling; they assumed the wind would have the predicted average for direction and magnitude. Finally the students were ready to write down some equations, using a variety of mathematical ideas. They had to find the angles between two planes when decribing the airplane's bearing. They needed to coordinatize the earth working with spherical and rectangular coordinates. They used the idea of a great circle when calculating the airplane's path. In order to avoid having the boxes break, they had to figure out at what height the parachutes on the boxes needed to open. This meant they had a two stage drop, so after they were all done with the equations, they had to calculate derivatives to make sure the pre and post parachute stages matched up. The students really enjoyed the experience. It was problem of current news interest. However, they mainly were happy that they could pick up their text books and turn thirty pages, on each page seeing at least one concept they had used in their solution. I asked Cutler to tell me how he thinks of these problems: "Keep your eyes open. You can see math everywhere. I always look for a problem that is interesting and deals with real life. It also needs to apply topics that the students have learned." He points out the window at a bridge spanning the Mississippi river. "Look at that bridge. I wonder what curve the underside of it traces. I can't tell off the top of my head, but with a few measurements and a bit of work, I bet we could make a good guess." From news3.cis.umn.edu!umn.edu!sander Fri Apr 9 16:47:38 CDT 1993 Article: 39 of geometry.pre-college Newsgroups: geometry.pre-college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Devaney Part 2: Mandelbrot Set Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: klein.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 9 Apr 1993 21:29:45 GMT Lines: 77 Status: OR The following is the second in a two part series of articles based on an interview with Boston University professor Bob Devaney. Educators often spend time considering what mathematics concepts are appropriate for high school students. According to Professor Devaney, the process of iteration should be taught in high school. The concept comes up in savings and interest problems and Newton's method. Chaos, fractals, talking about the Mandelbrot set: these are fun ways to introduce the iteration. Also, high school students often have seen a bit of complex arithmetic. The Mandelbrot set is a nice illustration of this concept. Professor Devaney thinks that detailed discussion of dynamics is most appropriate for after school math clubs, or as a last topic at the end of the term when there is often some dead time anyway. However, he says: "Students spend about two or three weeks learning how to factor cubic polynomials; there is no reason why they shouldn't also spend a couple weeks learning about the Mandelbrot set." As to the actual presentation of the Mandelbrot set appropriate for high school students, here is a general outline. It is perhaps even of interest to people who are not in high school but only know the Mandelbrot set from seeing pictures of it. The explanation should involve quite a bit of computer demonstration, but perhaps this description will give the idea. Consider the real one dimensional quadratic map f(x)=x*x+1. Iterate zero under this map. The first few iterates are 1,2,5,26. It is fairly clear that under successive iterations, one gets an arbitrarily large number. Now consider the map g(x)=x*x+0. In this case, g(0)=0, so there is a fixed point at the origin. For h(x)=x*x-1, h(0)=-1, and h(-1)=0, so zero is a period two point. The iterates of the origin under maps of the form of f, g, and h are generally hard to calculate. Putting these maps into a common form, add a parameter, and consider f(x,c)=x*x+c. Consider c=-1.1. If you try this in your head, you see that it is not so easy. At this stage, one needs a program which graphs iterates of the origin under f. Using this program, experiment with different values of c to see what sort of behavior occurs. At c=-2, the picture becomes chaotic. Now switch to the complex plane. In other words, consider the function f(z,c) defined above, except that now z and c can be complex numbers. For example, for c=i, I calculate the first two iterates of zero: f(0,i)=i and f(i,i)=-1+i. The Mandelbrot set is defined to be all the c values for which iterates of the origin stay bounded under iteration. At this point, one should use a computer program to draw the Mandelbrot set. Then, use a program with two windows representing copies of the complex plane: in one window, the user picks a complex c value with a mouse. After the user picks c, the other window shows iterates of the origin. This program gives the user an idea of what it means for iterates to get arbitrarily large. It is at this point possible to learn about the period of the periodic points in the Mandelbrot set based on the point's placement in one of the bulb shaped parts of the Mandelbrot set. For this and further discussion, I refer to Professor Devaney's books. The above explanation is a fairly simple and fun way to teach the concepts of iteration, complex arithmetic, and boundedness. It answers the concern that all this popularization of fractals and the Mandelbrot set is just a matter of looking at pretty pictures without any knowledge of the math; although this is not a complete explanation of the mathematics involved, it does seem to give a partial understanding with little needed background. From news1.cis.umn.edu!umn.edu!sander Sat Apr 10 11:21:30 CDT 1993 Article: 31 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: correction: Devaney Part 2: Mandelbrot Set Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota References: Date: Sat, 10 Apr 1993 16:15:13 GMT Lines: 5 Status: OR I made an error in the article on the Mandelbrot set: Under the map f(z)=z*z-2, the origin is eventually fixed. However, for any real number in the interval (-2,2) which is not an integer, the orbit of zero will be chaotic. From news3.cis.umn.edu!umn.edu!sander Wed Apr 14 14:23:14 CDT 1993 Article: 69 of geometry.forum Newsgroups: geometry.forum Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Glamour article: brains and gender Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 14 Apr 1993 18:36:48 GMT Lines: 22 Status: OR The April issue of Glamour has an article addressing the question of whether there is a real physical difference between the brains of men and women. Popular science articles in the last few years have given a lot of clout to the notion that a biological difference in the brains of men and women result in women's inability to do well in (well paid) technical professions. Specifically, articles claimed that women have less ability to do mathematics. This article disputes the above claim. The article is particularly interesting for those at the University of Minnesota, since it has a half page panel on the University of Minnesota Talented Youth Mathematics Program. I will soon be writing an article on this program here in the geometry newsgroups. The Glamour article is serious and well written. It is rather encouraging to see that a fashion magazine is willing to address this sort of issue. Of course, the article is towards the back of the magazine; it did not get mentioned on the cover, having been outranked in importance by "Ooh La La Lashes!" and "Topless Dancing: why I do it, why I like it." Things don't change overnight. From news3.cis.umn.edu!umn.edu!sander Mon Apr 19 17:47:36 CDT 1993 Article: 16 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Geometry Center Weekly Seminar Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Mon, 19 Apr 1993 22:43:07 GMT Lines: 5 Status: OR Starting this week, the Geometry Center will be holding a weekly seminar Mondays at 4pm. I'm sorry that this announcement is too late for this week's seminar. It was given by Dan Freed, regarding the principles of topological quantum field theory. From news3.cis.umn.edu!umn.edu!sander Tue Apr 20 13:54:40 CDT 1993 Article: 17 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: April Geometry Center Schedule Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Tue, 20 Apr 1993 16:26:34 GMT Lines: 92 Status: OR Here is a list of April announcements for the Geometry Center. As of a week or two ago, the list of seminars, the list of current geometry center visitors, job openings and various other cool information is available to everyone just by saying "finger geom@geom.umn.edu" April Visitors: --------- Dennis Roseman 4/11-4/24 Dan Freed 4/16-4/19 Igor Rivin 4/17- Harold Parks 4/18- Maria Klawe 4/26-4/27 Seminars: --------- 4:00 Monday April 19th - Prof. Dan Freed (U. of Texas, Austin and Geometry Center), Introduction to Topological Quantum Field Theory. (in the Classroom) For more info about the (informal) seminars, or if you would like to give a talk, please contact David Ben-Zvi, benzvi@geom 626-8304. Tours: ------ First and third Tuesdays of the month, 3:30-4:30, by appointment. Contact Tamara Munzner, munzner@geom 626-8325. Positions available: ----------------- Apprenticeships (students defer schooling for 6mo or more, very strong math and/or CS background required) Undergrad C Programmer (part time) Sr. Secretary Director of Technology Write admin@geom.umn.edu for more information. ===================================================================== ===================================================================== Other activities: April Geometry Lectures at U. of Wisconsin, Madison: --------------------------------------------------- The following are the abstracts and preview of the April lectures sponsored by The Center for the Mathematical Sciences, Computational and Visual geometry Working Group. Almost all of the lectures will include interactive computer graphics and demonstration of software on Silicon Graphics. All Wednesday and Monday lectures are intended for an audience with diverse background, in particular students. The Thursday lectures assume more mathematical background as indicated by the abstracts. For more information, please contact Amir Assadi, assadi@math.wisc.edu. Computational and Visual Geometry Seminar April 1993 1) Speaker: Ken Brakke (The Geometry Center, U of Minn) Time: 4:00pm Wednesday April 14 Place: TBA (most likely 901 vanVleck) Title: The Surface Evolver 2) Special Lecture (Please note change in time and place) Speaker: Ken Brakke The Geometry Center, U of Minn. Time: 2:30pm Thursday April 15 Place: B131 vanVleck Title: Soap Films and Covering Spaces 3) Speaker: Sanjay Tiwari (UW Madison) Time and Place: April 19, 4:00 pm, Mech. Eng. 174 (tentative room). (This is the lecture originally announced for March 31). 4) Speakers: Nathaniel Thurston (UC Berkeley and The Geometry Center) and David Ben-Zvi (Princeton and The Geometry Center) Titles : TBA Place: TBA Time and Date: 4:00 pm April 21 5) Speaker: Dennis Roseman (University of Iowa and The Geometry Center) Title: TBA Place: TBA Time: 4:00 pm April 28 6) Speaker: Dennis Roseman (Please note change in time and place) Title: TBA Time: 2:30 pm Thursday April 29 Place: TBA From news3.cis.umn.edu!umn.edu!sander Wed Apr 21 12:04:49 CDT 1993 Article: 41 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Geometry Center Movie Part 1 Message-ID: Summary: A mathematical description and history of sphere eversion. Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 21 Apr 1993 16:39:34 GMT Lines: 80 Status: OR The Geometry Center is currently making a movie called "Outside In," a visual exploration of sphere eversion. If you have seen "Not Knot," then you are familiar with the geometry film genre. This new movie will be of similar spirit; it uses one of the same directors and the same math consultants as the previous movie. The following article concentrates on the mathematical background. There will be a follow-up article about some of the people involved in making the movie. Sphere eversion has nothing to do with aversion to spheres (though this is the first impression several people had when I mentioned the term). Actually, eversion means turning the sphere inside out by pulling it through itself allowing for self intersections but without creating any holes, folds, or creases. In other words, a sphere eversion is a regular homotopy between two immersions of the sphere with opposite orientations, where a regular homotopy is defined to be a smooth deformation through immersions. Though sphere eversion is a simple problem to state, it turns out to be quite difficult to produce. The subject has an interesting history; in 1957, Stephen Smale came up with a proof about the general concept of embeddings of the sphere in three-space; the right side in and inside out spheres are examples of this concept. He proved that any two of these embeddings are regularly homotopic. People were initially quite skeptical of this result; the equivalent statement in the plane is simply false. That is, to be able to smoothly deform between two immersions of the circle in the plane, the immersions must have the same winding number. Winding number is just a count of the number of times that a loop goes around counterclockwise, with once around clockwise counting as minus one times around counterclockwise. In addition to the failure of the circle in the plane, an equivalent statement about embeddings of the torus in three space is also false; there are four distinct classes of embeddings of the torus which are not regularly homotopic to each other. After the initial skepticism, people believed the result but had no idea how to actually construct an eversion. Around seven years after Smale's result, Arnold Shapiro finally produced a concrete eversion for the sphere. In terms of popular awareness, in May, 1966, Tony Phillips wrote an illustrated article in Scientific American about regular homotopy of two immersions of the sphere, although not an actual eversion. The article also includes pictures and discussion of the four classes of tori mentioned above and a more complete explanation of the mathematics involved in Smale's theorem. According to George Francis, the article's effect on differential topologists was that "everybody" started looking for a better and clearer eversion [Francis, private correspondence]. I am not a differential topologist, and I think the article is quite good and fairly easy to read. Even for two of my non mathematician friends who read Scientific American at the time, the article was worthy; they each independently remembered it when I started describing the movie. Since the work of Shapiro, many other people have come up with eversions and illustrations of them. One notable source of illustrations of a large variety of eversions, as well as most of the historical information just mentioned, is A Topological Picturebook by George K. Francis. It is an enjoyable book for readers with any level of mathematical sophistication. "Outside In" describes an eversion created by Bill Thurston in the mid 70's. I recently got a preview of the movie, including the footage of the actual eversion. It is a quite impressive clip; the sphere deforms into something closely resembling an octopus, turns itself inside out and back into a sphere. One of my favorite eversions is a preliminary test in which the outside of the sphere has on it the face of Smale, but when the sphere everts, there on the inside is the face of Thurston. The potential date of completion for "Outside In" is the end of 1993. Though it is indeed "coming to a theater near you" if you happen to live in Minneapolis near the Minnesota Geometry Center, it may be necessary for others to procure a copy. When the movie is actually finished, the Center will post further information on how to get a copy. From news3.cis.umn.edu!umn.edu!sander Thu Apr 22 15:00:20 CDT 1993 Article: 18 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Discussion, computing problems in hyperbolic geometry Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 21 Apr 1993 17:03:43 GMT Lines: 4 Status: OR This Thursday at 4pm Minnesota Geometry Center, small seminar room Discussion on computing problems in hyperbolic geometry Led by Igor Rivin. From news1.cis.umn.edu!umn.edu!sander Thu Apr 22 15:17:12 CDT 1993 Article: 42 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Geometry Center Movie Part 2 Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota References: Date: Thu, 22 Apr 1993 20:05:42 GMT Lines: 69 Status: OR This is the second in a two part series on the movie "Outside In," a visual exploration of sphere eversion. This is a movie in progress at the Minnesota Geometry Center. The first part was a mathematical description of the plot; this part is a look at the people in charge. This is the second movie made at the Center, the first one being "Not Knot," a visual description of the basics of knot theory. The aim in making the two movies is to provide guidance and motivation for geometers, while at the same time presenting material at a low enough level that high school students can also learn something and enjoy it. To accommodate this diverse audience, the movies introduce the mathematics using professionally edited, fast paced animation. The Geometry Center is successfully making "Outside In" a movie with these features by judiciously choosing a pair of directors, one of whom is expert in artistic computer animation and visually pleasing graphics, the other of whom is able to work out the necessary math. There are many other people at the Center responsible for the success of the movie, but in this article, I concentrate on giving a description of these two directors and the experience they bring to the movie making process. Silvio Levy comes from the mathematical side of the spectrum. He got his PhD under the advisement of Professor Bill Thurston; currently he lives in Berkeley, working as a math editor and mathematician at large. Though at a remote location, he is a full time employee of the Geometry Center. Recently Levy edited and did research for an important new book by Epstein et. al. on the word problem and automatic groups. Levy has worked with Thurston in various capacities ever since graduate school. For years, he has been editing Thurston's book "Three-Dimensional Geometry and Topology." This is an expanded version of Thurston's 1978-9 unpublished lecture notes, available on an informal basis only, first through Princeton University and now through the Geometry Center. Despite the fact that they have not yet been published, the notes and draft of the new book have been influential. In the mathematics literature, there are many references to them. With the help of Levy and many others, Thurston is now sending the first volume to publishers. Levy is also in charge of a geometry lab at Berkeley which is doing computer animation of a two tetrahedron gluing discussed in this book. "Outside In" is also based on work of Thurston. Unlike Levy, joint director Delle Maxwell is not a mathematician but an artist and animator. Maxwell attended the Rhode Island School of Design, followed by graduate work at the Media Lab at MIT. After this, she worked in Japan, creating a computer generated commentator on a television series "Warnings from the 21st Century." More recently, she went to work for commercial animation company PDI. In Japan and at then more extensively at PDI, Maxwell learned how to do computer animation. The PDI animation system was not entirely automated, so everyone had to know a little about everything. Currently, Maxwell lives in New Jersey doing free-lance work. She consults part time for Silicon Graphics, helping with a graphics user interface on the update of the Indigo. She is also involved in the electronic arts community. For the past year, approximately one week out of a month, Levy and Maxwell meet at the Geometry Center to work on "Outside In." With the help of assistant director Tamara Munzer and the rest of the Center staff, they hope to finish the movie by the end of the year. For information on the behind the scenes work for "Outside In," see Maxwell's article in the next Geometry Center newsletter. For more on the mathematical content, such as a definition of sphere eversion, see the previous article. From news1.cis.umn.edu!umn.edu!sander Sun May 2 22:49:11 CDT 1993 Article: 75 of geometry.forum Newsgroups: geometry.forum Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: hyperbolic geometry software Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 28 Apr 1993 04:54:53 GMT Lines: 75 Status: OR There was a request for more information about last week's discussion on hyperbolic geometry software. In response, Oliver Goodman of the Geometry Center wrote the following: The Geometry Center is holding an ongoing series of discussions to determine how best to provide hyperbolic geometry software to the research and educational communities. The following represents some of the challenges the center faces in serving these communities. Research into hyperbolic geometry shows great potential for the use of computers. Many current problems involve mathematical objects which are too complex to be constructed by hand. The old tools of ruler and compass that for centuries have enabled mathematicians to experiment and discover in geometry are simply not sufficient for current research in three dimensional hyperbolic geometry. With the aid of computers we can begin to overcome these difficulties. While several researchers have already developed their own tools for tackling specific areas these have always involved starting from scratch and embarking on long software development projects. This is not ideal: There are many researchers who would like to use software tools but do not want to have to turn into programmers in order to do so. In future we would like to have an environment in which mathematicians can quickly find programs to help them in their research. Programs that require only the minimum of computing knowledge. Below that perhaps a level of easily re-used pieces so that the mathematician who has to turn programmer does not have to start from scratch. A large part of our discussion has been centered on finding strategies, technical and otherwise for bringing this ideal closer. The same problems facing researchers also face high-school teachers who would like to teach hyperbolic geometry. We believe very strongly that with the aid of software tools hyperbolic geometry can and should be taught in high school. Its discovery was a very significant step for mathematics. We think that the discovery of a geometry that breaks the rules by going counter to our everyday intuition about things will be as exciting to high school kids as it is to researchers. So in considering hyperbolic geometry software we have to consider the needs of teachers as well as researchers. Educational tools in general reach a much wider audience than research tools. This gives the authors of such tools a greater responsibility to make them easy to use, reliable and available on a popular platform. A goal here would be to emulate the success of the euclidean geometry tool Geometer's Sketchpad. While embarking on a project the size of Sketchpad remains beyond our current plans we are very interested in hearing from teachers who are considering teaching or have taught hyperbolic geometry to high school students. What would the ideal teaching aid for hyperbolic geometry look like ? One currently available package which enables users to carry out basic computations in hyperbolic geometry is the Mathematica (v 2.0) package "Hyperbolic" available from the Geometry Center at "geom.umn.edu" by anonymous ftp. It provides basic objects and functions for computations in 2 and 3 dimensional hyperbolic geometry. Pictures drawn with the aid of this package should help users to grasp the nature of hyperbolic space. To begin with one could show pictures of triangles whose angle sum is less than 180 degrees. Or a line and a point with several distinct lines through it none of which intersect with the first line - the counterexample to Euclid's parallel postulate which many students probably regard as 'obvious.' We would be very interested in hearing of the experiences of any teachers who have used this package. Comments on what they found difficult or awkward, what they liked and what new function they would like to see in the package would be very welcome. This is an exciting time for mathematics. Technical advances and networks are making possible new ways of learning and investigation that cross traditional boundaries between education and research. At the Geometry Center we are very pleased to be able to play a role in the advance of hyperbolic geometry into this new world. From news1.cis.umn.edu!umn.edu!sander Mon May 3 12:12:38 CDT 1993 Article: 19 of geometry.announcements Newsgroups: geometry.announcements Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: The Sciences: Minimal Surfaces Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Mon, 3 May 1993 16:55:06 GMT Lines: 13 Status: OR The May/June issue of The Sciences contains an article of interest called "Bubble, Bubble" by Robert Kanigel. It is a description of the life and work of Professor Jean Taylor. It specifically concentrates on her study of minimal surfaces and soap bubbles, done in collaboration with the other members of the "Minimal Surface Team" at the Minnesota Geometry Center. I did not think that the article was particularly well written; the author seemed to have trouble incorporating relevant biographical details in with the mathematics. However, he does a good job of describing some ideas of minimal surfaces as well as a bit of the history of the subject for a general audience. For this reason it is worth a look. From news1.cis.umn.edu!umn.edu!sander Fri May 7 16:59:51 CDT 1993 Article: 49 of geometry.college Xref: news1.cis.umn.edu geometry.college:49 geometry.research:32 Newsgroups: geometry.college,geometry.research Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Hamiltonian Systems Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: ricci.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 7 May 1993 21:51:50 GMT Lines: 171 Status: OR The following is a partial explanation of ideas of Hamiltonian Systems, an active research area within the more general category of Dynamical Systems. It has some interesting applications. However, one needs a bit of explanation in order to understand the ideas and how they relate. One of the applications of this area of research is to answer the question of whether the solar system is stable. In trying to answer this question Henri Poincare developed the ideas on which form the basis for the theory of Dynamical Systems. The story of Poincare's original essay on the subject is very interesting; I will tell it in a separate article. The mathematical study of the solar system is also called the n-body problem: given starting positions and velocities, try to determine the positions and velocities of n bodies after a certain amount of time, assuming that the bodies are only restricted by Newton's laws of gravitation. The question of stability of the solar system is interesting philosophically, but it is of no practical concern because any instability would take a long time even on an astronomical scale. However, a similar stability question is of practical interest to High Energy physicists when building the Superconducting Super Collider in Texas. The accelerator will be sending particles around a ring-shaped pipe of quite small diameter relative to the ring size. In order to accelerate the particles to full energy necessary to perform experiments, it will take approximately ten hours, during which time the particle will each travel 10^8 times around the pipe. There are many experimental imprecisions which could cause particles to escape from the beam tube. However, it is also possible than even the mathematical model might be unstable; this would mean that even with the best possible experimental precision, before the particles have reached the energy needed, the they would almost certainly escape from the tube. Therfore important to understand the model to see how they could escape before building the accelerator. The stability problems of the solar system and of particles in an accelerator are related. Trying to solve them leads to many different paths, including the study of Hamiltonian Systems. A Hamiltonian system is a system of differential equations of a special form. The most important example of this is the system of equations of position and velocity of particles satisfying Newton's laws of gravitation. In this case, solutions to the differential equations specify the position and velocity of particles at a given time. The space of all possible positions and velocities is called phase space; for s particles in m dimensions, an m dimensional vector is necessary to specify the point in space each of s particles. One needs another m dimensional vector to give each particle's velocity. Thus it seems that the phase space would be 2ms dimensional. However, one can reduce the problem by two dimensions. The reductions are a bit technical, but it is important to mention that one reduction changes the Hamiltonian system of differential equations into an associated iterated map. This associated map is called a symplectic map. In general, the dimension of the space of vectors necessary to specify the position of all the particles is called the number of degrees of freedom. In the previous paragraph, n=ms. From the above discussion, one can see that the dimension of the phase space for a n degree problem is 2n-2. A reformulation of the stability question is to ask whether solutions to a Hamiltonian system remain qualitatively the same for all time; in other words do solutions with starting point near p in phase space remain within some region close to solution starting at p? In particular, one considers this question for solutions starting close to fixed and periodic points; that is, solutions which stay fixed for all time and ones which keep coming back to the same point with after a certain period. Here is one way to insure stability of a fixed point p: Consider a set of initial values whose solutions stay within the same set for all time. This kind of set is called invariant. Now suppose that there is an invariant set L which encloses a region R. For example, in the plane, an invariant circle encloses a disc. Then a solution with starting point in R must stay in R for all time; for our example this means all solutions starting in the disc stay in the disc for all time. This is because in order to get out of the region R, the solution would have to pass through the invariant set L. We know the solution cannot do since L is invariant. Invariant sets which enclose regions therefore indicate whether there is stability. There is a very powerful theorem developed in the 60's which partially answers the stability question. This is called the KAM theorem; it states the existence of invariant sets for a Hamiltonian system. Each of these sets is a torus. In certain two dimensional cases, the tori surround a region containing fixed points exactly as described above. The theorem thereby solves stability in the case of two degrees of freedom. Unfortunately, KAM tori do not answer stability with more degrees of freedom. This is because in n dimensions, a torus is an n/2 dimensional object. But in order to enclose a region in n dimensions, a set must be n-1 dimensional. Therefore if n>2, a torus does not actually enclose any space. To get an idea of this, the circle encloses a disc in the plane but does not enclose a region in three dimension. A torus in the plane is a circle, so it encloses a disc. However in four dimensions, a torus is two dimensional, so it does not enclose a region. The smallest degree after two is three. This means we are working with a four dimensional phase space, since dimension=2*(degrees of freedom)-2. In this case, stability is not known and may very well not be true. In fact, researchers now think that chaos may occur. Consider a fixed point of the system; the points which exponentially go to this solution make up what is called the stable manifold. The points which exponentially leave the solution make up the unstable manifold. Consider a point which is on both the stable manifold of one fixed point and the unstable manifold of another. This is called a heteroclinic point, or in the case of the point being on the stable and unstable manifolds of the same point, it is called a homoclinic point. In other words, a heteroclinic point is a point which under the system goes backward in time to one periodic orbit and forward in time to another one. Existence of a heteroclinic point means that close to the first fixed point, there is a starting value which eventually arrives near the other fixed point. This is a qualitative change in the placement of the point in the phase plane under time. Thus it is clear that heteroclinic points are closely related to the question of stability. The results show more than just a small qualitative change; it has been shown that the mere existence of (transverse) homoclinic points implies that there is something called a Smale horseshoe in the system. This horseshoe always exhibits chaos. So finding these homoclinic points shows that the system is very far from stable. The study of heteroclinic and homoclinic points for symplectic maps (the reduction of Hamiltonian systems) is an active area of research. In particular, people are trying to solve the four dimensional case. Here at the the Minnesota Geometry Center Eduardo Tabacman is looking at this problem. He examines specific symplectic maps which are in some way representative and tries to computationally find heteroclinic and homoclinic points. The difficulties in research include finding ways to represent the stable and unstable manifolds and finding ways to distinguish heteroclinic and homoclinic points. Since the stable and unstable manifolds are surfaces which occur in four dimensional space, Tabacman has had to consider the issue of visualization of 4D and how to best comprehend 2D objects which occur in 4D. _____________________________________________________________________ There are many books on the Hamiltonian Systems. In particular I used the following references: Meyer, Kenneth, Hall, Glen R., Introduction to Hamiltonian Dynamical Systems and the n-body Problem, Springer-Verlag, New York, 1992. This is a good basic text. Poincare, Henri, New Methods of Celestial Mechanics, American Institute of Physics, 1993. This book has an excellent introduction by Daniel Goroff, which is interesting historically, as well as giving a good survey of the subject. Dumas, H.S., "The Role of Low-Dimensional Symplectic Maps in the Dynamics of Particle Accelerators," Imagine That!, the Minnesota Geometry Center Newsletter, Volume 1, Number 2, Winter 1992. Eduardo Tabacman is a graduate student at the University of Minnesota and a research assistant at the Center. He works with Geometry Center member and University of Minnesota Professor Richard McGehee. I may have mentioned in my welcome letter that I work with McGehee as well. I would like to thank him for helping me write this article. From news1.cis.umn.edu!umn.edu!sander Sun May 16 13:29:59 CDT 1993 Article: 52 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: The Truth About Poincare Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 14 May 1993 20:05:17 GMT Lines: 92 Status: OR THE TRUTH ABOUT POINCARE In 1885, King Oscar II of Sweden announced a mathematics contest in celebration of his sixtieth birthday. The prize would go to the essay that established the stability of the solar system. It was set up at the urging of Mittag-Leffler. The judges were quite a distinguished bunch: Mittag-Leffler, Weierstrass, and Hermite. Thus the winner would gain great prestige, and his essay would be published in the Swedish journal Acta Mathematica. The committee read the many entries from both mathematicians and astronomers and awarded the prize to Poincare for his essay on the three body problem. With the hindsight of a hundred years more research, he deserved the prize; this essay was where he first developed many of the fundamental ideas which led to the modern field of Dynamical Systems. However, though this is how people have told the story for many years, it is not the whole story. There were actually two essays by Poincare, only one of which he actually submitted to the contest. Perhaps due to time constraints, the judges did not read the essays carefully, and it was only after they awarded the prize to Poincare and published his original essay that Phragmen pointed out a flaw in Poincare's work. Poincare had assumed that stable and unstable manifolds do not intersect transversally. As mentioned in my article on Hamiltonian Systems, transverse intersections of these manifolds is the key to much of the interesting behavior. His mistake allowed him to conclude that he had solved the restricted three body problem. In 1887, he wrote to Mittag-Leffler: "In this particular case I have found a rigorous demonstration of stability and a method of placing precise limits on the elements of the third body."[Goroff] As soon as Mittag-Leffler heard that there was an error, he wrote to all of the subscribers to Acta Mathematica recalling the journal. He then destroyed all but one copy of the original journal; this copy still remains in a locked drawer in the archives of the Mittag-Leffler Institute. The recall of the journal was no secret. The other entrants, including many Swedish astronomers, now realized that the judges did not read the essays carefully. The written records of German math society meetings show quite a bit of debate about the prize scandal. Rather than picking a new winner in the contest, for the next year, Mittag-Leffler regularly wrote letters to Poincare asking him when he would finish the corrected version of the essay. After a year of this pressure, Poincare came out with a new essay, which Mittag-Leffler then published and sent to subscribers in the place of the recalled journal. The new essay did not even claim to have solved the original problem. However, it was a memorable work in which Poincare developed the important ideas for which people remember the contest. Though people still know the story in Sweden, it had been forgotten in this country. There was was a reference to it in 1912, when the scholar F.R.Moulton described the scandal in an article in Popular Astronomy.[Goroff] More recently, Richard McGehee found out about it during his stay at the Mittag-Leffler institute. He looked at many of the old documents from the archives, including copies of the letters that Mittag-Leffler wrote to Poincare and the one original Acta Mathematica journal. I first heard the story in long form from McGehee a few months ago in his class, and I wish that I could tell it as well. The class was based around this story. The course announcement described the contest, saying only that Poincare had won the prize without answering the stability question and promising to explain what was difficult about it. McGehee started the class by a brief description of Poincare's second essay. We then spent the quarter trying to understand all the theories on stability and why this is a hard question. The theory takes you quite far from the original question. On the last day we were still involved in some number theory that comes up in the KAM theorem, very far from anything to do with the solar system. During the break, one classmate mentioned to me that he thought it was unlikely that he would be able to tie all this in with the solar system again. It was then, after this three month set-up which we thought he would not be able to tie together, that McGehee told us what had really happened with the contest. Daniel Goroff also heard the story from McGehee. All the references are to his rendition of the story, published in his introduction to Poincare's book New Methods of Celestial Mechanics. From news3.cis.umn.edu!umn.edu!news Thu May 20 16:08:18 CDT 1993 Article: 56 of geometry.college Xref: news3.cis.umn.edu geometry.college:56 geometry.research:41 Newsgroups: geometry.college,geometry.research Path: news3.cis.umn.edu!umn.edu!news From: sander@diophantus.geom.umn.edu (Evelyn Sander) Subject: 4d Visualization Part 1 Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: University of Minnesota Date: Thu, 20 May 1993 21:00:38 GMT Lines: 62 Status: OR In the course of talking to people at the Geometry Center and observing their work, I have noticed many are trying to visualize four dimensional space. I noticed this common element when talking to Eduardo Tabacman, Richard McGehee, Bruce Peckham (who visited in the fall), and most recently Brad Barber. In addition to research, there is talk of higher dimensional graphics software. The Center's graphics program Geomview has four dimensional capabilities; both arbitrary projections and slices are possible. Last week there was a meeting to discuss the possibilities for software to accommodate k dimensional manifolds in n dimensional spaces. (There were no concrete conclusions as to how to actually do this, but the meeting resulted in a lengthy wish list.) When I asked people how well their work made them understand 4D, I got the following interesting responses: Tabacman and McGehee said they still didn't have the intuitive sense that one gets for 3D, and therefore did not feel they understood 4D. Barber felt he could visualize 4D, as it corresponded to theorems he knew. These answers made me curious for two reasons; first, they show a difference in opinion on what it means to visualize and understand. Second, it makes me wonder how much the understanding depends on the mathematical context. Inspired by these questions, I decided to approach the subject directly; namely, I have interviewed anyone here who looked at four dimensions in their research who was willing to discuss it. I am making these interviews into a series of articles, trying to specifically focus on 4D visualization. The content of the articles varies quite a bit since the answer to the second question above is that the understanding and even what it means to understand varies significantly with mathematical context. As a general guideline for the interviews, I asked the following list of questions. I would be interested to hear any comments that non Geometry Center people have on the subject. I hope that perhaps by linking these articles by a common theme I can inspire some discussion? Questions (meant as guidelines only): In what context did you use 4D space? Do you feel that you can understand or visualize 4D? If so, what did it take to make you feel that you understood? What are the mathematical ideas that help you understand it? If not, what would it take to make you feel that you understood? From news1.cis.umn.edu!umn.edu!sander Wed May 26 16:07:55 CDT 1993 Article: 59 of geometry.college Xref: news1.cis.umn.edu geometry.college:59 geometry.research:43 Newsgroups: geometry.college,geometry.research Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: 4D Visualization Part 2 Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 26 May 1993 21:05:02 GMT Lines: 59 Status: OR "I think the best way to understand four-dimensional space is by analogy with three dimensions," says John Sullivan of the Geometry Center. "This often gives a good idea of what happens in 4D, but the problem is that sometimes you miss things. "For example, I was talking to my Differential Geometry class about the fact that in 3D, there is only one direction tangent to a line and one direction normal to a plane. This means that by specifying a point on the sphere, you can uniquely specify a line by its tangent or a plane by its normal. In 4D, there is still one direction tangent to a line and one direction normal to a three dimensional hyperplane; however, a two dimensional object in 4D has two normal directions. I think in some ways this makes a 2D surface in 4D is harder to understand than a 3D hypersurface. "It is possible to describe a two-dimensional plane in 4D by a wedge product of two vectors in the plane. This product lies in a six-dimensional vector space, and by normalizing, one has a vector in the five-sphere. By analogy with 3D, since any vector on the two-sphere specifies a plane in 3D, one would think that any vector in this five-sphere would give a plane in 4D, but in fact this is not true. Only a vectors in a subset of the five-sphere called the Grassmanian actually specify planes in 4D. "The reason not all vectors in the five-sphere specify planes is related to another point missed by analogy; in 3D, all rotations fix a line. Even if you rotate around one line and then rotate around another line, there is some line in between which is fixed by the combined rotation. However, in 4D, there are many rotations that do not fix lines. Take, for example, a rotation in the x-y plane followed by a rotation in the z-w plane. Under this combined rotation, rather than tracing a circle, the orbit of a point might be dense on a torus." Aside from all these pathologies, Sullivan has gained quite a bit of understanding by analogy. For educational purposes and for fun, he has written a program which does a stereographic projection of 4D regular solids into 3D, where they become soap bubble clusters. This can best be understood by analogously considering the stereographic projection of, say, a cube or dodecahedron on the two-sphere into the plane. In his research on minimal surfaces, Sullivan has never actually done work which specifically applied to four dimensions. All the proofs work in arbitrary dimensions. However, even in this case, he always draws the picture in 3D. "I am just careful when I write the proofs to say things which are true in all dimensions." In contrast to Sullivan, parts of Ken Brakke's work on minimal surfaces is specific to 4D. I will give details of their research in another article. From news3.cis.umn.edu!umn.edu!sander Thu May 27 09:55:38 CDT 1993 Article: 60 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: 4D Part 3: Minimal Surfaces Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Thu, 27 May 1993 14:49:18 GMT Lines: 110 Status: OR Given a fixed boundary in nD, what n-1 dimensional hypersurface with this boundary has the minimum area? This is the question to answer in the field of minimal surfaces, studied by Ken Brakke and John Sullivan of the Geometry Center. In three dimensions, this question becomes: given a boundary in 3D, such as a loop or shape made out of wire, what is the surface with minimum area with this wire as its boundary? The answer should be quite familiar to everyone; the soap film formed by dipping a wire frame in soap solution gives a surface of minimum area with that boundary. Sometimes these surfaces are unexpected. For example, a tetrahedral frame makes a soap film with planes starting at each edge, all meeting at a point in the center, whereas a cubic frame makes a soap film with surfaces headed towards the center but which meet at an interior square with rounded edges. (Many science museums have soap bubbles and frames for people to play. It is also not difficult to make solution and wire frames.) Although it is easy to obtain minimal surfaces, they are mathematically difficult to describe. Even for the simple case of a cubic boundary, there is no known mathematical equation which describes the soap film. Even after gaining an understanding of the mathematics, researchers always are going back to soap films. "Whenever I get stuck, I take out the wire frames and soap solution again," says Brakke. Sullivan looks at n-1 dimensional minimal hypersurfaces in nD, where n is arbitrary. He observes that there are hypersurfaces which are locally minimal but not necessarily globally minimal; in other words, two different surfaces spanning a given boundary may have the property that each minimizes area with respect to perturbations in a small neighborhood of each point of the surface. For example, in 3D, given two parallel circles as a boundary wire, one can get two different soap films: namely, a flat disk inside each circle separately, and a catenoid connecting the two circles. The catenoid surface is the shape you get if you form a surface of rotation using the St. Louis Arch, rotating around a line above it and in its same plane. Sullivan finds general conditions to determine which of the locally minimal surfaces actually has the minimum area. His conditions apply any arbitrary dimension, so he does not actually have to work specifically with problems in 4D. Brakke looks at higher dimensional singularities in minimal surfaces. Singularities are just the possible ways in which a surface could be lacking in smoothness. For example, in the tetrahedral boundary case, the planes starting at the edges intersect three at a time along lines toward the center. These lines are singular. Also, the planes all meet at a singular point in the center. The two singularities above are both mathematical cones on some geometric object, where the cone on an object is defined as all the lines connecting that object to the origin. The first singularity was three planes meeting along a line at 120 degrees. This is the cone on the vertices of a triangle extended into 3D. The second singularity was six planes meeting at a point. This is the cone on a tetrahedral wire frame. Jean Taylor proved the two types of singularities described are the only possibilities in three dimensions. See the article on Taylor's work in the May/June issue of The Sciences. Thus Brakke is trying to classify singularities for minimal hypersurfaces in higher dimensions. Since each dimension builds on the singularities of the previous dimensions, Brakke concentrates his work specifically on 4D. He has been successful; using his program Surface Evolver to help guess possible surfaces with singularities, then checking his guesses analytically, Brakke proved that the 3D cone on the 2D hypercube frame in 4D is a minimal surface. In fact, in all dimensions higher than three the n-1 dimensional cone on the n-2 dimensional hypercube frame is a minimal surface with a singularity. The new singularity, along with singularities obtained by extending 3D surfaces to 4D hypersurfaces, is almost the whole story in the 4D case. Brian White proved that all possible singularities occur on flat sided cones in 4D; the result is specific to 4D and not true in 5D. Based on White's result, Brakke was able to show that there were only a finite number of cases of singularities in 4D minimal surfaces; for all but the simplex frame, hypercube frame, and one other case, he proved that these surfaces were not minimal. Brakke is currently trying to prove or disprove this last case; this would complete the 4D classification. Brakke says that even after his work in 4D, he is unable to really visualize it. His responded with skepticism to the idea that anyone can visualize 4D. He said, "Give them a test. Show them a bunch of projections of similar 4D objects and ask them which are pictures of the same object. Show them one 'side' of a 4D object, and have them describe the other 'side.' I think you'll find that nobody can really visualize 4D. The best they can do is try to understand a few theorems and see that the objects obeys these theorems." From news1.cis.umn.edu!umn.edu!sander Wed Jun 9 16:36:43 CDT 1993 Article: 66 of geometry.college Xref: news1.cis.umn.edu geometry.college:66 geometry.research:52 Newsgroups: geometry.college,geometry.research Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Hoops in three-space Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: gauss.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 9 Jun 1993 21:29:01 GMT Lines: 71 Status: OR A hoop is a geometric unit circle in three-space, or in other words, the result of applying a three dimensional rigid motion to the unit circle in the plane. Notice that a hoop is a circular curve and not a disk. Asimov considers what kind of three-dimensional regions one can fill with non-intersecting (but possibly interlocking) hoops. He is interested in the specific method of filling volumes with hoops, requiring that the hoops be placed in a continuous manner. Here is a precise definition of continuity of hoops in a region of three space: One can specify a hoop using a center point and a normal direction. If hoops fill a region, then there is a map which assigns to each point in the region the hoop passing through that point. A continuous placement of hoops is one in which this map is continuous. (Technical note: The center of a hoop is a point in three space, R^3. The normal is a point on the unit sphere, S^2 or on the projective plane, P^2, depending on whether hoops are oriented or unoriented. Thus the map goes from R^3 to either R^3 x S^2 or R^3 x P^2.) It is in fact possible to fill all of three-space with disjoint hoops. In 1964 J. Conway and H.T. Croft proved this using an Axiom of Choice argument. However, the placement of the hoops is not continuous or constructive, as is generally the case with the Axiom of Choice. One can use a continuous placement of hoops to cover a "unit torus," meaning a torus of revolution whose core circle has radius one. (An example of this is the result of revolving the circle (x-1)^2 + y^2 = r^2 in R^2 about the z-axis of R^3. The shape of a unit torus is determined by the radius r < 1 of the circle being revolved.) Notice the relationship between a unit torus and hoops; a unit torus is a surface of revolution around a hoop. In order to continuously fill a unit torus with disjoint hoops, each hoop must tilt so that it goes around the non unit circle of the torus exactly once as it goes around the unit circle of the torus. Using this covering method for unit tori with (non unit) circles of all radii between zero and r, it is possible to continuously fill a solid radius r unit torus with disjoint hoops. Fattening the torus as much as possible without having the hoops intersect, one obtains a solid unit torus of radius one (by slight abuse of terminology). This means the torus has no hole in the center. Asimov terms this shape a "bialy," after a kind of bread similar to a bagel. The bialy at least gives a lower bound of 2*(pi^2) for the possible connected volumes one can continuously fill with disjoint hoops. What about an upper bound? Asimov proved that for a set of hoops which fill a connected volume, any two hoops must link. Thus any volume continuously filled with hoops must fit inside a ball of radius three (whose volume is 36*pi). Although he does not yet have a least upper bound, it is clear that the restriction to continuous placement of hoops is indeed a severe restriction on the possible filled volume; it changes the possible region from all of three space to a region contained in a ball radius three! Asimov is currently trying to extend his results on hoops. He is considering a variety of dimensions of geometric unit spheres in higher dimensional spaces. From news1.cis.umn.edu!umn.edu!sander Tue Jun 15 15:00:08 CDT 1993 Article: 27 of geometry.announcements Newsgroups: geometry.announcements Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Geometry Center Welcomes Students Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: siegel.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Tue, 15 Jun 1993 18:17:11 GMT Lines: 27 Status: OR The Geometry Center Summer Institute for college students began yesterday. At first it was only one unfamiliar person calling the airport shuttle service to locate a forgotten backpack. Then a few others arrived, introduced themselves and asked if they could use the computers. At this point the Center staff came back from a trip to the dorm with the rest of the students. After some lectures, everyone took a break to meet each other at a pizza party. This week the summer students will attend a series of lectures on how to use the computers and other Center tools. They are also getting ideas for projects which to spend their summer. Summer projects will often be something like graphical animation of geometric concepts. The students are required to write papers on their projects at the end of the program, which will be compiled in a Geometry Center publication. This morning is the introduction to the Center computer system, which I expect will include instructions on reading news. Hopefully the Forum will see contributions from the institute participants this summer as they find interesting mathematical ideas they would like to share. Welcome to the Geometry Center and Forum! From news3.cis.umn.edu!umn.edu!news Sun Jun 20 12:05:02 CDT 1993 Article: 73 of geometry.college Xref: news3.cis.umn.edu geometry.college:73 geometry.research:58 Newsgroups: geometry.college,geometry.research Path: news3.cis.umn.edu!umn.edu!news From: sander@n3.math.umn.edu (Evelyn Sander) Subject: 4D Part 4: The 24-Cell and Klein Bottle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: n3.math.umn.edu Organization: University of Minnesota Date: Sat, 19 Jun 1993 00:00:10 GMT Lines: 117 Status: OR I want to make a correction and an addition to my article on hoops. First, the correction: the torus of revolution should revolve around the y axis, not the z axis. I also did not include any background information on the hoops researcher Dan Asimov; please see below. This previous article was based on an interview with Dan Asimov during his recent visit to the Center. Asimov works in the area of visualization at NASA Ames Research Center. The previous article focused on his current research in the area of "hoops" in three dimensions; this is a subject which some readers may have remembered from a puzzle that Asimov posted a few weeks ago in geometry.puzzles. In addition to the hoops research, I would like to briefly describe some observations that Asimov made about 4D. I wrote to him for details, and I have directly quoted large portions of his response (in quotation marks). For more information, also see Asimov's posted description of his n dimensional visualization program, called the "grand tour." In 3D, there are five regular solids (polytopes). In four dimensions there are six regular polytopes. In all other dimensions, there are only three basic types of polytopes. This means that three and four dimensions are exceptional; they have the three standard types of polytopes seen in higher dimensions, but on top of those they have additional polytopes. In four dimensions, one of the extra polytopes is the 24-cell, consisting of 24 octahedra. These octahedra are the "faces" of the four dimensional polytope, six meeting at each vertex. The object is self-dual. This means that the new polytope formed by connecting the centers of all the faces is again a 24-cell. In what sense is the 24-cell distinguishable from the three standard polytopes, and why does the self-dual property make the 24-cell special? In Asimov's words: "The 24-cell is best understood in the context of the classification of all regular polytopes in all dimensions: dim polytopes --- --------- 2 regular n-gons for n >= 3 3 the 5 Platonic solids = tetrahedron, cube,octahedron, dodecahedron, icosahedron = simplex, cube, cross-polytope, dodecahedron, icosahedron 4 simplex, cube, cross-polytope, 24-cell, 120-cell,600-cell >=5 simplex, cube, cross-polytope. The only self-dual polytopes in dimensions >= 3 are the simplex (which occurs in all dimensions) and the 24-cell (which occurs only in dimension 4). This gives the 24-cell a kind of symmetry unique to itself, with no analogue in any other dimensions." Asimov also observed that in 4D, it is possible to immerse the Klein bottle in a symmetric manner in the three-sphere (S^3). Remember that an immersion is only locally injective. In the case of this immersion, there is self-intersection. He used a stereographic projection of the three-sphere to get a surface in three dimensions. I was quite impressed with the beauty of this surface. I asked Asimov to give a more detailed explanation, which follows: "As for the Klein bottle, the surface I showed you is the stereographic projection of a particularly beautiful Klein bottle K that is immersed in S^3 as a minimal surface, and has a great circle in S^3 as its set of self-intersection. This surface K in S^3 has a great deal of symmetry: its group of isometries is a 1-dimensional Lie group. This Klein bottle can be described as those points (x,y,z,w) of S^3 which satisfy the polynomial equation w(x^2 -y^2) = 2xyz. In addition, K is the union of two embedded Mobius bands in S^3 whose intersection is two linked great circles in S^3, one of which is their common boundary. For the stereographic projection (used to send K from S^3 into R^3 where it could be viewed fairly well), its projection point was chosen to lie on the self-intersection circle of K, in order that as much as possible of K's symmetry be preserved after it is projected. After projection by stereographic projection S: S^3 -> R^3, the image S(K) represents the Klein bottle with two points removed, immersed in R^3 with the z-axis as its set of self-intersection. Unlike K in S^3, this image S(K) in R^3 is not a minimal surface. It is, however, conjectured by R. Kusner that it minimizes the integral of squared mean curvature. One of the Mobius bands that constitute half of K--call it M--can also be stereographically projected in the same way, of course, and the result is a surface S(M) in R^3 which depicts a Mobius band in R^3 whose boundary is not just a topological circle, but a perfect geometric circle. 'Depicts' is more appropriate than "is" here, since the projection point chosen to preserve symmetry lies *on* M in S^3, so it is missing from S(M). As a result, S(M) has to spread out to infinity in R^3. This Mobius band (actually a Mobius band minus a point) S(M) is the subject of a short computer graphics film, 'The Sudanese Mobius Band,' made by myself and Douglas Lerner for the 1984 Siggraph Film and Video Show." From news3.cis.umn.edu!umn.edu!sander Sun Jun 20 12:47:42 CDT 1993 Article: 74 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: They Didn't Eat Beans and Other Stories Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Sun, 20 Jun 1993 17:42:16 GMT Lines: 125 Status: OR I have been asked to write something which addresses the issue that there are no mathematicians who are household words. This means that there are no famous role models for kids to emulate. At first I was planning to do a series of detailed several page descriptions of the lives of a few specific mathematicians. However, after considering what would have intrigued me as a kid, this will only contain a some of the exciting parts about some great mathematicians and historical events. Many sources try to make mathematicians and mathematics sound far removed from the world, but this is not at all the case. There are politically active mathematicians, mathematicians who steal others' ideas, and even murder over theorems. There is passion and excitement, as in any history when the people involved really care about it. I am sorry for the lack of women in this description, but there really are not yet that many famous women mathematicians. Emmy Noether, Sophie Germain, and Sonya Kovalevsky are the only three that I can think of offhand. Born in around 532 B.C., the ancient Greek Pythagorus was the founder of a school of mathematicians and is credited with the discovery of the relationship between the lengths of the sides of a right triangle (although it is not clear Pythagorus actually deserves credit for this theorem). Pythagorus was also important politically; he founded the religious sect of the Pythagoreans, who became a major political force in Southern Italy, even gaining the rule of some of the cities. The major beliefs of the Pythagoreans included the transmigration of souls, that everything depended on whole numbers, and the sinfulness of eating beans. Other laws included not touching a white cock and not looking in a mirror beside a light.[Russell, Bertrand, "A History of Western Philosophy," Simon and Schuster, NY, 1945.] So great was the importance of whole numbers that the discovery that the square root of two is irrational remained a religious secret. It is said that when the Pythagorean Hippasus disclosed the secret, other members of the sect drowned him in the sea.[Eves, Howard, "An Introduction to the History of Mathematics," third edition, Holt, Rinehart and Winston, NY, 1964.] In the sixteenth century, mathematicians wanted to find a formulas like the quadratic formula for factoring third and fourth degree polynomials. The answers were first published by Cardan (1501-76), though it was not his work. He found out the secret of how to solve the cubic from Tartaglia (1500-57), who probably also did not discover it. Cardan's publication came after he promised Tartaglia that he would never reveal the secret. According to Boyer, it is probably Scipione del Ferro (1465-1526) who actually discovered the formula. He kept it a secret, revealing it to one student before he died.[Boyer, Carl, "A History of Mathematics," John Wiley & Sons, NY, 1968.] After the discovery of formulas to factor third and fourth degree polynomials, it is natural to wonder about five and beyond. In fact, it is impossible to write down a general formula to factor polynomials of any degree greater than four. It was Galois (1812-1832) who proved this result in the course of developing a branch of mathematics now called Galois theory. Through a series of unfortunate circumstances, Galois repeatedly was denied entrance to the Ecole Polytechnique, the most pretigious university in France, as well as never getting his work recognized in his lifetime, although two papers were published in 1830. This same year, Galois became a revolutionary, fighting for France to be a republic. Through this political activity (or perhaps over a woman), he was challenged to a duel. It was in this dual that he died at the age of twenty. According to legend, knowing that he would die, he wrote down many of his ideas in a letter to a friend the night before the duel. The letter and other partial manuscripts were finally published in the Journal de Mathematiques in 1846.[Boyer] I will not write much about Newton (1642-1727), but there are a few interesting things to mention. Newton was the first to discover calculus, but because he did not publish for more than ten years, Leibniz independently arrived at the same discovery and published first. The result was a terrible fight between the two, making the last part of Newton's life unhappy. In 1696, he was appointed Warden of the Mint and promoted to Master of the Mint in 1699.[Eves] He took the job seriously, saving the country money by introducing the idea of coin milling. This meant that people were no longer able to clip silver off the edges of the coins.[Barrow, John, "The World Within the World," Oxford University Press, 1990.] Credited with the invention of modern analysis, Euler (1707-83) is probably the most prolific mathematician ever. Spending the last seventeen years of his life blind did not slow down his productivity. He just dictated to his children. Aside from the mathematical content of his work, Euler standardized mathematical notation. He is responsible for the use of the letter e for exponential functions, the capital sigma for summation, i for the square root of minus one, and even for the use of the letter pi for the ratio of the circumference to diameter of the circle! [Boyer] Thus it is that we can write one of the most fundamental equations of modern mathematics, voted the most beautiful theorem by readers of the Mathematical Intelligencer.[Wells, David, "Are These the Most Beautiful?" Mathematical Intelligencer, Vol 12, No 3, 1990.] Namely: e^(i*pi)=-1 That mathematicians participate in the world is not something of the past. The contemporary mathematician Steve Smale, who is very important in many areas including Dynamical Systems, had to appear in front of the House Un-American Activities Committee and was active in the Free Speech Movement in Berkeley. He caused quite a bit of contraversy when he spoke against the U.S. and Soviet involvement in Vietnam in Moscow, 1966. The University of California denied him summer support; he then has his NSF proposal returned for political reasons.[Smale, Steve, "The Story of the Higher Dimensional Poincare Conjecture (What Actually Happened on the Beaches of Rio)," Mathematical Intelligencer, Vol 12, No 2, 1990.] Perhaps the most telling comment regarding the importance of mathematics comes from the algebraic geometer Alexandre Grothendieck, when he was teaching math in Vietnam in 1967. He says: "In general, I can attest that both the political leaders and the senior academic people are convinced that scientific research--including theoretical research having no immediate practical applications--is not a luxury, and that it it necessary ... starting now, without waiting for a better future."[Koblitz, Neal, "Recollections of Mathematics in a Country Under Siege," Mathematical Intelligencer, Vol 12, No 3, 1990.] I would like to thank Scott Carlson for sharing his knowledge and books with me for this article. From news1.cis.umn.edu!umn.edu!sander Wed Jun 23 10:53:00 CDT 1993 Article: 30 of geometry.announcements Xref: news1.cis.umn.edu geometry.announcements:30 geometry.research:59 Newsgroups: geometry.announcements,geometry.research Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Fermat's Last Theorem -- Rumors of a Proof Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 23 Jun 1993 15:49:20 GMT Lines: 21 Status: OR I received the following message today, and although I cannot vouch for its validity, it seems worth mentioning: >From sullivan Wed Jun 23 10:17:47 1993 Received: from warschawski.geom.umn.edu by cameron.geom.umn.edu; Wed, 23 Jun 1993 10:17:04 -0500 Date: Wed, 23 Jun 93 10:13:20 CDT From: sullivan Message-Id: <9306231513.AA08247@warschawski.geom.umn.edu> Received: by warschawski.geom.umn.edu; Wed, 23 Jun 93 10:13:20 CDT To: here, ima@ima.umn.edu, math@math.umn.edu Subject: rumors from Britain--Wiles proves Fermat Status: RO > Andrew Wiles just announced, at the end of his 3rd lecture here, > that he has proved Fermat's Last Theorem. He did this by proving > that every semistable elliptic curve over Q (i.e. square-free > conductor) is modular. The curves that Frey writes down, arising > from counterexamples to Fermat, are semistable and by work of > Ribet they cannot be modular, so this does it. From news1.cis.umn.edu!umn.edu!sander Fri Jun 25 23:28:18 CDT 1993 Article: 11 of geometry.institutes Newsgroups: geometry.institutes Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Geometry Center Welcomes Teachers Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 25 Jun 1993 16:29:07 GMT Lines: 17 Status: OR This week and next, there is a high school workshop for high school teachers at the Geometry Center. Starting next week, there will also be a similar workshop at Swarthmore. Hopefully through the Forum, it will be possible to establish a link between the two groups of teachers. Here at the Center, the teachers will each post a short description of themselves. It will probably include biographical details such as name, where they are from, and where they teach. It may also have comments on teaching, the workshop, the Center, or any other subject of interest, as well as questions for teachers at the Swarthmore workshop. We hope that the teachers at Swarthmore will respond, either privately with an individual electronic pal, or if they wish, through postings to the Forum. I hope the opportunity for communication proves fruitful. To all the teachers: Welcome to the Forum! From news1.cis.umn.edu!umn.edu!sander Mon Jun 28 12:26:08 CDT 1993 Article: 77 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: 4D Part 5: Grids in PDEs Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Mon, 28 Jun 1993 15:26:25 GMT Lines: 86 Status: OR "Instead of looking at 4-D as being very close to 3-D, you can often get a better feel for 4-D by thinking of it as quite a bit along the way to infinite-D. Understanding what happens in extremely high dimensions can clarify the emerging trends that make 4-D a _different_ place to live than 2- and 3-D," says Paul Burchard of the Geometry Center. Burchard is developing software to study Partial Differential Equations (PDEs) which are related to differential geometry. This article consists of observations on the problems encountered in higher dimensions when using one of the standard methods of PDEs, namely grids. Numerical calculations of solutions to differential equations often involve a rectangular or triangular grid, using only values of derivatives on the grid points to get an approximation to the function at those grid points. I will describe some of the problems with rectangular grids in higher dimensions. These same problem occur in all shapes of grid, not just rectangular ones. In a rectangular grid with k equally spaced grid points per unit length in each spatial direction, an n dimensional cube requires k^n grid points. In other words, the number of grid points grows exponentially with dimension. Thus using schemes with grid points very quickly becomes prohibitively costly in terms of storage and time as the dimension increases. Aside from the large number of points in a rectangular grid, Burchard observes another problem with this approach of having the grid points be the vertices of a hypercube. Namely, as n gets large, the corners of a n dimensional hypercube stick out more and more, making it a pointy and strangely shaped object. In terms of distance, the following shows that corners get further away: The length of a diagonal of a unit nD hypercube is the square root of n (sqrt(n)). (This can be deduced from the Pythagorean theorem and the fact that the length of each side is one.) The distance from the center to each corner is sqrt(n)/2. As dimension increases, the unit cube has corners which stick out more in linear distance. Another measurement of the oddness of the shape of the cube is to look at how much of the volume is in the corners. Compare the volume of the nD hypercube to the inscribed n dimensional ball. In some sense, the inscribed ball cuts out the corners of the cube, leaving only the middle. If the volume of this ball gets smaller, since the hypercube is always unit volume, the volume contained in the corners of the cube must be increasing as dimension increases. Through some calculations which I will describe separately, the inscribed nD ball has volume V(n)=(V(n-2)*pi)/(2*n). Note that the recursive formula contains an n in the denominator, so there is some kind of factorial decrease in volume. Thus as n increases, the volume of the inscribed ball quickly decreases. This means that more and more of the volume of the unit cube is contained away from the center, another indication that the corners of the high dimensional cube become more pointy. To give a more detailed picture, it is not only the corners which stick out in the hypercube; all of the "edges" stick out to progressively larger extent as the dimension increases. For example, in the three dimensional cube, the faces (two dimensional edges) are distance 1/2 from the center, the edges (one dimensional edges) are distance sqrt(2)/2 from the center, and the corners (zero dimensional edges) are furthest away, namely sqrt(3)/2 from the center. In general, the k dimensional edges of the n dimensional cube stick out distance sqrt(n-k)/3 from the center. By the above, perhaps grids are not the best method to solve high dimensional PDEs. In fact, even in relatively low dimensional cases, it is necessary to resort to other techniques. According to Burchard, "From the perspective of trying to write fast accurate code in differential equations, five is in a practical sense most of the way to infinity; in other words, it is practically impossible to write the code I would like and have it run in any reasonable amount of time. I think 4-D will still be feasible but slow; 5- or 6-D is where it starts to become practically impossible with grid-based techniques." From news1.cis.umn.edu!umn.edu!news Mon Aug 9 16:45:57 CDT 1993 Article: 86 of geometry.college Xref: news1.cis.umn.edu geometry.college:86 geometry.research:78 Newsgroups: geometry.college,geometry.research Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Ballistic Lunar Capture Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: fatou.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 9 Aug 1993 20:36:55 GMT Lines: 81 Status: OR The standard method to send a spacecraft to the moon calls for the craft to fire its engines get into a lunar transfer from the Earth, followed by a second firing of engines near the moon to slow the craft down enough to go into lunar orbit. W. Hohmann discovered this method in the 1920s, and this was the route used for many lunar missions, including the Apollo. Using this method, known as a Hohmann transfer, a craft takes three days to reach the moon. Since the 1920s, there has been a conjecture that it is possible to find another route to the moon in which the craft would naturally fall into a lunar orbit, without the need for a second firing. This lack of a second firing is known as ballistic capture. Though a ballistic capture would have to be quite slow, taking months rather than days to get to the moon, it would be quite good economically, due to savings on fuel, which in turn decrease size and weight, and allow for the use of smaller launch vehicle. Until recently, ballistic capture seemed an unworkable scheme. By the standard method the craft travels at one kilometer per second at the time of the second firing; this is such large velocity, that it seemed unlikely that anyone would find a ballistic capture that was physically feasible. However, in 1986 the problem was solved by Ed Belbruno, a current associate researcher at the Geometry Center. A mathematician by training, Belbruno started thinking about the problem while pondering the mathematical celestial mechanics problem of the non-existence of invariant tori. He began to study the stability of a craft's orbit based its position and velocity, (i.e., position in phase space). He came up with a five dimensional surface of instability (or chaotic region) in seven dimensional phase space. This surface is known as the fuzzy boundary. It is the dividing region, in which there is a change of influence as to whether the Earth, sun, or moon have more effect on a craft. Through the use of fuzzy boundaries between the Earth and moon and the Earth and sun, Belbruno discovered a ballistic capture. It involves initially moving towards the moon, followed by travelling four times that distance, all the way to the Earth/sun fuzzy boundary, then moving all the way around the Earth, and finally going through the moon's fuzzy boundary into a region where the moon captures the craft into orbit. Just looking at it, it seems like the craft goes incredibly out of the way to get to the moon. However, it is necessary to get to the Earth/sun fuzzy boundary to avoid going too fast when arriving near the moon. Through a very lucky accident, in 1991, merely five years later, a standard spacecraft successfully performed Belbruno's route. This fast a progression from theory to experiment is almost unheard of in the space industry. A craft launched by the Japanese had mechanical trouble, and it was not going to be possible to set it in lunar orbit by the standard means. Feeling they had nothing to lose, they attempted Belbruno's scheme. It was a success. As an answer to this unusual and rapid success, The Geometry Center will hold a conference on "Advances in Nonlinear Astrodynamics," November 8-10. The conference will focus on cost effective ways to move around the solar system using theories from the mathematical disciplines of celestial mechanics and dynamical systems. The conference attempts to bring together influential people in the space industry with researchers who study exotic orbits. This article was written based on an interview with Belbruno. For more information on his orbit, see the following article, written for a general audience: Belbruno, Edward, "Through the Fuzzy Boundary: A New Route to the Moon," The Planetary Report, Volume XII, Number 3, May/June, 1992. From news3.cis.umn.edu!umn.edu!news Tue Aug 17 16:11:25 CDT 1993 Article: 88 of geometry.college Xref: news3.cis.umn.edu geometry.forum:111 geometry.college:88 Newsgroups: geometry.forum,geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Convex Hulls Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: turing.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 17 Aug 1993 20:54:32 GMT Lines: 116 Status: OR It is often necessary to compute the convex hull of a set of points, meaning the smallest convex set containing the points. The problem arises in many branches of science. Brad Barber of the Geometry Center has designed an algorithm which computes the convex hull in arbitrary dimensions. It has improved the previous algorithms because it is as quick as is theoretically possible, while at the same time using effective storage techniques and taking into account possible imprecision in measurement. Here are some of the methods generally used to study convex hulls, talking specifically about the methods Barber used in his algorithm. One must decide how to store a convex hull, once found. It is especially tricky to do this in higher dimensions. Hulls are stored as a list of the highest dimensional facets (faces in two dimensions), the neighboring facets for each facet, as well as the vertices associated with the facets. This allows for a way to add extra points to a computed hull. So how do you actually compute a convex hull of a set of points S? I will talk about the two dimensional case; the n dimensional case is similar in spirit. Suppose you start with a hull and want to add some points. One way to do so is to pick a random point in S, see if it is inside the initial hull; if not, find the vertices of the new hull by deciding whether the old vertices are inside or outside the supporting cone of the new point. I never thought I would indulge in ASCII art, but here's an attempt to demonstrate the concept, where p1,p2,p3,p4 formed the old hull, and p1 is no longer part of the new hull which includes np. np p1 p2 p3 p4 Barber's algorithm improves the speed of just picking a random point; starting with a hull and some points to add, he assigns each point to one of the already known faces, by observing which half plane the point is in. For example above, np would be assigned to face from p1 to p2. Among all the points assigned to a given face, he then chooses the one furthest from the face. It is known theoretically that this point will be a vertex of the new hull. Now the process starts again, with the new hull, still only considering the set of points assigned to one face. In this way, the algorithm splits the global hull problem into a series of local problems, in this way improving the use of virtual memory. Another advantage of this approach is that after adding the furthest point from the face as a vertex, the new vertex will not be as far away. This means the change in convex hull is less for each addition. Thus it is fairly clear how to make an approximate convex hull, which stops considering new points once all the points are within some small distance of the hull. Barber has written the program data format in such a way that in three and four dimensions one can use geomview to look at the convex hulls. The pictures are quite beautiful; in 3D they incorporate the estimated error visually by having a lower bound surface and an upper bound surface for the surface of the hull. The four dimensional pictures are quite impressive. Barber says by working with hulls, he has gained a feeling for 4D. He said, "When I first looked at a projection of a 4D convex hull, I saw the picture and realized that it was just Delauney triangulation. I knew from the theory of Voronoi diagrams that the projection of an n-dimensional convex hull is an n-1-dimensional triangulation, so I felt like I could get a feeling for 4D." This gained understanding of 4D is only incidental to the convex hull program, but it influenced me to start work on a separate series of articles, looking at what it means to visualize 4D and who is doing it and why at the Geometry Center. This is a quite brief description of some of the features and advantages of Barber's convex hull algorithm. Hannu Huhdanpaa has implemented the algorithm. It is available from the Geometry Center by anonymous ftp. A Macintosh version is in the Geometry Forum. Huhdanpaa is now working on imprecise convex hulls. For further information, please see: Preparata, Franco, Shamos, Michael, Computational Geometry,Springer-Verlag, New York, 1985. This is a general discussion of algorithms on convex hulls and related topics. Barber, C.B., Dobkin, D.P., Huhdanpaa, H., "The Quickhull algorithm for convex hull," Technical Report GCG53, The Geometry Center, University of Minnesota, July 30, 1993. To get the program from a Unix computer that is attached to Internet: mkdir qhull; cd qhull; ftp geom.umn.edu; user: anonymous; password: ; cd pub; get qhull.tar.Z; exit; uncompress qhull.tar.Z; tar xf qhull.tar; make To get the program from the Geometry Forum: See Annie's instructions. From news3.cis.umn.edu!umn.edu!news Tue Aug 17 16:11:39 CDT 1993 Article: 89 of geometry.college Newsgroups: geometry.announcments,geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: SIGGRAPH Meeting Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: turing.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 17 Aug 1993 21:01:46 GMT Lines: 67 Status: OR Last week was the meeting of SIGGRAPH, the conference of the Graphics Special Interest Group of the Association of Computing Machinery (ACM), the professional association of computer scientists. The conference was the largest of the ACM meetings, with 7000 people attending the papers and panels technical sessions and a total attendance of 30,000. SIGGRAPH includes an exhibit of computer graphics art, an electronic theater showing video clips from commercials, movies, art films, and scientific visualization, and an exhibit floor in which computer companies show their newest merchandise. Several people from the Geometry Center attended. Here are some of the highlights and trends that they noticed this year in graphics. As a result of the dinosaurs from Jurassic Park, which were designed on Silicon Graphics (SGI) machines, as well as the location of the conference in L.A., the prevailing image in demonstrations was the dinosaur. SGI had a virtual reality ride where the viewer flew on the back of a prehistoric bird; the wait to see it was over an hour. In terms of trends, the popular idea this year is multimedia. The SGI Indy, a new low end machine, includes a built in video camera. Sun and Dec have new machines with video cameras as well; the Indy camera is mounted on top of the monitor whereas the Sun camera is in a small hole at the end of a flexible tube so that the user can move it around easily. The addition of video is intended to improve electronic communication. For example, it will enable easy video electronic mail. One of the art displays used a video camera as well; it was a flight simulator in which a video camera sensed the movement of the user's outstretched arms, which controlled the motion of the plane. Other notable products included the SGI top end machine Onyx, which combines the power of a supercomputer with reknowned SGI graphics, such as Reality Engine hardware texture mapping; that is, hardware for mapping a textured look onto 3D objects. Also of note were the improved virtual reality machines, stereo HD-TV, and force feedback data gloves. Mark Phillips attended a one day course on design of large scale graphics systems. This is of special interest to the staff at the Geometry Center because they have created the interactive three dimensional graphics package Geomview. Of particular note was a talk by Paul Strauss of SGI, one of the leaders of the team that developed IRIS Inventor. His presentation included a discussion of many of the same design issues that were involved in writing Geomview. Charlie Gunn, formerly from the Geometry Center, gave a talk titled "Discrete Groups and Visualization of Three-Dimensional Manifolds". He discussed the program Maniview, a 3-manifold visualization tool which works in conjunction with Geomview. The almost universal method used in 3D graphics is to create a 3D model inside the computer which is viewed with a virtual camera. Tamara Munzner was particularly excited by a new idea from two researchers at Apple: 3D graphics derived from a series of two dimensional images and an interpolation method known as morphing to give the viewer the idea of 3D, without ever building an internal 3D model. The conference is a chance to find out what is new in graphics. By the end of the week, it gave everyone I talked to a sense of "visual overload." In addition to the visual excitement, Daeron Meyer said he likes the fact that people at the conference have such varied backgrounds and objectives. Whether vendors, commercial users, artists, or graphics researchers, SIGGRAPH gives everyone a chance to communicate with others in the field of computer graphics. From news1.cis.umn.edu!umn.edu!news Wed Aug 25 15:28:02 CDT 1993 Article: 90 of geometry.college Xref: news1.cis.umn.edu geometry.pre-college:146 geometry.college:90 Newsgroups: geometry.pre-college,geometry.college Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Volumes in nD Using Basic High School Geometry Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Wed, 25 Aug 1993 20:17:37 GMT Lines: 108 Status: OR This is a description of a geometric means to calculate the volume of a n-ball inscribed in a n dimensional hypercube. In two dimensions we know that the disk inscribed in the unit square has radius 1/2 and therefore area pi*(1/2)^2=pi/4. In three dimensions, the ball inscribed in the unit cube again has radius 1/2, where sqrt means take the square root. Thus the volume of this ball is 4/3*pi*(1/2)^3=pi/6. Paul Burchard, a postdoc at the Geometry Center, showed me how to extend these results to n dimensions (denoted R^n) without using more than basic high school geometry and a few pictures. The extension turns out to be a recursive relation based on the two and three dimensional results: the n-ball inscribed in the unit hypercube has volume equal to pi/(2*n) times the volume of the (n-2)-ball. Perhaps this gives a good way to introduce a high school geometry course to higher dimensional spaces. In the course of reading the article, please see the associated figures, available from the Forum by anonymous ftp in the directory hs.geometry.article By describing the ball as a cone over an object and using some general properties of volumes of cones, I will be able to find the volume of the ball. Here is a relationship between spheres and solid balls using cones: By definition, a cone over an object (not intersecting the origin) consists of all the lines connecting that object to the origin. A closed n-ball is the boundary and interior of an (n-1)-sphere. More precisely, the sphere S(n-1)={x in R^n:|x|=1}, and the ball B(n)={x in R^n:|x|<=1}={t*x:x in S(n-1), t in [0,1]}. Thus the n-ball is actually a cone over the (n-1)-sphere. Thus to find the volume of the ball (Vol(n)), we need an equation for the volume of a cone over a sphere. Note that in fact Vol(n) is the volume of the ball of radius one, when really we wanted the volume of a ball of radius 1/2. This means we need to divide by 2^n at the end of the calculation. We can think of the cone over the (n-1)-sphere as a union of cones over infinitesimal n-1 dimensional cubes. Note that the volume of a cone over a flat object only depends on the distance from the plane of the object to the origin (see Figure 1). The volume varies linearly with the distance from the object to the origin, since only varying one dimension makes the volume change linearly. As an analogy, think of the cube (see Figure 2). The n dimensional cube consists of exactly n identical cones over n-1 dimensional cubes. To see this, note that a vertex V of a cube intersects exactly n faces and has n faces disjoint from it. Almost every point of the cube is in exactly one cone between a disjoint face and V, the cube consists of the union of cones from each of the n disjoint faces to V (see Figure 3). Given an arbitrary point P in the cube, draw a ray from V through P. A point on this line intersects a disjoint face. This face is unique for all but a set of zero volume. Here is a slightly technical proof: Assume that we are discussing the unit cube. If we make V the origin, and all the other vertices vectors consisting of 0's and 1's, then the cube is contained entirely in the first quadrant. The faces intersecting V are the coordinate (n-1)-planes. Exclude points in these coordinate planes; also exclude points on rays from V to the (n-2) edges (these are the intersections of adjacent faces). Both these are sets of zero volume. Any remaining point P in the cube is in the first quadrant; therefore a ray through the origin must intersect one of the faces of the cube disjoint from V. The choice is unique by avoiding edges. The volume of the cube of height h is h^n, and the cube consists of n equal cones over a (n-1) dimensional face, which is really just an (n-1)-cube. Thus volume of a height h cone over an (n-1)-cube of side length h is (h^n)/n. Thus by the linearity, the volume of the cone of height one over an (n-1)-cube of side length h is (h^n)/(n*h)=1/n*volume of the (n-1) cube. Since we think of the sphere as made up of infinitessimal (n-1)-cubes, the volume of the cone over the unit sphere=Vol(n)=(1/n)*surface area of the sphere=A(n-1)/n. All that is left is to calculate the area of the sphere in n dimensions=A(n-1). First consider the 3D case for the sphere of radius one. Since the surface area of the region from phi1 to phi2 in spherical coordinates is 2*pi*(cos(phi1)-cos(phi2)), which is also equal to the surface area of the portion of the cylinder of radius one which surrounds this region (see Figure 4). For a more elementary proof involving similar triangles, see Figure 5. Now we only need the surface area of the cylinder. Thus A(2)=2*pi*2. In nD, the generalization of the statement is true. Using n dimensional spherical coordinates, the same pair of similar triangles show that the (n-1)-sphere has the same area as the cross product of the circle with the (n-2)-ball. That is, A(n-1)=2*pi*Vol(n-2). So finally, since Vol(n)=1/n*A(n-1), we have Vol(n)=2*pi/n*Vol(n-2). All these results were for radius one spheres and balls. To get the volume of the inscribed ball in a unit cube VI(n), just divide by 2^n. In other words, VI(n)=pi/(2*n)*VI(n-2). This was a result quoted in the beginning of the article. From news3.cis.umn.edu!umn.edu!sander Mon Sep 6 15:31:43 CDT 1993 Article: 94 of geometry.college Xref: news3.cis.umn.edu geometry.pre-college:163 geometry.college:94 Newsgroups: geometry.pre-college,geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Midge Cozzens: N.S.F. and Pre-college Education Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Mon, 6 Sep 1993 20:26:12 GMT Lines: 66 Status: OR "I always knew I wanted to be in mathematics and education," says Midge Cozzens. "In third grade, my intention was to teach arithmetic. Of course my goals shifted a bit, but they always have involved teaching and math." Cozzens has accomplished her goal; she is currently the director of the Division of Elementary, Secondary, and Informal Education at the National Science Foundation. For twelve years previous to this job, she was a professor/chairperson and research mathematician at Northeastern University in Boston. Why did she decide to go to the N.S.F. instead? "I have always said that changes need to be made in education. Turning fifty has an effect on the way you think and what you feel you can accomplish. I thought I could make an impact in education, and I needed to act on my convictions. "It is the responsibility of researchers at the university level to pay attention to K-12 education. The researchers are the ones who have the interest in science and math. In addition, whether or not it should be true, researchers have more clout than educators. I have a particular advantage being an applied mathematician. It gives me understanding for a large community of mathematicians and scientists, as well as their respect." What effect has the N.S.F. had on education? "Since the N.S.F. is in charge of the funding, we really control what happens in mathematics and science education. The National Council of Teachers of Mathematics (N.C.T.M.) published their standards, and the science standards will be ready in six months or a year. These standards need implementation. Through the Teacher Enhancement Program, the curriculum development efforts, and the creation of model programs for the math standards, we are establishing the method of implementation for both math and science. "As an example of the implementation, the N.C.T.M. standards require teaching of data analysis and statistics at all levels. Most of the teachers have never seen this material. Through the Teacher Enhancement Program, the teachers will learn the content, and as they learn the material, they will learn a different teaching style. They will become accustomed to the new interactive learning. This is done through problem solving, rather than the traditional talking head environment." "One of the important changes we are making in education is the recognition that there are different ways to learn. Even as young as two years old, studies show that different children have different strategies to approach a problem. It is almost impossible to change the basic method of attack used by a given child. For example, some people learn best by listening to a lecture or by reading a book. I am a visual learner. I have to get my hands on things, see them, and touch them. I always thought lectures were boring. If teachers are aware that people learn in different ways, it will change their method of teaching. "When I started this job two years ago, many people told me that it was impossible to make an impact on education in this country. I think we can and have made an impact. The many changes in teaching methods and subject matter will give all children a chance to learn and enjoy math." From news3.cis.umn.edu!umn.edu!sander Thu Sep 16 09:34:19 CDT 1993 Article: 96 of geometry.college Xref: news3.cis.umn.edu geometry.research:87 geometry.college:96 Newsgroups: geometry.research,geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Dendritic Growth Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Thu, 16 Sep 1993 14:26:12 GMT Lines: 152 Status: OR Dendritic Growth In a solution of liquid or gas at approximately the temperature of solidification which contains a small crystal. What shape forms as the crystal continues to grow? For example, how do ice crystals form in a container of water or in a cloud? One possibility is the creation of dendrites, objects such as snowflakes with complex spiny structure, as pictured in the figures. (The figures are available by anonymous ftp. They are all made by Andy Roosen.) The boundary between liquid and crystal has two main governing forces on it; first, there is the energy of surface tension. This means that there is a tendency for the area of the surface of the crystal to decrease. Therefore the rate of solidification depends on the curvature of the surface. the cold temperature of the liquid means that it has a tendency to freeze. However, the process of freezing liquid produces a latent heat of fusion. Thus every time freezing takes place in some area, the solution surrounding the area gets warmer and less likely to freeze further until the heat has diffused throughout the liquid. As a result of the force of temperature, if there is a small bump on a crystal, the heat of fusion near the bump diffuses quicker since it is protruding into the liquid, thereby increasing the rate at which crystal forms. If the only force on a crystal was the temperature, these bumps in the crystal would become arbitrarily thin. However, since there is also surface tension, the bumps stay a finite thickness. Since in a natural physical system there is always noise, there are always enough small perturbations to make the spiny structure characteristic for dendrites. This week at the Geometry Center, a group of researchers known as the Minimal Surfaces Team has convened with the object of studying dendrites. Formed five years ago, the team meets at the Center to work on the mathematics of minimal surfaces. Although Jean Taylor of Rutgers University and Fred Almgren of Princeton University are the only remaining members from the original team, the aim remains the same. This year the eight members of the team come from a variety of universities, as well as from the National Institute of Standards and Technology (N.I.S.T.). They are studying the precise growth methods of dendrites as described above. The interest is both theoretical and practical; the study of dendrites has applications in areas such as metallurgy. Starting with basic assumptions about the laws governing the dendritic growth, the members work on applying computational and mathematical techniques. I talked to Almgren as well as Andy Roosen, a researcher from N.I.S.T. Each told me about the particular aspects of dendrites that they are studying. Here is a brief description of their research. To date, most of the research about dendrites has been about smooth two-dimensional objects. Almgren has extended the results to a three-dimensional model. It was a challenge to create a practical three-dimensional algorithm because the model of crystal growth is so costly in terms of time and storage. In order to conserve these quantities, he used a special efficient partition of the space. A standard method to partition space is to use an evenly spaced three-dimensional grid. However, it is not necessary to have this kind of precision far away from the crystal, since there will only be large changes near the interface of solution and crystal. Almgren's method is to divide the space in large blocks far from the crystal and finer blocks close to the crystal. Since the structure is constantly changing, it is necessary to repartition the space at every step. Almgren has worked on this with Princeton graduate students Nung Kwan Yip and David Caraballo and Rutgers undergraduate student Sharon Caraballo. Together they have constructed a practical algorithm to accomplish the repartition. Roosen works on another generalization of the smooth planar case: rather than extending the dimension of the dendrites, he studies dendrites which are not smooth. Starting with some work of Taylor on faceted dendrites, Roosen has incorporated temperature effects. He restricts the crystal boundary to a limited number of possible directions; in other words, the boundary consists of straight lines, and these lines can only have certain specified directions. Using the restriction of linear sides in a limited number of directions, Roosen imposes a change in temperature across a side. Experimentally, it is known that a flat side with a temperature change along it will not remain flat; it will break into segments. Roosen has constructed a model of this behavior. He determine mathematically where the model will break into several linear segments. The model appears physically accurate. One interesting point about the theoretical model is that if the grid is too fine, the model it fails to produce the spiny structures always seen with actual dendrites. In other words, in order to model the physical system, one must either be inaccurate or reintroduce noise. It is the noise which characterizes the system. See figures 1 and 3. The study of dendrites is of important practical use as well as interesting mathematics. It is most intriguing after seeing some of the pictures. Thus there are six figures to accompany the article. Although not all of them have been described specifically in the article, the captions should be sufficiently descriptive to make them worth a look. The figures are located in the pictures/articles/dendritic.growth directory on the Geometry Forum (forum.swarthmore.edu). To get them, use an anonymous ftp. Captions for the figures (by Andy Roosen): 1) A "frilly" computation that has a lot of numerical noise because it was on a coarse temperature grid and a low surface energy. 2) A "boring" computation on a fine temperature grid. Notice that no side-branches appeared. 3) An effort to reintroduce side branches by perturbing the velocities of edges depending on their distance from the center. In this and the previous figure, several interface positions at various times are shown. 4) A computation different from the others: instead of moving each edge by the average temperature along it, an edge is moved by the low temperature along it. notice that the dendrite is much more blocky. It really appears crystalline. 5&6) Two computations in which perturbations were introduced by regularly raising the temperature at the tip of the dendrites a bit (imagine a lazer repeatedly shining on the tip of the dendrite). Color (in figure 6 only) indicates temperature with redder = hotter and bluer = colder. The temperature ranges from -.5 to 0 with 0 as the melting temperature, and the latent heat of fusion is 1. From news3.cis.umn.edu!umn.edu!sander Fri Sep 17 19:10:33 CDT 1993 Article: 183 of geometry.pre-college Newsgroups: geometry.pre-college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Algebraic Geometry for High School Students Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 17 Sep 1993 21:39:23 GMT Lines: 58 Status: OR Starting Tuesday, University of Minnesota Professor Vic Reiner will teach an introductory course on algebraic geometry for ten to twelve high school students. Rather than attempt to develop deep theory, Reiner has chosen a computational approach to the subject. The students will become familiar with the mathematics software package MAPLE, as well as learning some of the algorithms that MAPLE uses. In order to have the proper computational facilities, the class will meet at the Geometry Center. Reiner is a Combinatorist, so I asked him how he had decided on algebraic geometry as the topic of his course. He explained that he chose the topic because of a well written new textbook on the subject. The book, "Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra," by David Cox, John Little, and Donal O'Shea, was just published in 1992. It combines three courses, developed independently by the authors. Reiner first saw the book at an NSF Regional Geometry Institute at Amherst. He likes it particularly well for this course because it manages to develop the basic concepts of algebraic geometry with a minimum prerequisite knowledge. The only necessary previous mathematics background is linear algebra. The book discusses the subjects that one cares about in algebraic geometry, namely manipulation of polynomials and basic ideas of varieties, but it does not get into deep theoretical material; the authors choose instead to show practical algorithms. As quickly as the second chapter, the book introduces Groebner bases, the underlying idea in numerical algorithms for finding roots of polynomials. Here are a few of the highlights that Reiner hopes to cover in the class. To begin they will learn some basic concepts. For example, they have not had any complex analysis. However, using a heuristic topological argument involving winding numbers, Reiner plans to justify the fundamental theorem of algebra, thereby motivating use of complex rather than real spaces. He will also discuss basic ideas of fields. The students will learn to numerically solve a system of polynomial equations with MAPLE and graph the answer with graphics packages from the Geometry Center. In order to remove the mystique as to how MAPLE works, Reiner will break down the algorithms into small steps. He also plans to discuss Pythagorean triples and Fermat's Last Theorem. Reiner's course is part of a larger program called the University of Minnesota Talented Youth Mathematics Program, abbreviated UMTYMP. UMTYMP includes an accelerated high school math curriculum, followed by a calculus sequence. The first three years of calculus are in depth versions of the same classes usually taught to calculus students, although taught differently than at a college, as UMTYMP pioneers experimental teaching methods and curricula. The fourth year of calculus is actually a topics class, the subject of which varies from year to year. Last year the class covered probablility and combinatorics. Next year will be differential equations and dynamical systems. This will be the first year of the algebraic geometry class. Reiner says, "These students are very good. The students could handle this class at Mt. Holyoke, Amherst, and Holy Cross, the places where the courses were first taught by Cox, Little, and O'Shea; I'm sure these high school students will have no trouble." From news1.cis.umn.edu!umn.edu!sander Mon Sep 20 14:18:20 CDT 1993 Article: 97 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: David Epstein: Innovative Teaching Methods Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Mon, 20 Sep 1993 14:57:13 GMT Lines: 72 Status: OR "I have been interested in teaching for a long time. For many years I was worried about the ineffectiveness of my teaching. My sister was a primary school teacher and is now a lecturer in sociology. She has always complained to me about the quality of mathematics teaching at university. The problem was I never knew what to do about it," says David Epstein, professor of mathematics at Warwick University in England. "I first saw how I could improve my teaching at a summer course two years ago at the Geometry Center." "The course instructors were all famous mathematicians: John Conway, Bill Thurston, Peter Doyle, and Jane Gilman. The students ranged in age from 15 to 65. Many of them were high school students. Some of them were famous research mathematicians. However, despite their diverse backgrounds, everyone learned something. "The class consisted of discussion and problem solving; there were no formal lectures. Through unusual presentation techniques, the instructors managed to present some quite advanced mathematics usually first encountered at the graduate level. For example, to describe curvature of a torus, they made use of peelings from potatos. We often made models and cut objects out of paper." "The Geometry Center summer course inspired me to try to improve my own teaching. I have been trying some new techniques with a course on metric spaces at Warwick. The students are polarized on whether they like the outcome. The unusual format of the class means they have to work harder. Some of them do not like this, but others have told members of the department to make their classes as hard as mine. In general I think it is good for the students. The course is in its third year, and it improves each time." Here are some of the teaching techniques in Epstein's class: Despite having approximately 160 students, Epstein uses a discussion format rather than a formal lecture style. In addition, Epstein attempts to concentrate on understanding rather than tests. To this end, in past years he has tried giving a variety of extra credit questions, with monetary awards for those who have answered the most extra questions correctly. Epstein plans to continue his tradition of giving out examples of all the standard errors made on past tests and making the students work out flaws in the reasoning. This year, the most important change in Epstein's class is the syllabus; "I like to annoy my collegues by pointing out that our syllabus design is based on a never ending process; we always design a course as a fundamental building block for more advanced courses. We never reach a goal. Using a technique explained to me by Regine Douady, an expert in elementary education, this year my class has a syllabus different from any book on metric spaces. The students start with a list of problems which they can understand and attempt before the class begins. However, solving these questions is quite difficult without some powerful theory. The object of the class is to design the tools necessary to answer these problems. This means that there is motivation for the material. When they see the key to solving one of the problems, it is a relief and often a surprize." The questions for Epstein's class will be available from either the Geometry Center or from the Geometry Forum by anonymous ftp in postscript form. They are in the directory: pictures/articles/epstein.innovative.methods Though some problems involve quite advanced material, they are definitely worth trying. If anyone has further questions about the course, email to David Epstein at dbae@geom.umn.edu From news1.cis.umn.edu!umn.edu!sander Wed Sep 22 09:56:48 CDT 1993 Article: 98 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: TA Orientation, Univ of Minnesota Math Department Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Tue, 21 Sep 1993 22:40:32 GMT Lines: 77 Status: OR This past week, I helped with an orientation program to prepare the new teaching assistants in the University of Minnesota Math Department for their duties as instructors. The program was run by Professor Don Kahn, Director of Graduate Studies and four advanced TAs, including me. Throughout the week, we attempted to not only inform the new TAs of all of the procedures involved in teaching recitation sections, but also prepare them for all of the unexpected incidents that often occur, and give them an opportunity to practice teaching for a friendly audience. This orientation program runs every year. In my personal experience, as well as the experience of those who I have asked, the program is quite successful. I hope the following description will prove useful to teachers at other universities. Perhaps you will have occasion to run a similar program. Maybe this will be interesting to others who have wondered about teacher preparation techniques at the university level. The orientation had one component of standard introduction to the University of Minnesota and to the Math Department. This took approximately one hour each day. It consisted of welcome remarks from the Chair, Assistant Chair, and Director of Undergraduate Studies. It also consisted of instructions on the library and computer system. For another hour each day, we discussed different aspects of teaching. The subjects were: lecture techniques, what to expect in office hours, how to proctor exams, and what to do on the first day of class. We tried to review all the possible situations that might occur and how to avoid some of the more unpleasant ones. This was perhaps the most fun for the advanced TAs, since in order to put everyone at ease, we started each discussion with a ten minute theatrical demonstration. To give an example, in order to demonstrate lecture techniques, Kahn gave his rendition of the world's worst lecture. The new TAs wrote down as many mistakes as they could find. Nobody had much trouble detecting some errors; he started his lecture wearing sunglasses, taking pills, lighting a cigarette, and writing in the lower right hand corner of the board. One of the advanced TAs then proceeded with a good lecture, and we discussed some tips on how to give good lectures. The new TAs seemed relaxed to ask questions after they had a chance to laugh at our demonstrations. The third component and most important part of the orientation program was the practice teaching sessions. For two hours each day, each advanced TA was in a classroom with five new TAs. For about thirty minutes, each new TA would conduct a recitation section on a preassigned section of a calculus book. The rest of us behaved as if we were calculus students. At the end of the thirty minutes, everyone would make constructive suggestions. Some comments were about teaching in general, such as board technique, eye contact, voice volume, and avoiding nervousness. Others pertained only to mathematics, such as how to more clearly explain a particular concept, how to make it clear when you are giving a definition and when you are stating a theorem, whether to show intermediate steps on a problem, how much mathematical notation to use, and what level of generality to use. Of course, the answers to many of these issues are matters of personal preference. The small groups discussed the advantages to a variety of approaches. There were many comments I could make based on experience, but there were also helpful suggestions from the other new TAs. I think everyone benefited from the suggestions and the practice. One person told me, "It was great to have an opportunity to make mistakes in front of a friendly and tolerant audience. Now I won't have to make them in a real classroom situation." It has been a pleasure to get to know some new fellow graduate students and help them prepare to teach. I hope that other universities have similar programs. It enables TAs to both be more at ease and more qualified in their new role. From news1.cis.umn.edu!umn.edu!news Tue Oct 12 15:30:42 CDT 1993 Article: 112 of geometry.announcements Newsgroups: geometry.announcements Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Geometry Center Seminars Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 12 Oct 1993 15:47:15 GMT Lines: 34 Status: OR The Geometry Center fall seminars start on October 18. The seminar list follows. For updates and further details about the Geometry Center, finger geom@geom.umn.edu. If you need to know more information about seminars, or if you will be here and would like to give a talk, please contact Paul Burchard, burchard@geom.umn.edu, (612) 626-8319. Seminars: --------- Mon., Oct. 18, 2:30 pm Oliver Goodman (Geometry Center), ``Andre'ev's theorem and circle packings'' Geometry Center Conference Room (1300 2nd St. South, Rm. 538) Mon., Nov. 1, 2:30 pm Leonidas Palios (Geometry Center), ``Tetrahedralizing the space between a convex polyhedron and a convex polygon'' Geometry Center Classroom (1300 2nd St. South, Rm. 531) Mon., Nov. 8, 2:30 pm Alfred Gray (University of Maryland), ``Surfaces'' Geometry Center Classroom (1300 2nd St. South, Rm. 531) Mon., Nov. 22, 2:30 pm Dave Witte (Williams College), ``Tessellations of Solvmanifolds'' Geometry Center Conference Room (1300 2nd St. South, Rm. 538) From news3.cis.umn.edu!umn.edu!sander Thu Oct 14 10:52:46 CDT 1993 Article: 113 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Forum Gets Noticed Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Wed, 13 Oct 1993 22:39:52 GMT Lines: 8 Status: OR Forum Gets Noticed Gene Klotz has written an article about the Geometry Forum in the October 1993 issue of Notices of the American Mathematical Society. It is in the "Computers and Mathematics" column, page 985, along with an article about Geometry Center software Geomview. Although everyone who sees this is already a regular subscriber, perhaps it will still prove entertaining and informative. From news1.cis.umn.edu!umn.edu!sander Fri Oct 15 13:21:25 CDT 1993 Article: 105 of geometry.college Xref: news1.cis.umn.edu geometry.college:105 geometry.research:92 Newsgroups: geometry.college,geometry.research Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Quantum Field Theory Message-ID: Summary: A link between mathematics and physics, as described by Dan Freed Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Fri, 15 Oct 1993 15:23:01 GMT Lines: 62 Status: OR "This is a period in which mathematics and physics are closely linked," says Geometry Center visiting professor Dan Freed. "This has happened quite frequently in the past. For example, the Greeks developed trigonometry to try to describe the stars' placements in the sky. Newton invented calculus while trying explain Kepler's work about astronomy. Gauss studied the geometry of surfaces because he was trying to understand and describe the curvature of the earth. Einstein's theory of general relativity gave importance to the subject of Riemannian geometry. Now it is the work of physicists in quantum field theory which is giving mathematics a new push." Dan Freed is currently working in the mathematical discipline of topological quantum field theory which relates closely to the physical results. Using quantum field theory, physicists have brought new insight to unsolved problems in mathematics by using a physical intuition for the problem not available to mathematicians. For example, here is a mathematical situation which became more clear by physical intuition: Four-dimensional objects are have some problems not seen in any other dimension. There are some four-dimensional manifolds which are indistinguishable by traditional means, such as classical homotopy theory. They are homeomorphic but not diffeomorphic. Therefore, the mathematician Donaldson came up with a series of computable quantities which are invariant for a given manifold. These quantities are called Donaldson invariants. They enable one to distinguish between the four-dimensional manifolds which were previously indistinguishable by any computable means. Physicist Ed Witten fit these Donaldson invariants into a general framework using quantum field theory. Just this week Freed heard that Witten used quantum field theory techniques to "prove" (in the physicist's sense) an interesting result about the Donaldson invariants, generalizing recent work of Kronheimer and Mrowka. In another case, mathematicians were trying to compute the number of a certain kind of rational curves of a given degree. They had computed some of the numbers of curves, but it was quite difficult and cumbersome. Then the physicist Candelas and his collaborators came up with a generating function for the number of curves. The numbers did not agree with those that the mathematicians had calculated in one case. This resulted in a great deal of debate. However, it turned out that the mathematicians had made a mistake, and Candelas' function has been successful in predicting the answers. After the results on the Donaldson invariants, Witten did some further work in which he linked quantum field theory to the Jones polynomials, an important tool for knot theorists. While creating this link, Witten discovered a new invariant in three-dimensional manifolds. Though this has not been earth-shattering for topology, it is a rare and thus exciting discovery. The physicists' work is not what mathematicians would call rigorous. Many of the methods used are cannot be justified mathematically. Yet the results seem to work mathematically. Freed says, "I would not say that the physicists are doing things that are wrong, but certainly they are using tools which do not belong to the toolkits of mathematicians. Though to mathematicians certain steps do not make sense, the physical framework of quantum field theory serves as an excellent prediction and motivation for mathematical results. Over the past decade we have seen many instances where these predictions--often numerical and very concrete--have proved correct. The challenge for mathematicians is to create the new mathematics behind these physical insights." From news1.cis.umn.edu!umn.edu!sander Tue Oct 19 18:28:21 CDT 1993 Article: 115 of geometry.announcements Newsgroups: geometry.announcements Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Math Birthday Party Message-ID: Summary: Announcing the Hirsch Symposium Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: Geometry Center, University of Minnesota Date: Tue, 19 Oct 1993 23:10:22 GMT Lines: 20 Status: OR Thursday through Sunday, at the University of California, Berkeley, there will be a special meeting of the Midwest Dynamical Systems Seminar in honor of Professor Morris Hirsch's 60th birthday. Hirsch recieved his PhD from the University of Chicago in 1958. He has been on the faculty at the University of California, Berkeley since 1965. He is known for his contributions in topology of manifolds and dynamical systems. The topics covered during the conference reflect some of Hirsch's research interests; Thursday and Friday's talks will concentrate on Neural Nets and Monotone Systems; Saturday and Sunday will concentrate on Applications of Geometry and Topology to Dynamical Systems. I will attend the conference. Assuming email access in Berkeley, I will post details on interesting talks during my stay. Otherwise look for postings early next week. The postings will be in geometry.college and geometry.research. From sander@geom.umn.edu Thu Oct 28 17:29:55 CDT 1993 Article: 107 of geometry.college Xref: news1.cis.umn.edu geometry.research:96 geometry.college:107 Newsgroups: geometry.research,geometry.college Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Image Homotopy Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: turing.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Thu, 28 Oct 1993 22:01:11 GMT Lines: 128 Status: OR Image Homotopy Given a series of mathematical objects, one often classifies them by an appropriate choice of criterion for equivalence. In particular, consider the case when the objects are smooth, locally one-to-one maps of surfaces into R^3. These maps are called immersions of surfaces into R^3. Examples of immersed surfaces include common geometric shapes such as cylinders, spheres, and tori. The standard immersion of the torus is a surface that looks like the outside of a bagel or donut. It is the surface of revolution generated by rotating a circle in three-space around a line that lies in the plane of the circle but does not intersect it. The general torus is the cross product of two circles. This is an abstract surface not contained in any space. There are a variety of possible immersions of the torus. For example, if you twist the standard torus, or even if you cut the standard torus through the small circle, twist the resulting cylinder around itself into a knot, and glue the circle back together, you still have an immersion of the torus. One would like to find an equivalence relation for these immersions. This is the discussion of one such relation, known as image homotopy. This article describes image homotopy and some of the results about this relation. It contrasts this concept with a similar and perhaps more familiar concept of regular homotopy. It is the result of a recent discussion with Geometry Center postdoc Davide Cervone, who uses a version of it in his research. >From the example of the torus, one can see that there are a large variety of immersions of a given surface. We wish to establish an equivalence relation for immersions of one surface. One common relation used to equate immersions of a surface is called regular homotopy. Two immersions F and G of a surface are regularly homotopic if there is a smooth map H of the surface through time such that at each fixed time, H is an immersion of the surface, H at time zero is equal to F, and H at time one is equal to G. In other words, two immersions are regularly homotopic when one can be smoothly deformed into the other while always maintaining an immersion. There is a disadvantage of regular homotopy; sometimes two immersions that look identical are not regularly homotopic. For example, consider the torus as a product of two circles mapped to the standard torus of revolution, in which one circle rotates around a line, and its path traces out the other circle. Now exchange the roles of the two circles to get a different immersion. Although they look identical, these immersions are not regularly homotopic. A smooth map from one to the other would have to interchange the two circles, turning the torus inside out. This is not possible without cutting a hole in the torus or using the fourth dimension. Unlike the torus, the sphere and the inside-out sphere are regularly homotopic. Much research, and even several movies have been devoted to showing the regular homotopy between the two immersions. See my article "Geometry Center Movie Part 1, A mathematical description and history of sphere eversion," geometry.college, April 21, 1993, which is available by ftp from forum.swarthmore.edu. In order to overcome the problem that two immersions might appear identical without being in the same regular homotopy equivalence class, one can use another equivalence related to regular homotopy called image homotopy. Immersions F and G of a surface are image homotopic if there is a diffeomorphism S from the surface to itself so that F and G composed with S are regularly homotopic. Thus all regularly homotopic immersions are image homotopic, but image homotopy makes fewer distinctions among immersions. In particular, immersions which appear identical are image homotopic. It turns out that there are two image homotopy classes for immersions of the torus, as opposed to four regular homotopy classes. One of the classes contains the standard torus. Here is Cervone's description of a representative of the second class: "Start with a rectangle, and curl the top edge forward and down toward the middle, and the bottom edge back and up toward the middle, then glue these two edges together (the surface will have to pass through itself to do it). This forms a "tube" with a cross section in the shape of a figure-8. Bring the two ends of this tube together, but before gluing, rotate one of the ends a full 360 degrees (this puts a twist in the tube, just as rotating your wrist when you hold the end of a belt puts a twist into the belt). Now glue the two figure-8's together, and you have a torus with twists in several different directions. The self-intersection can not be removed while maintaining an immersion throughout." What is known in general about image homotopy classes of immersions of surfaces? According to Cervone, in 1985, Ulrich Pinkall classified all surfaces up to image homotopy. "Given a specific surface, we can consider all possible immersions of that surface into R^3, and can ask which of them are equivalent under image homopoty. Pinkall's result was to determine these equivalence classes, using algebraic means, for an arbitrary surface. The structure of the set of these equivalence classes turns out to be that of a semi-group with only four generators and a small number of defining relations. The generators of the semi-group are the two classes of tori [previously described] together with left- and right-handed versions of an immersion of the real projective plane called the Boy surface. The sphere is the identity." Davide Cervone recieved his PhD from Brown University. This fall he started as a postdoc at the Geometry Center. Cervone works with piecewise-linear surfaces (ones made out of flat triangles) rather than smooth surfaces; he looks for immersions in each image homotopy class that are made from the smallest possible number of triangles, and has generated several examples that were previously unknown. From sander@geom.umn.edu Fri Oct 29 12:12:26 CDT 1993 Article: 119 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Re: Geometry Center Seminars Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Fri, 29 Oct 1993 02:51:07 GMT Lines: 35 Status: OR Geometry Center Seminars (revised) TIME: Mondays, 2:30 pm PLACE: The Geometry Center, FMC Building (1300 2nd St. South), 5th floor Nov.1 Center Postdoc Leonidas Palios ``Tetrahedralizing the space between a convex polyhedron and a convex polygon'' Geometry Center Classroom (1300 2nd St. South, Room 531) Nov. 8 Professor Alfred Gray, from U. of Maryland ``Surfaces'' Geometry Center Classroom (1300 2nd St. South, Room 531) Nov. 15 ### SPECIAL TIME, THIS WEEK ONLY: 1:00 pm ### Professor Marjorie Senechal, from Smith College ``Quasicrystals and Geometry'' Geometry Center Classroom (1300 2nd St. South, Room 531) Nov. 22 Professor Dave Witte, from Williams College ``Tessellations of Solvmanifolds'' Geometry Center Conference Rm (1300 2nd St. South, Room 538) For more info, contact Paul Burchard Email: burchard@geom.umn.edu Phone: 612-62618319 From sander@geom.umn.edu Fri Oct 29 14:23:00 CDT 1993 Article: 111 of geometry.college Xref: news1.cis.umn.edu geometry.college:111 geometry.pre-college:223 Newsgroups: geometry.college,geometry.pre-college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Myths of Mathematics Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Fri, 29 Oct 1993 17:05:36 GMT Lines: 1004 Status: OR "Myths of Mathematics," by Professor Morris Hirsch Last week, I attended a birthday party for Morris Hirsch, professor of mathematics at UC Berkeley. I shall soon write more about the conference; however, I wanted to make the following available to everyone. It is a LaTex file of notes from a lecture that Hirsch recently gave about myths in mathematics. He gave the lecture at Howard University; it was addressed to a general undergraduate audience, not restricted to math or science majors. Hirsch has kindly allowed me to distribute this file of his lecture notes. He plans to eventually publish a version of the notes. In the meantime, he would be grateful for feedback. If you have comments, email or write them to: Professor Morris W. Hirsch Department of Mathematics University of California Berkeley, CA 94720 USA e-mail: hirsch@math.berkeley.edu If you have LaTex, you can use it on the following. Otherwise, it is possible to read the text and ignore the control sequences. I had to comment out the line "\input{howpreamble}" in order to sucessfully run LaTex. Originally, the two % symbols were not on this line, and you may wish to remove them if LaTex still works. Hirsch also mentions: "You may have to change the style to [12pt] if you don't have 'remark' or 'oldlfont' styles. I don't think I use them substantially, if at all. And I don't use most of the macros in the preamble." Here is the file: %%%% LaTeX preamble file "preamblehow.tex", for input into the main %%%% file "Myths of Mathematics" \documentstyle[12pt,remark,oldlfont]{article} \def\bq{\begin{quote}} \def\eq{\end{quote}} %% \input{howpreamble} \evensidemargin .5in \oddsidemargin .5in \topmargin 0in \textwidth 6in \textheight 8in \setcounter{section}{-1} \setcounter{secnumdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newremark{example}[theorem]{Example} \newremark{remark}[theorem]{Remark} \newremark{conjecture}[theorem]{Conjecture} %%%TEX MACROS %\def\mylabel#1{\label{#1}} %\def\label#1{\marginpar{#1}\label{#1}} %%fails for equations. \def\mylabel#1{\marginpar{#1}\label{#1}} \reversemarginpar %%puts label in LEFT margin \def\bc{\begin{center}} \def\ec{\end{center}} \def\bthm{\begin{theorem}} \def\ethm{\end{theorem}} \def\bcor{\begin{corollary}} \def\ecor{\end{corollary}} \def\bprop{\begin{proposition}} \def\eprop{\end{proposition}} \def\blem{\begin{lemma}} \def\elem{\end{lemma}} \def\bex{\begin{example}} \def\eex{\end{example}} \def\brem{\begin{remark}} \def\erem{\end{remark}} \def\prf{{\bf Proof }} \def\bdes{\begin{description}} \def\edes{\end{description}} \def\ita{\item[(a)]} \def\itb{\item[(b)]} \def\itc{\item[(c)]} \def\itd{\item[(d)]} \def\ite{\item[(e)]} \def\itf{\item[(f)]} \def\iti{\item[(i)]} \def\itii{\item[(ii)]} \def\itiii{\item[(iii)]} \def\itiv{\item[(iv)]} \def\itv{\item[(v)]} \def\itvi{\item[(vi)]} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\ben{\begin{enumerate}} \def\een{\end{enumerate}} \def\beqar{\begin{eqnarray}} \def\eeqar{\end{eqnarray}} \def\beqarr{\begin{eqnarray*}} \def\eeqarr{\end{eqnarray*}} \def\mm#1{$#1$} %variable math \def\RR{{\bf R}} % bold R \def\R#1{{\bf R}^{#1}} %bold superscripted R-variable \def\Rp{{\bf R}_+} %bold R-plus \def\QED{\hspace{.1in}{\bf QED}\\[2ex]} \def\qed{\hspace{.1in}{\bf QED}} \def\prf{{\bf Proof }\hspace{.1in}} \def\tt#1{{\it #1}} %variable italic \def\m#1{$#1$} %variable math mode \def\ZZ{{\bf Z}} %bold Z \def\NN{{\bf N}} %bold N \def\rar{\rightarrow} \def\vecx{(x_1,\ldots,x_n)} %x1,...,xn \def\vec#1{(#1_1,\ldots,#1_n)} %variable n-vector \def\sb#1{_{#1}} %variable subscript \def\sp#1{^{#1}} %variable superscript \def\phit{\Phi_t} \def\eps{\epsilon} \def\lam{\lambda} \def\del{\delta} \def\sig{\sigma} \def\part{\partial} \def\prt#1#2{\frac{\partial #1}{\partial #2}} \def\d#1dt{\frac{d#1}{dt}} %%variable ODE left side \def\hrar{\hookrightarrow} %%% Above to be used only in math mode!! %%% \def\tt#1{{\it #1}} %variable italic %%%%%%%%%%%%%%%%%%% Main LaTeX file %%%%%%%%%%%%%%% \begin{document} \title{MYTHS OF MATHEMATICS\\A lecture delivered at Howard University October 13, 1993} \author{Morris W. Hirsch\\Department of Mathematics\\University of California at Berkeley} \maketitle \section*{Introduction} A myth is a tale associated with a system of belief which adds meaning to the articles of belief and provides guidance to the believers. A myth need not be objectively true to be effective. Nor is a myth necessarily false. I mean myth in a very general sense. ``Belief system'' and ``Article of faith'' come close to my meaning. A myth often takes the form of a story, rather than a set of principles. , Stories are rarely examined as to to their correctness--- they are just stories. Yet some stories have been heard so often that their truth is taken for granted--- hence their power. Such stories are myths. It is always easier to examine--- and debunk--- other people's myths than our own. A good example of a myth that is useful, even though it is generally held to be false by most authorities, is the {\it Myth of the Tooth Fairy}. It has the useful effect of reassuring a child that there is nothing wrong with a tooth falling out--- in fact it is a Good Thing because in place of the tooth the Tooth Fairy leaves money. This myth is part of a constellation of tales, part of the very important {\it Myth of Childhood}, that share a similar function: They reassure the child that there are good creatures looking out for the child's safety, that the world is benevolent, that the future is good provided certain rules are followed. Santa Claus and the Easter Bunny are other myths in this group, but we could also include many religious and patriotic lessons. It is probably psychologically important for a child to believe these things when they are very young. But surely Science and Mathematics are free of myths? The easy answer is to reply: That is merely part of the Myth of Science! While this is true in a sense, what is more useful is the realization that it is difficult if not impossible for any institution, organization or scientific discipline to function without myths. A myth is not necessarily a bad thing; in fact it seems impossible to live without myths. But sometimes the basic myth in some area can acquire such a grip on our minds as to interfere with further progress. Sometimes the discrepancy between myth and reality becomes so obvious that that it forces a crisis and paradigm shift. By making myths explicit--- which is not necessarily to debunk them--- we can explore the hidden assumptions governing our activities; this may lead to greater insight into what we are doing. In this spirit I propose to examine the myths of mathematics. \section*{Early Mathematical Myths} Mathematics is an enormous and complex endeavor. We know of no society in which mathematics is totally absent; every culture has developed a body of mathematical knowledge suitable to its needs. Mathematics is much older than all currently practiced religions, older than all other sciences, older than all other academic disciplines except perhaps philosophy. Probably symbols of reckoning preceeded written language. According to the mathematician and historian Dirk Struik, there is a piece of bone at least 30,000 years old on which there are 55 marks in groups of 5. While this may be mere doodling, or decoration, perhaps someone needed to tally 55 things. It seems reasonable to believe that mathematics is coextensive and coeval with homo sapiens. But perhaps this is just another Myth! In any case, mathematics has certainly changed in many ways over the centuries, and its myths have changed accordingly. Waht a pity we don not know more about Pythagoras; he was surely a great mathematician, and surely one of the most mystical. We are told that he (or his followers) believed that the world is based on number. We don't know exactly what this means, but a reasonable interpretation is \bq {\bf The Myth of Measurability:} {\it The world can be described and explained in mathematical terms,} \eq one of the most powerful myths in science. Closely related is \bq {\bf The Myth of Necessity:} {\it We must use Mathematics to describe the world.} \eq These myths are given great power by the most important mathematical myth of all, \bq {\bf The Myth of Certainty:} {\it Mathematical knowledge is the most certain form of knowledge.} \eq One of Pythagoras' great discoveries was the relationship between the length of a stretched cord, and the musical pitch the cord emits when plucked: If the length is halved, the pitch goes up an octave; while if the length is reduced to a third, the pitch rises by an ocatve and a fifth.\footnote{%% In the pure untempered scale--- don't tune your guitar this way.} %% This must have been an astonishing revelation, hard to appreciate in these days of omnipresent music. For it reveals a tight connection between the concrete physical phenomena of strings and sounds, and the rather high-level mathematical abstraction of {\it ratios}. The idea of ratio is not a simple thing to explain. It is not a thing, but a relation between quantities--- which are also abstractions. The ancient Greeks considered only ratios between pairs of similar quantities--- two lengths, two areas, two numbers.\footnote{ %% The Greeks would never mix or compare kinds of quantities. The Babylonians, on the other hand, saw no problem in adding the measure of a length to that of an area.} %% They did not think in terms of abstract real numbers, but in terms of ratios. But although dissimilar quantities did not have a ratio, two ratios could be compared, even though one was a ratio of, say, two lengths and the other was a ratio of two areas. Thus they knew that the ratio between the areas of circles equaled the ratio of the squares of their diameters. Ratios were the closest they came to our modern concept of real numbers. The Greeks faced the difficult problem of explaining what a ratio {\it is} They never did this, but they felt forced to say something about the nature of ratio. Long before Euclid they explained how to calculate the ratio of two line segments. The ratio of two line segments, according to this explanation, is the same as the ratio of two whole numbers, obtained as follows: Find some small line segment out of which each of the given segments can be built by putting copies of the tiny segment end to end and exact number of times. For example, if one of the line segments takes 339 small segments, while the other takes 11, then the ratio is 339 to 11. Similarly for the ratio of two areas, two volumes or two angles. This treatment of ratio is based on the assumption that for any two line segments, there is some small line segment out of which each can be constructed in the manner described. The lengths of such a pair of segments are called {\it commensurable}. Thus these early Greek mathematicians believed that all lengths are commensurable. \section*{A Dead Myth} Implicit in the early Greek theory of ratio is \bq {\bf The Myth of Commensurability:} {\it Any two comparable quantities are in the same ratio as two whole numbers.} \eq This is a very plausible principle. For the Greeks, who had to contend with many diferent kinds of quantities, it certainly clarified and simplified the concept of ratio. Why do I call this a Myth and not a Principle? Because it is false! It is just not the case that any two lengths are in the ratio of two whole numbers--- the Pythagoreans discovered that the Myth of Commensurability conflicts with another, better established mathematical principle, namely the Pythagorean Theorem. It is a peculiarity of mathematics that it cannot tolerate conflict. Physics can live, not happily but quite successfully, with conflicts between Newtonian mechanics, relativity theory and quantum mechanics. But Mathematics insists on consistency between all its parts. If two mathematical statements contradict each other, one has to go. This is closely related to some other myths we will examine. Here is the conflict that killed the Myth of Commensurability. Consider two line segments: One, having length $S$, is the side of a square. The other, of length $H$, is a diagonal of the same square. The Pythagoreans discovered that the side and the diagonal of a square are not commensurable. Two adjacent sides and the diagonal form a right triangle whose two perpendicular sides are $S$, and whose hypotenuse is $L$. The Pythagorean Theorem--- which was discovered in many cultures, and may well predate Pythagoras--- tells us that $H^2=S^2 + S^2$, or more simply, $H^2=2S^2$. If we accept the Myth of Commensurability, then the ratio of $H$ to $S$ equals the ratio of two whole numbers, call them $A$ and $B$. Thus there are two whole numbers such that $H/S=A/B$. So $H^2/S^2=A^2/B^2$. But $H^2/S^2=2$, so $A^2/B^2=2$, or equivalently, $A^2=2B^2$. (We--- but not the ancient Greeks--- would say $ \frac{A}{B}=\sqrt{2}$.) Summarizing the argument so far: If we combine the Myth of Commensurability with the Pythagorean Theorem, we conclude: {\it There are two whole numbers $A,B$ such that $A^2=2B^2$.} Now if you try to find such numbers $A$ and $B$ you will not succeed. You can find numbers that {\it almost} work, such as $7^2=5\times 2^2 -1$, or $17^2=2\times 12^2 +1$. But you will never find whole numbers $A$ and $B$ such that $A^2=2B^2$, no matter how long you look.\footnote{ %% To find arbitrarily many number pairs $A,B$ such that $(A/B)^2$ is close to $2$, start from any pair $A,B$ and define a new pair $P,Q$ by $P=A^2+2B^2,\,Q=2AB$. Then do the same starting with the new pair, etc. If we denote the rational number $A/B$ by $x_0$ this leads to an infinite sequence $x_0,\,x_1,\ldots$ such that ${x_n}^2$ converges to $2$, where each term is obtained from the preceding one by the formula $$x_{n+1}=\frac{1}{2}(x_n+\frac{2}{x_n}).$$ Try it!} %% How can we know this? It is not just experience--- even if we checked a thousand or a trillion numbers, that would not logically prevent someone from discovering two other numbers that worked. The reason that we can be absolutely certain is that there is a simple logical {\it proof} showing that such numbers cannot exist. The proof is by contradiction: We assume there are two numbers such that $A^2=2B^2$, and then we derive a logical contradiction. To start the proof we first observe that if there is a pair $A,B$ of whole numbers for which $A^2=2B^2$, then there is one such pair with $A$ as small as possible. And notice that $B$ is even smaller than $A$. Next we notice that $A^2$ is an even number, because it equals $2B^2$. This means that $A$ must be even, because if a number is odd then its square is odd.\footnote{ %% An odd number $X$ has the form $X =2Z+1$ for some whole number $Z$. Then $X^2=4Z^2+4Z+1$, which is also odd.} %% Therefore there is some whole number $M$ such that $2M=A$. From this follows $4M^2=A^2= 2B^2$. So $B^2=2M^2$. But now the pair $B, M$ has the same property as the pair $A, B$--- namely the square of the first equals twice the square of the second--- but $B$ is clearly less than $A$. This contradicts the assumption that $A$ was as small as possible for such a pair. Since Mathematics cannot tolerate contradictions, one of our original assumptions was wrong. Since the Pythagorean theorem is well-established, this kills the Myth of Commensurability. It is said that the Pythagoreans were required to keep this secret, on pain of death. It is a terrible thing to destroy a myth. The Myth of Commensurability was replaced by Eudoxus' new and extremely subtle definition of ``same ratio''--- very close to the modern definition of ``real number''--- which unfortunately we have no time to explore here. \section* {The Myth of Existence} Many, perhaps most mathematicians hold a Platonic view of the objects mathematical thought--- numbers, functions, sets, spaces, and so on. The strongest form of this belief is: \begin{quote} {\bf The Myth of Existence:} {\it Mathematical objects have an existence independent of humans, and independent of physical reality.} \end{quote} To many mathematicians, for example, it is obvious not only that the number three exists, but it has always existed, and will always exist, even if all life disappears from earth, even if the universe collapses to a point. \footnote{ %% This of course applies only to that small minority of mathematicians who see philosophical questions as anything but a regrettable waste of time.} It is just ``out there'' somewhere, somehow. Other mathematicians, when questioned, angrily refuse to discuss such things, asserting--- perhaps wisely--- that they are meaningless. A corollary of the Myth of Existence is that mathematicians are engaged in {\it discovering truths}. On this view, a mathematical statement is not just an empty logical tautology; rather it is an assertion about reality. This in turn has interesting corollaries we will discuss next, such as the Myths of Truth and Consistency. Despite its simplicity and popularity, there are severe philosophical difficulties with the idealist or Platonist outlook expressed my the Myth of Existence. Those who subscribe to it have no explanation for the what Wigner called the ``unreasonable effectiveness'' of the applications of mathematics to the physical world. \section*{The Myth of Truth} Many mathematicians hold to a mystical notion of mathematical {\it truth} as something apart from or more basic than {\it proof}: \bq {\bf The Myth of Truth:} {\it Every mathematical statement is either true or false.} \eq This myth is a corollary of the Myth of Existence, since if mathematical objects exist, then mathematical propositions, being statements about reality, hence are either true or false. The Myth of Truth gives a simple justification of the technique of indirect proof, which we used above: To prove that some mathematical statement $S$ is true, it is sufficient to show that the assumption that $S$ is false leads to a logical contradiction. Since contradictions are not tolerated, we can then eliminate the possibility that $S$ is false. The Myth of Truth then assures us that $S$ must be true. Let us test this myth on a few mathematical statements: \bdes \item[Statement A] $\sqrt{2}$ is not a rational number. \item[Statement B] The decimal expansion of $\sqrt{2}$ is nonrepeating. \item[Statement C] The decimal expansion of $\sqrt{2}$ contains $1000$ consecutive $7$s. \edes Statement $A$ is true because we proved it--- it is the modern formulation of what the Pythagoreans proved about the diagonal of a square not being commensurable with the side. Statement $B$ is is likewise true--- because (it can be proved that) every repeating decimal is a rational number, so $\sqrt{2}$ cannot be repeating. What about Statement $C$? Like Statement $B$, it involves the the entire decimal expansion of $\sqrt{2}$. But unfortunately there is, at present, no mathematical theory that can prove Statement $C$, or that can disprove it. At present Statement B is neither proved nor disproved. The Myth of Truth, however, is not about proof--- it is about truth. It asserts that either Statement $C$ is true, or its opposite is true. The opposite of Statement $C$ is: \bdes \item[Statement $C'$] The decimal expansion of $\sqrt{2}$ does not contain $1000$ consecutive $7$s. \edes Now at first glance it may seem obvious that either $C$ or $C'$ must be true, even if we do not yet know which. But what could this mean if we have no proof of either $C$ or $C'$? Some possible answers are: \bq Someday someone will find a proof of $C$, or of $C'$. \eq This is not a very helpful definition of what it means for a statement to be true. What if the world blows up before any such proofs are found? Another frequently given explanation for the truth value of B is: \bq Imagine a computer running forever, grinding out the whole decimal expansion of $\sqrt{2}$. This infinitely long print-out either does, or does not, contain $1000$ consecutive $7$s. \eq I have trouble with the concept of a computer running ``forever'', followed (when?) by inspection of the printout (by whom?) of an infinite print-out. And who is to guarantee the correctness of the computer program? An older world view explained the truth value of $C$ by referring to the mind of God. While I find this explanation preferable to postulating magical computers, it has the disadvantage of requiring prior agreement on theology, which is philosophically more controversial than mathematics. If we dispense with the notion of mathematical Truth, are we forced to give up Proof by Contradiction? Recall that to prove some mathematical statement $P$ in this way, we argue that $P$ is either true or false, and then show that the assumption that $P$ is false leads to a contradiction, which leaves only the possibility that $P$ is true. But if ``true'' and ``false'' do not apply to P, how can we argue in this way? The answer is, we just do it! That is, {\it we agree to admit proof by contradiction as a valid technique for proving theorems}, without taking a stand on whether every mathematical statement, even in the absence of proof, must be either ``true'' or ``false''. This in fact is the way almost all mathematicians behave when they do mathematics, whatever they may say about mathematical truth. Lurking behind this agreement is another myth: \bq {\bf The Myth of Consistency:} {\it No proofs of both a statement and its opposite will ever be found.} \eq Put succinctly: Mathematics is consistent. This is a reformulation of the principle alluded to earlier, that Mathematics cannot tolerate contradictions. But the Myth of Consistency goes further, in that it makes a prediction about human behavior. In fact, as we saw with the Myth of Commensurability, it sometimes happen that two contradictory statements {\it are} obtained in mathematical reasonsing. But we also saw that when this occurs, the very foundations of mathematics are revised to eliminate one of them. Thus the Myth of Consistency really asserts that we will always be able to satisfactorily revise the foundations of mathematics in order to eliminate any given contradiction that has been found. As a prediction of the future, this Myth is of course uncertain. But {\it when we do mathematics, we behave as if we believe it}. For there is no point in proving a theorem, or adding up a grocery bill, if we really think the result is likely to be contradicted.\footnote{ %% There are other activities in which we behave as if we hold certain beliefs, without necessarily subscribing to them. For example Daniel Dennett pointed out in his book ``The Intentional Stance'' that when we play chess with a computer, we ascribe intelligence and desires to the machine--- ``It wants to trap my queen''--- even though we know the computer is merely responding to a program. In other words we behave {\it as if} we believe the computer thinks and and desires.} \section*{The Myth of Proof} I argued above that the working mathematician does not need the concept of mathematical truth, only of mathematical proof. Most mathematicians subscribe to: \bq {\bf The Myth of Proof:} {\it There is a clear concept of mathematical proof, independent of time and place, understood and accepted by all mathematicians.} \eq One would expect that so fundamental a tool as proof would be the subject of basic courses given wherever mathematicians are trained. Someplace there must by a set of rules by which the correctness of a proof can be ascertained. But there are no such courses, and no such rules! The working definition of ``correct proof'' is ``an argument that convinces most mathematicans''.\footnote{ %% More precisely, most mathematicians who work in the particular subject under discussion.} %% The proofs that fill mathematics journals contain language like ``It is easy to see that\ldots It is well known that\ldots A similar argument shows\ldots'' For the most part these are useful ways to shorten what would otherwise be intolerably tedious. If pressed, the authors could {\it almost} always replace them with a correct formal argument. There are several kinds of exceptions: \bdes \iti Mistakes are made! Occasionally a ``proof'' is published for a theorem that subsequently is shown to be false. \itii Very often the author is not really familiar with the proofs of all the previously proved theorems quoted in the proof, and therefore could not fill in all the gaps (although someone else could). \itiii A proof might involve a computer calculation. The philosophy of such proofs has yet to be worked out. Is the author responsible for demonstrating the correctness of the computer program, including handling of round-off errors? The reliability of the hardware? How does the author know that the operating system of the computer facility has not introduced errors into the calculation? \itiv The proof may involve thousands of pages of articles by scores of authors. Who can guarantee the correctness of such a proof? Who could fill in all the gaps in all these papers? \itv The proof may involve new principles, perhaps unconsciously invoked, that have not yet been accepted by the mathematical community. \edes All the above have happened. Even worse violations of the Myth of Proof have occurred. The great Gauss, arguably the greatest mathematician since Newton, published a widely admired proof of the Fundamental Theorem of Algebra,\footnote{ %% This theorem states that every algebraic equation has at least one solution in the complex number field.} %% based his proof on a lemma which he did not prove, but only promised to prove in a later paper. He did not keep this promise, although correct proofs were later found. Even Euclid made errors in his proofs. His very first proposition, that every line segment has a perpendicular bisector, was not correctly proved: Euclid's proof assumed that two circles intersected, but we now no that his axioms were insufficient to prove this. Yet for 2000 years Euclid's work the very model of rigorous proof. \section*{The Myth of Formalism} In the background of notion of ``proof'' is an important modern myth, a submyth of the Myth of Proof, namely the \bq {\bf Myth of Formalism}: {\it Every mathematical proof can be expressed in a formal mathematical system.} \eq Such a system is a list of symbols and axioms, and formal rules for deriving statements from the axioms or previously derived statements. These derived statements are the Theorems, and the derivations are the proofs. The symbols and axioms have interpretations in the relevant mathematical area (number theory, calculus, etc.), but appeal to the interpretation is prohibited in the derivations. It is this last property which makes the system ``formal'': Only the {\it form} of mathematical statements counts in assessing their correctness, not their meaning. In principle a computer could check whether a sequence of statements expressed in the formal system constitutes a proof. There is no disagreement about the Myth of Formalism. Indeed, any alleged proof that is not expressible formally would be extremely suspect. A formal system is a way of making absolutely clear and explicit exactly what rules we utilize in our proofs. Yet the very notion of a formal mathematical system is a creation of this century. Why then do we not simply agree on a formal system in which all our proofs could, if necessary, be expressed? The reason is a celebrated theorem by Kurt G\"odel about formal systems. It says that {\it no consistent formal system can be adequate for all the theorems that mathematicians can prove}. More precisely: {\it Given any consistent formal system $S$ that is adequate for arithmetic, there is a theorem that can be expressed in $S$, and which can be proved, but which has no proof expressible in $S$.} As an example, it has been shown that in one of the standard formal systems (Zermelo-Frankel set theory with the Axiom of Choice, referred to as ZFC), there is a certain formula for a real-valued function $f(x)$, with the following properties. The formula is built up out of the usual operations of $+$ and $\times$, the whole numbers, the number $\pi$, the function ``absolute value'', and the function $sin x$. It can be proved that $f(x)=0$ for all real numbers $x$; but in the formal system there can be no proof of this based on the formula. G\"odel's theorem means that the notion of ``correct proof'' cannot be formalized, at least not in any formal system of the kind we currently understand. The best we can say is that for any correct proof there is some reasonable formal system in which it can be expressed; but no single formal system will work for all proofs. It should be pointed out, however, that most ``natural'' mathematics seems to be doable in ZFC. The Myth of Consistency comes in here, because an inconsistent formal system would seem to be useless as a setting for mathematics: If two contradictory statements both have proofs in the system, than every statement expressible in the system has a proof in the system. For example, the statement $1=0$ has a proof in any inconsistent system. The question naturally arises of whether for any chunk of mathematics there is a {\it consistent} formal system in which it can be expressed? For example, is ZFC consistent? Nobody knows. But G\"odel proved that the consistency of ZFC cannot be proved within ZFC! The Myth of Formalism interacts with the Myths of Truth and Existence in an interesting way. According to the latter myths, every mathematical proposition is a statement about about reality, and hence is either true or false; in particular, the proposition is meaningful. But the Myth of Formalism inplies that while the meaning of the proposition may certainly be important to the process of {\it discovering} a proof, the meaning is irrelevant in the final mathematical verification of the truth of a proposition, i.~e. in its written proof. \section*{The Myths of Timelessness and Universality} The content of most sciences and scholarly disciplines change over time. The truths of Physics in Aristotle's day differed from those of the middle ages, which in turn conflict with those of the nineteenth century. Einstein's Physics conflicts with Newton's. But according to a widespread belief, this is not so for Mathematics: There is widespread belief in \bq {\bf The Myths of Timelessness and Universality}: {\it The theorems of mathematics are eternally and universally valid.} \eq Thus Euclid's theorems are still accepted today as theorems. No one can conceive of theorems proved in Africa to contradict theorems proved in Asia. How then does Mathematics handle the occasional errors and inconsistencies that occur? For example it was realized early in this century that the proof of the first proposition in Euclid is incomplete, and in fact considerable work has to be done to complete it. The solution was to fix it up by adding a considerable number of new axioms. It is not unusual to look at a piece of mathematics done a century or two ago and discover that the proof is entirely inadequate by today's standards. Yet we do automatically not reject it--- we can almost always fix up the proof (sometimes this may take many years). If one accepts the Myth of Existence, or its corollary, the Myth of Truth, then the universality and timelessness of mathematics is an immediate consequence.\footnote{ %% Provided of course that Truth is believed to be timeless and universal.} %% It is hard to doubt the Myth of Universality: Is it conveivable that the mathematics developed by the inhabitants of Jupiter contradicts ours? Could $2+2=5$ on Jupiter? The Myth of Timelessness is more problematical since as a matter of historical fact, the content of mathematics, our interpretations of it, and the idea of what constitutes a correct proof, have changed over the centuries. Consider for example the Pythagorean Theorem: Until the middle of the last century this, like all geometry, was considered a fact about reality. Thus geometry was a branch of physics. The discovery of non-Euclidean geometry changed that perception: Other geometries were discovered (or invented), based on different axiom systems. The axioms and theorems of these new geometries conflicted with those of Euclid. In one geometry, for example, there are no parallel lines. It was shown that these new geometries were just as valid mathematically as Euclidean geometry--- their theorems followed logically from their axioms. At that point the role of geometry changed: Instead of one geometry which described reality, there were many possible geometries, and the question (asked by Riemann) was: Which one of them described reality? Later Poincar\'e said that even this was the wrong question: {\it Any} geometry is compatible with Physics--- the search should be for the geometry which give the simplest mathematical expression of physical laws. The discovery of non-Euclidean geometry, and the geometries of more than 3 dimensions, raised serious problems with the Myths of Truth and Existence. If Euclidean Geometry is not about physical space, then what is it about? What is 4-dimensional geometry about? Apparently the existence of something like Plato's ideal forms is required to save these Myths. A similar problem arose with the development of algebra in the nineteenth century. Originally algebra was viewed as merely an abstract form or arithmetic, which in turn was a formalization of counting. Thus algebra was directly tied to reality. The letters used in algebraic formulas were understood to stand for numbers, so whatever algebraic identities held for numbers must also hold for letters. For example, $xy$ must equal $yx$, since that holds for numbers. Thus, numbers being ``real'', algebra described reality. But eventually there was a need for other algebraic systems. Hamilton invented the system called Quaternions, and applied it to physics. In this system the commutative law is not valid: $xy$ need not be equal to $yx$. Hamilton said there was no need for algebraic symbols to stand for numbers, or for anything--- they could be just ``objects of thought''--- and thus they could obey entirely new algebraic laws. At this point the Myths of Truth and Existence are again challenged. If algebra is not about numbers, what is it about? What part of reality does it describe? In spite of the undeniable fact that the content and concepts of Mathematics changes over the years, there is still {\it something} about its theorems that seems to be permanent. While we can object to the correctness of Euclid's proofs, we still accept his theorems as correct. It is extremely rare for a long-accepted theorem to be found to be false. Accepted mathematical theories may become uninteresting with time, but they are rarely if ever rejected as being wrong. This seeming permanence of Mathematical results needs an explanation which goes beyond the Myths of Existence and Truth. My own inclination is to look for the explanation based on the principle that Mathematics is a creation of human brains, and these brains have been formed through evolution of the species and development of individuals, as a response to interaction with the physical world. But an explanation of this kind will need to rely on a reliable theory of human intelligence and consciousness, which unfortunately does not yet exist. \section*{Mathematics as Modeling} The nature of mathematics, mathematical truth, and mathematical proof are indeed knotty philosophical problems, and it is probably fortunate for both mathematics and philosophy that most mathematicians prefer to spend their time on mathematics. It is useful, however, occasionally to step back and consider just what it is we are doing. And I think it is possible to reach a more reasonable position than the unworldy view. I start from the obvious fact that Mathematics is a recognizable activity of a recognizable group of human beings, these people being called mathematicians. (It doesn't affect the discussion much if you want to include animals or computers.) I take Mathematics to be what mathematicians do. But this is not necessarily the same as what they say they do! What, then, do mathematicians do when they are "doing mathematics"? They think, talk, write, type, draw pictures, or use calculators and computers. Of course only sometimes is the result of this activity called mathematics. Whether a particular line of {\it thinking} is mathematics can only be ascertained by asking the thinker. The test of the {\it other} activities, on the other hand, is to ask someone familiar with mathematics. The foregoing, while being a practical way of determining whether a given activity is mathematics, does not help in characterizing mathematics abstractly. Such a characterizaion does not seem to be necessary, however; and I doubt that it is is possible. In any case, it would entail a large chunk of ontology and epistemology, or perhaps psychology, sociology and biology, and is better left to professional philosophers. It is useful to divide mathematical activities into two types, private and public. {\it Private mathematics} is that which is never communicated to anyone else; all the rest I call {\it public mathematics}. A moment's thought shows that private math is very different from public math: by definition no one else can check its correctness, find it interesting, learn it, use it, or examine it in any way. Thus its role is quite distinct from that of public math, so different that henceforth we exlude it from consideration. Mathematics, then is public--- something communicated, or at least communicable (like a disease). It does not seem possible to communicate something without making use of the physical world (even mindreading presumably makes use of brains). But this means that {\it the belief that "mathematics is independent of the physical world" can not be valid.} Physical reality is involved at least to the extent of marking paper, vibrating the air or aether, or typing at a computer. There is further close connection between physical reality and mathematics: The concepts we call "mathematical" exist in our minds only because we have abstracted them from perceptions of the physical world. We can understand natural numbers only because we have learned, through years of experiencing the real world, to make the abstraction of "individual object"; and similarly for other mathematical ideas such as line, infinity, continuity, etc. Newborn infants do not do mathematics. Thus both the activity and the content of mathematics depend upon physical reality. It is not surprising, in view of these connections, that mathematics is connected to reality in another way: {\it Mathematical statements often have predictive value.} Only if ignore the origins of mathematical abstractions can we be surprised that the theorem "2 + 2 = 4" accurately predicts the outcome of combining two pairs of apples. Or perhaps we should say that the successful application of mathematics is no more mysterious (but no less) than the connection between our thoughts and the outside world. Rather than taking Mathematics as a description of reality, I find it more useful to look at a mathematical theory as a simplified {\it model of some process or activity}. By this I mean a conceptual system which captures certain the features of that activity. We can then investigate the properties of this system, and hope that our discoveries apply to the original activity. In this view Arithmetic is a model of what we do when we count, or calculate. We want to abstract certain general features from the physical process of counting. One of the things we notice about counting is that if we count the same set of objects twice, we get the same answer.\footnote{ %% This invariance is something we learn through experience; very young children are not surprised if they get different answers.} %% This leads us to postulate familiar rules such as the commutative, associative and distributive laws, and to derive other rules, called theorems. Do these theorems accurately describe the activity of counting and calculting? To a certain extent they do. But it is questionable, for example, that counting the grains of sand in the Sahara, or even in a small sandpile, will always gives the same result. After all, arithmetic is only a {\it model} of a piece of reality. In the same way, Geometry can be viewed as a model of certain activities of measuring, drawing and seeing. There is no reason to think that there is only one such model--- therefore we should not be surprised that different geometries are possible. They capture different intuitions about measuring, drawing and seeing. For example, straight railroad tracks appear to meet at the horizon. But they also appear to meet in the opposite direction. Thus there should be a geometry in which ``straight lines'' meet at {\it two} points.\footnote{ %% There is such a geometry: on the surface of a ball, ``straight lines'' are great circles, any two of which meet at two antipodal points.} %% Basic to Geometry is the notion of a {\it rigid motion}, such as a rotation or translation. When we do Geometry, sometimes we are imagining combininations of rigid motions, and of other kinds of transformations, such as a uniform stretching or shrinking. There is a part of mathematics that {\it models} the combining of such transformations, namely Group Theory. (It is also used to model other mathematical things, such as symmetries, and algebraic substitutions.) In this way Group Theory is a model of certain concepts that arose in geometry. Today it is common to base all mathematics on Set Theory. Now sets are ubiquitous in mathematics:\footnote{ %% This is a strictly modern phenomenon.} %% the set of solutions to an equation, the set of points inside a sphere, and so on. But in doing mathematics we never deal with {\it abstract} sets. We only deal with sets of mathematical objects--- points, lines, functions, spaces, etc. But Set Theory deals precisely with abstract sets. In this sense Set Theory is a model of what mathematicians do when they deal with sets: Form unions, intersections, Cartesian products, and so on. But a model should not be confused with the reality it is modeling! Thus Set Theory does not coincide with mathematics, nor does mathematical logic. The paradoxes that have arisen in Set Theory are not contradictions in mathematics--- they are the result of inadequate modeling.\footnote{ %% A famous such paradox comes from considering the set $S$ comprising every set which is not a member of itself. By the law of the excluded middle, either $\S\in S$ or $S\not \in S$; but each leads to a contradictions.} %% In fact there is no reason to believe that all of mathematics can be modeled on Set Theory, useful as that subject is for many things. In a similar way, mathematical (or symbolic) logic is a model of what mathematicians do when they reason when they reason deductively. A branch of mathematical logic deals with the structure of mathematical proofs. Using it we reason precisely--- mathematically--- about certain aspects of what we do when we prove theorems such as the kinds of axiom systems do we use, the kinds of logical derivations are permissible, the consistency (or lack of it) between our theorems. Although there have been many attempts to base all of mathematics on logic, there is no reason to think this is possible. If we view mathematics as a human activity which models parts of reality, then the Myths of Existence, Truth and Proof are no longer relevant. It is true that we lose the beautiful certainty that stems from those myths--- but that certainty was itself never more than a myth, and one we are better off without. In place of that specious certainty we acquire a more realistic understanding the nature of mathematics. \end{document} From sander@geom.umn.edu Fri Oct 29 14:48:07 CDT 1993 Article: 121 of geometry.announcements Newsgroups: geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Guibas Talk on Computational Geometry Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Fri, 29 Oct 1993 19:30:33 GMT Lines: 32 Status: OR Dr. Leonidas Guibas, one of the leading researchers in Computational Geometry, is visiting the University of Minnesota and the Geometry Center on November 15 and 16. He is going to give a talk on campus during his visit: Date: Monday November 15 Time: 2:30 pm Place: 108 Mechanical Engineering Building Pre-colloquium reception is at 2:00pm in the 5th floor lounge of the EE/CS Dept. Abstract: RANDOMIZED ALGORITHMS IN COMPUTATIONAL GEOMETRY Leonidas J. Guibas Stanford University The major intellectual contribution of the area of algorithms to Computer Science as a whole over the past ten years has been the introduction of the use of randomization as a fundamental tool of algorithm design. Randomization not only yields some of the asymptotically most efficient algorithms -- it does so through algorithms that are amazingly simple and therefore eminently implementable and practical. In this talk I will explore a number of different paradigms for developing efficient algorithms in computational geometry using randomization. Among others, I will illustrate the techniques of partitioning by random sampling, randomized incremental construcions, and randomized re-weighing. From sander@geom.umn.edu Mon Nov 1 15:04:45 CST 1993 Article: 112 of geometry.college Newsgroups: geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Sharkovskii's Theorem Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Mon, 1 Nov 1993 16:52:28 GMT Lines: 49 Status: OR Sharkovskii's Theorem There is a theorem which is both beautiful and important in the history of dynamical systems. Thus it deserves its own article. It involves a somewhat strange way to order the integers, stated at the end of the article. I write n ">" k to mean n is greater than k in this strange ordering. Here is the theorem: Sharkovskii's Theorem (1964): If f is a continuous map from an interval to the real line, then if f has a point of least period n, and if n ">" k, then f has a point of least period k. This says that just knowing one periodic point can indicate the existence of many other periodic points, without you ever having to find those other points. In particular, a continuous real-valued map of an interval with a point of period three has points of every other period. This is the subject of one of the most famous papers in dynamical systems, Li and Yorke's "Period three implies chaos," 1975. This paper is the first time that the word chaos was used mathematics; it was a well-received paper that provided the name of a new branch of math. Though it turned out that the result in the paper had been proved in stronger form previously (Li and Yorke did not know of Sharkovskii's paper, as it was in an obscure journal in the Soviet Union), the paper stated the results in such an exciting and beautiful way, that it is still quoted to this day. Strange Ordering of the integers: 3 is "largest," followed by 5 ">" 7 ">" 9 ">" all odd numbers, backwards from the standard ordering of the integers. Next largest are, 2*3 ">" 2*5 ">" 2*7 ">" in backward order all integers of the form 2 times an odd integer. Then: 4*3 ">" 4*5 ">" 4*7 ">" . . . . . . 2^n*3 ">" 2^n*5 ">" 2^n*7 ">" . . . 2^(n+1)*3 ">" 2^(n+1)*5 ">" 2^(n+1)*7 ">" . . . . . . Finally, . . . 2^n ">" 2^(n-1) ">" . . . ">"4 ">" 2 ">" 1. From sander@geom.umn.edu Mon Nov 1 19:40:24 CST 1993 Article: 114 of geometry.college Xref: news1.cis.umn.edu geometry.research:98 geometry.college:114 Newsgroups: geometry.research,geometry.college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Re: Image Homotopy Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Tue, 2 Nov 1993 01:31:39 GMT Lines: 23 Status: OR Clarification In my article on Image Homotopy, my definition of regular homotopy is not clear enough. I said: "Two immersions F and G of a surface are regularly homotopic if there is a smooth map H of the surface through time such that at each fixed time, H is an immersion of the surface, H at time zero is equal to F, and H at time one is equal to G. In other words, two immersions are regularly homotopic when one can be smoothly deformed into the other while always maintaining an immersion." To be more precise, let I denote the interval [0,1]; given a surface M and two immersions F and G of the surface, if there is a smooth map H:MxI->R^3, such that H(p,0)=F(p), H(p,1)=G(p), and for each t, the map taking p to H(p,t) is an immersion of M, then F and G are regularly homotopic. It seems that I did not make it sufficiently clear that H must be a smooth map on MxI. It is not sufficient to only insist that fixing one variable, H is smooth as a function of the other. With a map of this sort, one can deform a figure eight into a circle by pinching off one loop. From sander@geom.umn.edu Tue Nov 2 10:38:38 CST 1993 Article: 115 of geometry.college Xref: news1.cis.umn.edu geometry.college:115 geometry.pre-college:227 Newsgroups: geometry.college,geometry.pre-college Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Mathematics Software Demonstration Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: cameron.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Tue, 2 Nov 1993 01:34:35 GMT Lines: 54 Status: OR Mathematics Software Demonstration Last week, I had a request from Loki Jorgenson, for suggestions of available mathematics software that he could show to a general audience the following day. Dr. Jorgenson is Research Manager of the Centre for Experimental & Constructive Mathematics at Simon Fraser University in Burnaby, Canada. Fortunately, the Geometry Center has software which is good for demonstrations, and ftp makes it possible to instantaneously retrieve software located in Minnesota, and use it in a demonstration in British Columbia the next day! At my request, he has kindly provided the following description of the day of demonstrations for the Forum: We are happy to report a successful day of demonstrations at the Centre for Experimental & Constructive Mathematics, thanks in part to the helpful people (like Evelyn Sander) at the Geometry Centre. Tuesday October 19th was a busy day as we hosted a visit from winners of the British Columbia Science Fairs in the morning and then moved our equipment downtown to the reception ceremony for the director of the Centre who is also the new Shrum Chair of Science at Simon Fraser University. At both events we demonstrated the fast growing world of mathematical visualization using platforms made available over the Internet by the Geometry Centre. We transferred, built and installed the software kali, geomview, geometer, evolver and CRSolver, all from the anonymous ftp site geom.umn.edu. We used these platforms in the morning to inspire (we hope!) a future generation of experimental mathematicians who had come to Simon Fraser University as part of their prize for winning in the provincial Science Fair. They were encouraged to take a hands-on part in the demonstrations through the interactive interfaces built in to the platforms. From all accounts, we made an excellent impression. Later that afternoon, we wowed academic dignitaries in attendance at the Shrum Chair reception for our director, Dr. Jonathan Borwein. This is an honour which is bestowed once every five years to a leading Canadian scientist who will blaze a new trail for the SFU science community to follow. Part of our purpose at the reception was to give some indication of the work already going on in graphic mathematics research and what could be expected from the Center for Experimental & Constructive Mathematics which Dr. Borwein had established as part of the Chair. Our intention is to do work of similar calibre to that already done at the Geometry Centre in experimental mathematics. We thank the Geometry Centre for their part in making our presentations a success. Loki Jorgenson, Research Manager loki@cecm.sfu.ca From sander@geom.umn.edu Tue Nov 2 12:55:37 CST 1993 Article: 122 of geometry.announcements Newsgroups: geometry.announcements Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Astrodynamics Conference Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: euler.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 2 Nov 1993 17:25:32 GMT Lines: 20 Status: OR Conference Announcement: Advances in Nonlinear Astrodynamics Ballistic capture and weak stability boundary transfers, gravity assist, halo orbits with applications to operational spacecraft: Hiten, ISEE-C. Date: November 8-10, 1993 Place: The Geometry Center, University of Minnesota, Minneapolis, MN Fees: Excepting the banquet and evening receptions, there will be no registration fee for any students or any University of Minnesota employees. For more information, contact Edward Belbruno, belbruno@geom.umn.edu. For background information on ballistic capture and the Hiten spacecraft, see my article, available by ftp from forum.swarthmore.edu: "Ballistic Lunar Capture," geometry.college,geometry.research, August 9, 1993. From sander@geom.umn.edu Tue Nov 16 13:04:34 CST 1993 Article: 124 of geometry.college Xref: news3.cis.umn.edu geometry.college:124 geometry.research:113 Newsgroups: geometry.college,geometry.research Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Quasicrystals Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Mon, 15 Nov 1993 19:38:30 GMT Lines: 106 Status: OR Quasicrystals Starting in 1912, X-ray diffraction made it possible to study crystal structure. Although it does not show the actual crystal, diffraction indirectly indicates some of the structure, by giving information about the Fourier transform of the density distribution. If the density in a substance has a lot of symmetry, the X-rays will reinforce or cancel, as they stay in phase or go out of phase with each other; as a result, the diffraction pattern will exhibit symmetry as well. If all of the density in a substance were to occur at the vertices of a lattice, the diffraction pattern would exhibit hexagonal, square, or rectangular symmetry. Until 1984, these were the only three observed symmetries. Using this and further evidence, physicists thought that crystals had a lattice structure at the atomic level. In fact, they considered this to be one of the defining characteristics of a crystal. In 1984, a paper of Shechtman, Blech, Gratias, and Cahn disproved the assumption that substances arranged themselves in lattices; they found a metal alloy with with five-fold symmetry in its diffraction pattern. Since this is not a possible symmetry for the diffraction pattern of a lattice structure, the alloy must have a different structure. This discovery gave rise to the new research area of quasicrystals. In the last ten years, there has been much progress in this area, but there are still many open questions; even the structure of this first metal alloy is still an open question. The study of quasicrystals uses many different branches of mathematics and physics, including the study of Penrose tilings. This is a nonperiodic way to tile the plane or three-dimensional space. It uses only two rhombus-shaped objects in two dimensions or two rhombohedrons in three dimensions. The tilings must obey certain strict matching rules, only allowing for a finite number of different ways to put the two kinds of tiles together. Although a Penrose tiling is not periodic, it has something called quasiperiodicity. This means that patterns repeat themselves quite often; in fact, given any pattern in the tiling, there is an R>0 such that it is possible to find a repeat of the pattern within any ball of radius R. Thus Penrose tilings still retain a lot of order. It is quite amazing that this abstract geometric construction of Penrose tiles relates to the study of quasicrystals; using the vertices of a certain three-dimensional Penrose tiling as the points of density of a theoretical substance, the resulting diffraction pattern (in other words, a planar slice of the square modulus of its Fourier transform), exhibits five-fold symmetry and looks quite similar to that of the alloy discovered in 1985. Here is an even further amazing fact related to the diffraction of Penrose tiles; in 1982 N.G. de Bruijn showed that all Penrose tilings of the plane result from the projection of part of a five-dimensional lattice onto the plane. Likewise, all Penrose tilings in three dimensions result from the projection of part of a six-dimensional lattice onto three-dimensional space. Thus the new alloy, although no longer a lattice, has a diffraction pattern similar to that of the projection of part of a six-dimensional lattice. This discovery of nonperiodic tilings with symmetry in the diffraction pattern resulted in the following mathematical question: how ordered does a pattern of densities have to be so that the diffraction pattern still has bright spots? Physically, we know that there must be some order; otherwise the waves will have no strong reinforcements and cancellations needed for dark and bright spots. What are the conditions for these reinforcements and cancellations to occur in the Fourier transforms of projections of portions of arbitrary dimensional lattices? In what ways can one predict a diffraction pattern based on knowledge of the densities? Mathematics professor Marjorie Senechal tries to answer these questions. In order to simplify the question, she only looks at diffraction patterns of point densities in the plane. This is still a rich subject with many open questions. Senechal is currently writing a book called "Quasicrystals and Geometry" and is at the Geometry Center for a month to prepare the illustrations. One chapter of the book will consist of an atlas of diffraction patterns of tilings of the plane. Some are Penrose tilings, but others are more general. The atlas is meant to supplement a book called "Atlas of Optical Transforms." To help Senechal prepare these illustrations, Center staff have developed software to compute the Fourier transform of an arbitrary set of point densities and to project portions of lattices in high dimensions onto the plane. The software to compute projections of high dimensional lattices will be available by ftp soon. Senechal's book should be available in about a year from Cambridge University Press. References: G. Harburn, C.A. Taylor, and T.R. Welberry, Atlas of Optical Transforms, G. Bell & Sons, Ltd., London, 1975. Marjorie Senechal and Jean Taylor, "Quasicrystals: The View from Les Houches," The Mathematical Intelligencer, Vol. 12, No. 2, 1990. The original article on quasicrystals: D. Schechtman, I. Blech, D. Gratias, J. Cahn, "Metallic phase with long range orientational order and no translational symmetry," Physical Review Letters, Vol. 53, 1984, p. 1951-1954. Also see: Richard Kenyon, "Self-Similar Tilings," Geometry Center Preprint 21, 1990. From sander@geom.umn.edu Wed Nov 24 10:26:45 CST 1993 Article: 122 of geometry.puzzles Newsgroups: geometry.puzzles Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Triangle Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Wed, 24 Nov 1993 16:14:55 GMT Lines: 16 Status: OR Triangle Puzzle Pick an arbitrary point in the interior of an equilateral triangle. Now go halfway towards one of the three vertices (you get to pick which vertex). Now you are at a new point in the triangle. Again go halfway towards any vertex. Continue this process. What is the long term behavior? In other words, what points can you reach after repeating this process an arbitrarily large number of times? I am interested in using this problem to demonstrate the use of computers in teaching. Please let me know if you solve it using computers and what software you use. I will post an answer next week. Happy Thanksgiving! From sander@geom.umn.edu Mon Nov 29 17:31:44 CST 1993 Article: 129 of geometry.puzzles Newsgroups: geometry.puzzles Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Triangle Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Mon, 29 Nov 1993 21:34:58 GMT Lines: 23 Status: OR I have gotten a number of questions about the triangle puzzle. I will answer all of them at once: 1. When I ask "what points can you reach after repeating this process an arbitrarily large number of times," I had in mind: "what points are the limit of infinite sequences of applications of this process?" Thank you to Dan Asimov for pointing this out. This is also the answer to Art Mabbot's question: I do not want you to save points along the way; just look at where they tend to cluster if you could apply the process forever. 2. In answer to John Conway's question: at each stage, you are free to choose which vertex from which to halve your distance. You do not have to do so in cyclic order. 3. In answer to Art Mabbot, in accordance with 1 above, it is not important whether your point starts in the interior or on the boundary of the triangle. However, I do not allow for points in the exterior of the triangle. Hopefully this clarifies everything. I will post an answer on Wednesday. From sander@geom.umn.edu Wed Dec 1 10:13:18 CST 1993 Article: 135 of geometry.puzzles Newsgroups: geometry.puzzles Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Triangle Puzzle Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Distribution: inet Date: Wed, 1 Dec 1993 00:16:44 GMT Lines: 171 Status: OR The Puzzle: Triangle Puzzle Pick an arbitrary point in the interior of an equilateral triangle. Now go halfway towards one of the three vertices (you get to pick which vertex). Now you are at a new point in the triangle. Again go halfway towards any vertex. Continue this process. What is the long term behavior? In other words, what points can you reach after repeating this process an arbitrarily large number of times? I am interested in using this problem to demonstrate the use of computers in teaching. Please let me know if you solve it using computers and what software you use. I will post an answer next week. The Solution: As some people pointed out, the points which you can reach even after an arbitrarily long time form something called a Sierpinski gasket. This is the part of the equilateral triangle remaining after repeatedly removing inverted equilateral triangles of half the height. If you have never seen this before, it will probably be difficult to understand without pictures. Thus, there are pictures available by ftp; there is a picture of the removal of the first four sets of triangles, as well as an approximate picture made by having the computer play the game above. This includes points that are not contained in the gasket, but it looks approximately like the gasket. There is an important distinction between looking almost the same and really being the set. Please see a comment on this by John Conway. If I had chosen a starting point in the gasket, I would have the gasket, rather than adding any points. I also have enclosed the program used to generate this gasket so that if you are skeptical, you can fiddle with the program and convince yourself. One way to solve the problem is to consider what happens to the entire triangle when it goes halfway to a vertex. The result will be a triangle starting at that vertex, but with half the side length of the original. Thus no matter which of the three vertices you pick, you will miss the middle equilateral triangle. Therefore the set of points points you can reach after going halfway to a vertex once is exactly the same as the first step in constructing the Sierpinski gasket (see the pictures). Likewise, looking at what happens when each of these three triangles goes halfway to each vertex gives the second step in constructing the Sierpinski gasket. In fact, at each step, the set of all possible points after going halfway to the vertices is the same as the set of points left at that step of computing the gasket. I wanted to add a comment about what happens if instead of allowing you to pick the vertex, I specify a particular sequence of vertices; in particular, what if the vertices must be chosen in a cyclic permutation? I tried this, and only saw three limit points on the screen. At first I thought there was a mistake in the code, but then I realized that this follows from the contraction mapping principle. This suggests a more general principle at work regarding the limit points of any specified sequence. I do not wish to elaborate in this article. It is similar to the relationship between middle third Cantor sets and infinite sequences of zeros and ones. For more information, try Devaney's book Introduction to Chaotic Dynamical Systems. The Sierpinski gasket is fairly trendy; in addition to being aesthetically pleasing, it is a fractal. Although there are many explanations of this idea, let me briefly explain. The Sierpinski gasket has fractional Hausdorff dimension; here is an intuitive definition of Hausdorff dimension. It is not meant to be at all rigorous. I will give a reference for a more precise definition. Given a straight line segment of length one in the plane, it fits exactly inside an equilateral triangle of height one. How many triangles of height 1/2 does it take to cover the line segment? Exactly two. How many triangles of height 1/4 does it take? Exactly four. Thus the height of the triangle is inversely proportional to the number of triangles it takes to cover. We call a shape which eventually has this property dimension one. Now consider an equilateral triangle of height one in the plane. How many equilateral triangles of height 1/2 does it take to cover the shape now? Exactly four. How many triangle of height 1/4 does it take to cover? Sixteen. Thus the square of the height of the triangle is inversely proportional to the number of triangles needed to cover. We call a shape which eventually has this property dimension two. Now consider the Sierpinski gasket. It takes one triangle of height one to cover, three triangles of height 1/2, nine triangles of height 1/4, and etc. Thus the dimension of the gasket must be somewhere between one and two. In fact, it must be dimension log(3)/log(2). Thus the gasket is a fractal. Thank you to everyone who asked questions and took such an interest in this puzzle. I will have to post more in geometry.puzzles. Hope you enjoyed it too. Figures: Available by ftp from forum.swarthmore.edu in the /pictures/articles/triangle.puzzle directory. References: M. Barnsley 'Fractals Everywhere' M. Schroeder 'Chaos, Fractals, and Power Laws' Here is a computer program which plays the game. The output is in Postscript. #include #include #define LEFTMARGIN 180. #define WIDTH 5760. #define BASEMARGIN 1400. main() /* Sierpinski Gasket */ { float a1[3], a2[3], x1, x2; int i, k, max; long r; max=50000 ; /* max is the number of points; make this larger to see a clearer picture. (50,000 points gives a very nice picture, but it may take a long time.) */ a1[0]=0.0; /* The points (a1[i],a2[i]) (i=0,1,or 2) are the three vertices of triangle. */ a2[0]=0.0; a1[1]=1.0; a2[1]=0.0; a1[2]=0.5; a2[2]=sqrt(3.0)*0.5; /* (x1,x2) is the starting point: change this to see that the picture is independent of x. */ x1=0.5; x2=0.5; /* Some Postscript stuff. */ printf("%%!PS-Adobe-2.0\n"); printf("%%%%BoundingBox: 0 0 612 792\n"); printf("%%%%Pages: 1\n"); printf("%%%%EndComments\n"); printf("%%%%Page: 1 1\n"); printf("/dot {2 0 360 arc fill} def\n .1 .1 scale\n"); for(k=0;k 2*(pow(2,31)-1)/3) i=2; else i=1; x1=fabs(0.5*(a1[i]+x1)); /* This goes halfway to the chosen vertex. */ x2=fabs(0.5*(a2[i]+x2)); /* printf("%f %f \n", x1,x2);*/ /* This prints the new point. */ printf("%d %d dot\n", (int)(x1*WIDTH + LEFTMARGIN), (int)(x2*WIDTH + BASEMARGIN)); } printf("showpage\n"); printf("%%%%Trailer\n"); } From sander@geom.umn.edu Thu Dec 16 22:57:13 CST 1993 Article: 119 of geometry.research Newsgroups: geometry.research Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Kelvin Conjecture Overthrown Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Mon, 13 Dec 1993 16:02:05 GMT Lines: 44 Status: OR I received the following email from Ken Brakke: I've got some news from the wonderful world of soap bubbles that I'd like to publicize on the Geometry Forum. Here's the news: A long-standing conjecture about soap bubbles has been overthrown. In 1887, Lord Kelvin pondered how to partition space into cells of equal volume with the least area of surface between them, i.e. the most efficient soap bubble froth. He came up with a 14-sided space-filling polyhedron with 6 square sides and 8 hexagonal sides. The faces have to curve a bit where they meet to form the proper soap film angles. Now Denis Weaire (dweaire@vax1.tcd.ie) and Robert Phelan (rphelan@alice.phy.tcd.ie) of Trinity College, Dublin, have beaten Kelvin. The Weaire-Phelan structure uses two kinds of cells, a dodecahedron and a tetrakaidecahedron with 2 hexagons and 12 pentagons. The surface area is 0.3% less than the Kelvin structure, which is a whopping big amount to all of us who have tried to beat Kelvin over the years. The Weaire-Phelan structure has cubic symmetry, and the fundamental region is a 2x2x2 cube. The 8 cells start as Voronoi cells on centers 0 0 0 1 1 1 0.5 0 1 1.5 0 1 0 1 0.5 0 1 1.5 1 0.5 0 1 1.5 0 There are two dodecahedra (centered at (0,0,0) and (1,1,1)) and six tetrakaidecahedra. The cells must be adjusted a bit to get exactly equal volumes, and the faces must curve a bit to get the proper soap film angles. The tetrakaidecahedra stack on their hexagonal faces to form three sets of perpendicular interlocking columns, with the interstices filled by the dodecahedra. Ken Brakke brakke@geom.umn.edu From sander@geom.umn.edu Tue Dec 21 14:42:25 CST 1993 Article: 125 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Squaring the Circle 1 Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 21 Dec 1993 20:26:40 GMT Lines: 143 Status: OR Squaring the Circle Part 1 The following is a proof that given an arbitrary circle, it is impossible to construct a square of the same area using only straight edge and compass. People started to try to solve the classical Greek problem of squaring a circle by construction around 200 BC. It was finally proven impossible in 1882, when Lindemann proved that pi was trancendental. (I will omit the proof of this here.) Here are the basic ideas of the proof, following closely the discussion by Courant and Robbins in "What Is Mathematics." Given an arbitrary circle, let us define our unit of measure to be the radius of the circle. That means that the circle has area pi. Therefore, in order to construct a square of equal area, we need to construct the side of the square, which must have length square root of pi. I show that this is impossible. First, I must explain what it actually means to say that pi is transendental. This means that pi does not satisfy any rational polynomial equation. In other words, there is no equation of the form a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0)=0 with a's all rational, which holds for x=pi. In particular, this means we cannot find pi by a finite number of applications of the operations of addition, subtraction, multiplication, division, and taking nth roots of rational numbers. Note that it is also impossible to retrieve sqrt(pi) in this way. Now I show that these operations are the only ones available using construction. Given segments of lengths x and y, what are all the possible lengths of new segments we can construct? Here are five possibilities: It is possible to make a segment length x+y by lying the two next to each other. Likewise, drawing segment x starting at the right end of y and heading left, the distance from the left endpoint of y to the left endpoint of x is x-y. Through a more sophisticated construction, one can make segments of length xy, x/y, and the square root of x or y. (See accompanying figures.) Thus we can add, subtract, multiply, divide, and take square roots. We can find a right angle to a segment, so in fact, we can think of the five operations as acting on coordinates of points in a coordinate system in the plane. The five constructions above cover five powerful operations on the coordinates of points in the plane. However, there are other operations not mentioned, such as taking nth roots where n>2, taking trigonometric functions of x and y. Are any additional operations possible by construction? The answer is no. I will show this, and in addition that these five operations are not sufficient to get a segment with length square root of pi. First I give an exhaustive list of techniques in construction. Then I will show that using this list of techniques, one can only perform the five operations mentioned above. Using a straight edge, we can: 1. Draw a line through two points. 2. Find the intersection of two lines. Using a compass, we can: 3. Draw a circle of a given radius with a given center. 4. Find the intersection of a circle with another circle or line. Given one segment of unit length, the above constructions allow us to reach all points with rational coordinates. Starting with these points with rational coordinates, and using 1-4, what points can we reach? If two points have rational coordinates (a,b) and (c,d), the line between them, i.e. points solving x(b-d)-y(a-c)=(bc-ad), is of the form jx+ky=m (j, k,m rational). Given two nonparallel lines of this form: jx+ky=m and nx+py=q (j,k,m,n,p,q rational), the lines intersect at the unique point x=(mp-kq)/(jp-kn), y=(jq-mn)/(jp-kn), which again has rational coordinates. Thus 1 and 2 do not give any additional points. A circle with rational center (a,b) through rational point (c,d) are the points solving the equation (x-a)^2+(y-b)^2=(c-a)^2+(d-b)^2. This is of the form: x^2+y^2+jx+ky=m (j,k,m rational) Given a circle of this form and a line through two points with rational coordinates: x^2+y^2+jx+ky=m and nx+py=q (j,k,m,n,p,q rational), their intersection points will be solutions to the quadratic equation, and thus are of the form: (a+b(sqrt(c)), d+e(sqrt(f))), where a,b,c,d,e,f are rational. This result follows from the fact that solving for x in the second equation and substituting into the first, we obtain a quadratic equation. Similarly, solving for the intersections of the two circles x^2+y^2+ax+by=c and x^2+y^2+dx+ey=f (a,b,c,d,e,f rational), is equivalent to solving for intersections of the system: x^2+y^2+ax+by=c and (a-d)x+(b-e)y =c-f. This is intersection of a circle and a line through a point with rational coordinates. We already showed that the result of this intersection. Therefore, the only additional points obtained from 3 and 4 are points with coordinates containing square roots of rational numbers, i.e., coordinates of the form d+e(sqrt(f)) (d,e,f rational). >From the above two paragraphs, we see that starting with rationals, it is only possible by construction to perform the five operations above. In other words, we can only add, subtract, multiply, divide, and take square roots of coordinates to get new coordinates. But we know that it is impossible to ever get the square root of pi from rational numbers in this way. Thus it is impossible find the length of the side of the square of the same area as a given circle. This proves that it is impossible to square the circle!! From sander@geom.umn.edu Tue Dec 21 14:42:36 CST 1993 Article: 126 of geometry.college Newsgroups: geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Squaring the Circle 1 Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 21 Dec 1993 20:30:02 GMT Lines: 3 Status: OR I forgot to mention that the figures are available by anonymous ftp from forum.swarthmore.edu in file: /pictures/articles/squaring.the.circle/circle.eps. From sander@geom.umn.edu Tue Jan 11 10:09:16 CST 1994 Article: 58 of geometry.institutes Newsgroups: geometry.institutes Path: news1.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: 1994 Park City/IAS Mathematics Institute Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Mon, 10 Jan 1994 17:26:41 GMT Lines: 67 Status: OR In case you did not see the announcement last month, the Park City/IAS Mathematics Institute announced their 1994 Summer Session (July 10 - July 30) on Gauge Theory and the Topology of Four-Manifolds. I had a chance to talk to Dan Freed, one of the founders of the institute, called the Park City Regional Geometry Institute until this year. He told me about the advantages of the program over a more traditional kind of mathematics meeting. Perhaps this description will help you decide whether you are interested in the program. The institute is vertically integrated, including both research and education. It consists of four separate but simultaneous programs; these are for researchers, graduate students, undergraduate students, and high school teachers. Freed suggested that the additional support from the Institute for Advanced Study should send a message to the mathematical community that this vertical integration is important. The programs for graduate students and researchers are respectively based on Les Houches, a summer school for physics in France, and the Aspen Institute for Theoretical Physics. The announcement reads, "The Graduate Summer School consists of a set of intensive short lecture courses given by leaders in the field. They are intended to bridge the gap between general graduate courses and research seminars." The students get to know a lot of people in their area. It is also a chance to get research ideas and perhaps thesis problems. In addition to other graduate students, they get to know researchers there for the research program. The fact that the institute is in the mountains, rather than on a university campus, puts everyone on more equal footing, making it easier to establish these contacts. The research program has little structure, with usually only one formal session per day. In the past, some of the researchers have sat in on the summer school lectures. The program for college students gives them a chance "to enhance their interest in mathematics in general and geometry and topology in particular." The students will do a computer project, and attend a series of lectures. There is a formal mentoring program, in addition to the chance to meet math teachers, graduates, and researchers in a relaxed setting. The program for high school teachers runs quite differently. It is based on the model of "sites." The idea is to get many teachers from one location and teach them some new ideas for teaching high school geometry. After the summer, the teachers continue to meet in discussion groups. They then come back again the following year. This has in the past been quite successful. For example, in Austin, Texas, the teachers who had attended the summer program formed their own institute to teach junior high school teachers in the Austin area. University of Texas faculty, working with some of the teachers, have formed a Saturday morning program for high school students. Faculty have also started visiting high school classes. (This year teachers will return to Park City, so there will be no new high school teachers accepted to the program.) Running all these programs simultaneously gives people from different parts of the mathematical community a chance to get acquainted and gain an understanding and respect for each other's professions. All four programs end at 3 pm every day, giving people a chance to attend interactive programs. For example, there are education seminars, mathematical culture seminars, and evening programs. Some examples of past evening programs include screenings of math videos, panel discussions about women in science, talks on mathematical physics, and discussions of employment. There are also pizza parties, a coffee house, health club, and chances to hike in the mountains. Freed says that the interaction has been quite successful. By anecdotal evidence, people do keep in touch after the summer. Most of the subsequent contact has been of a professional nature, but once there was even a marriage! From sander@geom.umn.edu Wed Jan 19 11:19:37 CST 1994 Article: 129 of geometry.college Xref: news1.cis.umn.edu geometry.forum:135 geometry.college:129 Newsgroups: geometry.forum,geometry.college Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: QuasiTiler: New Software Message-ID: Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 18 Jan 1994 22:18:25 GMT Lines: 66 Status: OR QuasiTiler: New software to create quasiperiodic tilings of the plane Penrose tilings are a beautiful nonperiodic tilings of the plane using only two different rhombi. They have a variety of applications. For example, they arise in the study of quasicrystals; see Evelyn Sander, "Quasicrystals," geometry.college, November 15, 1993. The tilings have a rich mathematical structure: Despite their nonperiodicity, they exhibit five-fold symmetry, obey certain strict matching rules, and only allow for a finite number of different ways to put the two tiles together. They also have the property of quasiperiodicity; namely, given any pattern in the tiling, there is an R>0 such that it is possible to find a repeat of the pattern within any ball of radius R. In 1982 N.G. de Bruijn showed that all Penrose tilings of the plane result from the projection of part of a five-dimensional lattice onto a plane. In fact, by specifying the "slope" and "offset" in five dimensions of the plane upon which to project, one can specify a tiling. If this plane is invariant under five-fold rotation about the body diagonal of the unit cube, and if we chose carefully which part of the lattice is projected, then the tiling is a Penrose tiling. More specifically, we start with the five-dimensional integer lattice and its connecting 2-facets. Pick a plane E upon which we will project facets and points from the lattice to create the tiling. We choose E orthogonal to the line through the body diagonal of the unit cube. Here is how we decide which are the allowed vertices in the tiling of E: Project the unit cube into the three-dimensional orthogonal complement of E. The resulting object is a rhombic icosahedron which we call K. The allowed vertices are exactly those which land inside K under projection, and the allowed facets are those for which all four associated vertices are allowed. Our restrictions on E and allowed vertices forbid overlaps and gaps in the tiling. Eugenio Durand, a Geometry Center programmer, has written the program QuasiTiler to find the described quasiperiodic tilings of the plane. He originally wrote it to help Marjorie Senechal with her work on quasicrystals. The program allows the user to specify the "slope" of the plane E, using a mouse to modify a picture of the five-dimensional unit cube. There are three degrees of freedom for the offset of the plane. The user uses three sliders to change the offset. One of the offset directions specifies whether the tiling is a Penrose tiling. The program then shows the tiling that the user has specified. In addition, the user may specify a lattice dimension other than five. It is an easy program to use, and the results are beautiful. Please anonymous ftp the figures in /pictures/articles/quasitiler on forum.swarthmore.edu for some examples of pictures you can make with QuasiTiler. The figures are as follows: 1. A standard Penrose tiling. 2. A slight change of slope. 3. An offset away from Penrose. 4. Offset while remaining Penrose. QuasiTiler runs on Next machines. It is available by anonymous ftp in the /u/ftp/pub directory of geom.umn.edu. If you do not have a Next, it is also available as an interactive program from the Geometry Center on WorldWideWeb using mosaic. (I am in the process of writing an article about WWW and mosaic, so do not worry if you don't yet know about them.) The following reference describes Penrose tiles and briefly mentions the approach used in QuasiTiler: Ivars Peterson, "The mathematical tourist: snapshots of modern mathematics," New York, Freeman, 1988. From sander@geom.umn.edu Thu Jan 27 16:34:43 CST 1994 Article: 130 of geometry.announcements Xref: news3.cis.umn.edu geometry.forum:143 geometry.announcements:130 Newsgroups: geometry.forum,geometry.announcements Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: World Wide Web and Mosaic Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Thu, 27 Jan 1994 20:11:54 GMT Lines: 72 Status: OR World Wide Web and Mosaic World Wide Web (WWW) and the recently released browser for it, Mosaic are exciting innovations in Internet communications. This revolutionary software has been getting quite a lot of press. In December, 1993, the subject received a page in the Guardian and a page and a half in the New York Times business section. WWW and Mosaic is a hypertext based system, which allows you to explore resources from all over. It allows you to choose to look at text, images, and sounds by a click of the mouse button. When you run Mosaic, you see something similar to a page of a book with color illustrations. The book page has lots of cross references, indicated by highlighted words; if you see an interesting reference, you can click on it to see a different page of text, illustrations, and perhaps sounds and other interesting things. To carry the analogy of a book page further, each computer system which is on WWW can create their own book for everyone else in the world to see. The result is a library throughout the world which is yours to browse. In addition to being like a library, WWW has software for you to try and download. The creaters of Mosaic and WWW carefully integrated the former methods of Internet communications within the program. Thus you can read news groups, use Gopher, and even send email from within Mosaic. In fact, the WWW pages often have cross references to Gopher and news groups. By the nature of WWW, it is an excellent documentation system. Thus there is a wealth of self-documentation. I do not want to duplicate too much of this information. I will tell you a few places to look to find out more. From the initial hypertext page, in order to find out about the history of WWW, you click on the highlighted word WWW. This takes you to a new hypertext page with general information. Clicking on the word history takes you to yet another page with a full history starting with the first project proposal in March, 1989. In order to learn about the people involved, you can click on any highlighted name. This gives you a hypertext page with photos of all the people involved and their biographies. Two other places for information are: To get started: in the NCSA Mosaic Home Page (http://www.ncsa.uiuc.edu), look at Starting Points for Internet Exploration. For a complete explanation of Mosaic, WWW, and how it all works, look at http://www.geom.umn.edu/docs/guide/www.guide.html The Geometry Center is a part of WWW. There are biographies and photos of all the Center staff, information on upcoming events, a software library with programs available to download, an image library containing many of the images created at the Center, and all of my Forum articles and figures (to see these, from the Geometry Center title page, click on "Geometry Forum". Then on "Articles with Figures"). For example, the image library contains a copy of the image developed here that appeared on the cover of Scientific American in October, 1993. In addition to making software available to download, the Center has several interactive programs on WWW. This is not just a demo of what you could do were you to download; you actually use the software, vary parameters involved, and see the results. This means that before you download, you can see what the programs actually do. It has the added advantage that you can try hard to install or incompatible software without hassle. I highly recommend exploring "the web." It is the best piece of software I have ever seen. Gene Klotz says it is the best thing since sliced bread; I would have to agree. It has become so popular, that perhaps it will result in the Information Highway traffic jam of the century. However, if you can try it at an odd time, or look at data from something other than the most popular sources, it will not be particularly slow. Mosaic and WWW software are available for a large variety of machines, including PCs and Macintoshes. They are available by anonymous ftp from: ftp.ncsa.uiuc.edu in the directory Mosaic. Read the README files for instructions. The Geometry Center's WWW entry is in the file http://www.geom.umn.edu/ From sander@geom.umn.edu Fri Jan 28 10:29:04 CST 1994 Article: 130 of geometry.college Xref: news1.cis.umn.edu geometry.pre-college:376 geometry.college:130 Newsgroups: geometry.pre-college,geometry.college Path: news1.cis.umn.edu!umn.edu!csus.edu!wupost!ukma!forum.swarthmore.edu!news2.cis.umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Gallery of Mathematicians Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 25 Jan 1994 22:14:40 GMT Lines: 52 Status: OR Gallery of Mathematicians Nancy Goldberg, math teacher in UMTYMP and at Eden Prairie High School (in Minnesota), noticed that all textbook pictures of mathematicians are of pre-twentieth century mathematicians. This gives the illusion that mathematics research is finished and now only a subject studied in school. In order to destroy this illusion, she has started a project in conjunction with Arnie Cutler at the Geometry Center called the Gallery of Mathematicians. The Center asks its many visitors to allow themselves to be part of the Gallery. If they agree, the Center staff take their photo. They also supply a statement for high school students. These statements can be biographies, advice, description of research, or anything else that might be interesting to the students. Already there are over forty contributors to the Gallery. They are quite a distinguished group. Many of the people mentioned in previous articles appear in the Gallery, such as Fred Almgren, George Francis, Jean Taylor, Ken Brakke: "Although they are familiar to everybody, ... soap films still hold surprises," Dan Freed: "I kept taking more and more math courses, ... and before I realized it, I was becoming a professional mathematician!" and Gene Klotz, creator of the Geometry Forum: "I've been a member of the Swarthmore College Department of Mathematics since the dawn of man, when I wrote my dissertation on Lie algebras." In addition to mathematics researchers, there are a wide range of others who use math in their work. For example, there are high school teachers, education researchers, software developers, and physics professors. At current time, the Gallery is just a collection of photos and statements. The plan is to make either a poster set (around 25) or an academic year calendar available at cost to those interested. Advertising will start in April, taking orders first, and having everything ready for the beginning of the 94/95 school year. The cost is potentially $20 but perhaps less. If you are interested in this project, please give your input: Would you prefer a poster set or a calendar? Is it better to have a $20 photo quality product, or would you prefer to pay less for a copy quality product? If $20 is too high, what is the maximum you would be willing to pay? Do you have any other suggestions? Send all questions, comments, and advice to Arnie Cutler (cutler@geom.umn.edu). From sander@geom.umn.edu Wed Feb 2 10:50:56 CST 1994 Article: 131 of geometry.college Xref: news1.cis.umn.edu geometry.forum:145 geometry.college:131 Newsgroups: geometry.forum,geometry.college Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Hyperbolic.m: hyperbolic geometry software Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 1 Feb 1994 20:58:29 GMT Lines: 106 Status: OR Hyperbolic.m: A general-purpose hyperbolic geometry package "The Elements," a 325 BC publication of Euclid, records Euclid's principles of geometry. For many centuries, these principles remained the fundamental basis of the study of geometry. There are five postulates of particular importance: 1. A straight line may be drawn from any point to any other point. 2. The straight line may be produced to any length. 3. Around any point as a center, a circle of any radius may be described. 4. Any two right angles are equal. 5. Given a line and a point not on the line, there is exactly one line through the given point not intersecting the given line.[B] This fifth postulate, also called the parallel postulate, receives special attention. Mathematicians did not like it because it possesses a different character from the first four, being global rather than local in nature. Since intersection of lines can take place at an arbitrary distance, there is no way to actually check whether lines intersect. Thus starting as soon as the 4th century AD, there were attempts to prove that 5 follows from 1-4. All attempts to prove this failed, since in fact the parallel postulate is independent of 1-4. It was not until the middle of the 19th century that Bolyai and Lobachevsky separately showed that in fact it is possible to construct a consistent geometry in which the parallel postulate does not hold. In this geometry, there are infinitely many non-intersecting lines through a given point. It is called hyperbolic geometry. Mathematicians found it difficult to accept hyperbolic geometry. In the middle of the 19th century, most were unable to imagine or accept any geometric system which did not fit with Euclid's principles. As an indication, even Bolyai's father, a mathematician himself, was unable to believe or understand the work of his son because of his belief in the absolute truth of Euclidean geometry.[B] Soon after the work of Bolyai and Lobachevsky, others constructed models of hyperbolic geometry within Euclidean space. By renaming certain geometric objects from Euclidean space as the lines and circles, people developed systems in which all of Euclid's postulates hold except that there were infinitely many non-intersecting lines, violating the parallel postulate. These models, all isometric, verify the existence of hyperbolic geometry. As Courant and Robbins state in "What is Mathematics," "This must, eo ipso, be just as consistent as the original Euclidean geometry, because it is presented to us, seen from another point of view and described with other words, as a body of facts of ordinary Euclidean geometry."[C] The modern approach to geometry is somewhat different from that of "The Elements." However, a (geodesically complete Riemannian) geometry on a surface is essentially a structure on the surface obeying the modernized versions of postulates 1-4. Each surface has a unique associated geometry locally isometric to the sphere, the plane, or the hyperbolic plane. The sign of a special constant called the Euler characteristic determines which of the three kinds of geometries a surface possesses. There is a complete classification of all compact connected surfaces. By computing the Euler characteristic of all these surfaces, one can draw an amazing conclusion; of these infinite number of possible surfaces, only seven have non-negative curvature, implying that all but seven are hyperbolic. For example, all n-holed tori with n>1 have an associated hyperbolic geometry. Among the quite common class of compact connected surfaces, a geometric structure that took over a millennium to discover is overwhelmingly the most frequent. Since every surface with a hyperbolic geometry is locally isometric to the hyperbolic plane, the modern study of hyperbolic geometry involves studying models for the hyperbolic plane which are embedded in Euclidean space. If you just want a general idea of them, look at the figures. I briefly describe the properties of the Klein, Poincare, and upper half plane models in a separate article. Oliver Goodman, a postdoc at the Geometry Center, has written a general-purpose hyperbolic geometry software package called Hyperbolic.m. Operating within Mathematica, Hyperbolic.m allows the user to manipulate hyperbolic objects (in any dimensional hyperbolic space) such as points, lines, vectors, polyhedra, and other structures. The user enters these structures within any of five hyperbolic models. The package calculates angles and lengths for vectors and distances between points, convert between different models, and apply isometries. It also computes triangle groups. Hyperbolic.m allows the user to display the results of manipulations using Mathematica graphics. Please see the figures for examples. Although there are several programs which, when combined, do what Hyperbolic.m does, there are none that do everything. The package's versatility allows the user to use only one program, and to take advantage of other functions within Mathematica. Hyperbolic.m is available by anonymous ftp from geom.umn.edu in the file hyperbolic.tar.Z in the pub/software directory. Any questions or comments on the package should be sent to Goodman (oag@geom.umn.edu). To see figures made using Hyperbolic.m, use anonymous ftp from forum.swarthmore.edu in the /pictures/articles/hyperbolic directory. Figure 4 is a regular icosohedron in the Poincare model of 3D hyperbolic space. If you have Mosaic, you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum/figures.html This article was written based on interviews with Goodman, as well as material from the following references: [B] Janos Bolyai, "Appendix," Originally published in 1832, North Holland, 1987, Introduction by Ferenc Karteszi. [C] Courant/Robbins, "What is Mathematics?" Oxford University Press, London, 1941. From sander@geom.umn.edu Tue Feb 8 16:23:47 CST 1994 Article: 133 of geometry.college Xref: news3.cis.umn.edu geometry.pre-college:411 geometry.college:133 Newsgroups: geometry.pre-college,geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: More Math Needed in Science Museums Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 8 Feb 1994 02:30:16 GMT Lines: 43 Status: OR I thought Forum readers would enjoy the following by John Sullivan: I went to the Ontario Science Center in Toronto last August. This is a modern science museum, with a number of interactive exhibits like those at the Exploratorium in San Fransisco, but also large sections on more applied things like food+nutrition or transportation. I looked for exhibits about math, and was pretty disappointed. The only one they seemed to think was math was a collection of pocket calculators with signs suggesting that you do various "tricks" along the lines of "Take your house number, multiply by 2, add three, multiply by 5, subtract 5, erase the last digit, subtract one; look you have your number back". There was a whole room devoted to sports (the materials used in sports equipment, etc). One exhibit was about judging things like gymnastics, and was trying to make the point that a judge who gave everyone low scores didn't adversely affect the results. Somehow they didn't seem to explain this well. They had a system set up where 5 people could judge videotaped events, but this didn't seem to teach much science. There were some computers running LOGO available, and a quite nice double pendulum, but again no attempt to explain what was going on in these exhibit. They did have a good exhibit on optical illusions, with many of the standard ones done nicely with sliding tranparent panels set up so that you could check that whatever didn't look true was in fact true. Along with these were a couple of copies of the room with distorted perspective that I've seen elsewhere. (You look with one eye from one particular viewpoint, and it looks cubical even though it isn't.) One such room had marbles seeming to roll up hill, and figures seeming to change size, but my favorite exhibit was another such room that you could walk into. Hanging on one wall was a TV screen connected to a camera at the magic viewpoint--you could walk around this funny-shaped room, but see yourself walking in a "normal" room, your size changing. I think we need to get more math into the museums. -John Sullivan University of Minnesota From sander@geom.umn.edu Mon Feb 14 10:26:26 CST 1994 Article: 134 of geometry.college Xref: news3.cis.umn.edu geometry.forum:156 geometry.college:134 Newsgroups: geometry.forum,geometry.college Path: news3.cis.umn.edu!umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Hyperbolic.m: hyperbolic geometry software Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Fri, 11 Feb 1994 00:30:29 GMT Lines: 85 Status: OR Models of the Hyperbolic Plane This is part 2 of an article I wrote last week: Evelyn Sander, "Hyperbolic.m: hyperbolic geometry software," geometry.college, Feb 1, 1994. The following is a description of some of the models for the hyperbolic plane. In order to understand the descriptions, refer to the figures. They may seem a bit strange. However, a result due to Hilbert says that it is impossible to smoothly embed the hyperbolic plane in Euclidean three-space using the usual Euclidean geometry. (Technical note: In fact it is possible to have a C^1 embedding into R^3, according to a 1955 construction of Nicolaas Kuiper, but according to William Thurston, the result would be "incredibly unwieldy, and pretty much useless in the study of the surface's intrinsic geometry."[William Thurston, "Three Dimensional Geometry and Topology," Geometry Center Preprint, 1991, p.43.]) Since there is no such smooth embedding, any model of the hyperbolic plane has to use a different geometry. In other words, we must redefine words like point, line, distance, and angle in order to have a surface in which the parallel postulate fails, but which still satisfies Euclid's postulates 1-4 (stated in the previous article). Here are brief descriptions of three models: Klein Model In the Klein model of the hyperbolic plane, the "plane" is the unit disk; in other words, the interior of the Euclidean unit circle. We call Euclidean points the "points" for our model. We call the portions of Euclidean lines which intersect the disk "lines." See the Klein model in Figure 1. From the above, the model seems similar to Euclidean space. However, there are differences; since we insist that postulate 2 holds, which says that we can make every line infinitely long, we must have a new description of "distance." We define the "distance" between two points as follows: If (x,y) and (u,v) are the Euclidean coordinates of two points, then the hyperbolic "distance" between them in the Klein model is arccosh{(1-xu-yv)/sqrt[(1-x^2-y^2)(1-u^2-v^2)]}. Thus postulate 2 holds in this model, since the "distance" from any point in the disk to the unit circle is infinite. Also, the way we define "angles" in the model is not the same as Euclidean definition of angles. To see this, look at the regular pentagon in the Klein model in Figure 2. As you can see, although all five of the "angles" are the same, the Euclidean angles are not the same. Upper Half Plane Model The upper half plane model takes the Euclidean upper half plane as the "plane." Now the "lines" are portions of circles with their center on the boundary, as shown in Figure 1. The "distance" between two points with Euclidean coordinates (x,y) and (u,v) is arccosh{1+[(x-u)^2+(y-v)^2]/2yv}. Thus the "distance" from any point to the x-axis is infinite, ensuring that postulate 2 holds in this model. Unlike the Klein model, "angles" are the same as Euclidean angles. Remember, you have to draw tangents to the "lines" to calculate the Euclidean angles. See Figures 1 and 2. Poincare Model In the Poincare model, the "plane" is the unit disk, and "points" are Euclidean points. "Lines" are portions of circles intersecting the disk and meeting the boundary at right angles. The "distance" between two points with coordinates z and w in the complex plane is: 2 arctanh{|(z-w)/(1-wz*|}, where z* is the complex conjugate of z. Thus as in the Klein model, the "distance" to the boundary of the disk is infinite, and postulate 2 holds. Like the upper half plane model, the "angles" for the model are the same as Euclidean angles. See Figures 1 and 2. Also compare the isometric octahedrons in Figure 3. It is also possible to have higher dimensional hyperbolic spaces. There is a Poincare model for three-dimensional hyperbolic space. In this case the "space" is the unit sphere, "lines" are portions of circles intersecting the boundary of the unit sphere at right angles, and "planes" are portions of spheres which meet the unit sphere at right angles. Figure 4 shows a regular icosohedron in this Poincare model of 3D hyperbolic space. The figures are the same ones that I referred to in part 1 of this article. To see them, use anonymous ftp from forum.swarthmore.edu in the /pictures/articles/hyperbolic directory. If you have Mosaic, you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum/figures.html From sander@geom.umn.edu Mon Mar 7 16:19:29 CST 1994 Article: 144 of geometry.college Xref: news3.cis.umn.edu geometry.research:152 geometry.college:144 Newsgroups: geometry.research,geometry.college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Groups and Symmetry Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Sat, 5 Mar 1994 22:31:58 GMT Lines: 122 Status: OR Groups and Symmetry "Group theory is the abstract study of symmetry," says Peter Webb, University of Minnesota and Geometry Center professor. "This is not the way you are first taught to look at groups. Classes usually first teach the definition involving multiplication laws for elements within the group. What could this have to do with symmetry?" Webb explains that by looking at groups another way, the symmetry becomes clearer. Instead of looking at multiplication structure within the group, look instead at how a group can act on another object. For example, the group of nonsingular n by n matrices with real coefficients acts by matrix multiplication on n-dimensional vectors. In this way, the group actions are all the invertible linear transformations on n-dimensional space. The study of group actions on a set is a starting point for many different areas of study. Geometers use group actions on geometric objects to find out more information about the structure of the object. In contrast, group theorists, such as Webb, use group actions on a set to study the structure of the group. Often, the actions of the group are symmetries of the object. Thus group theory becomes the study of symmetry. The advantage of looking at group actions is that the group may be very large, consisting of perhaps thousands of elements. Thus writing down a group multiplication table would be time-consuming and not very illuminating. There is often a much smaller set on which it acts. This means it is easier to see properties of the group by looking at the set on which it acts. One of the methods that Webb finds useful in his research is to consider the actions of a group on a simplicial complex, the high dimensional version of a graph. Just as a graph consists of points and edges joining points of the graph, a simplicial complex consists of points, edges joining points of the complex, 2-D faces joining edges, 3-D faces joining 2-D faces, and generally a k-D face joining (k-1)-D faces. A solid triangle and a triangulation of a sphere are two-dimensional simplicial complexes. A tetrahedron is a three-dimensional simplicial complex. Webb is particularly interested in a certain simplicial complex which was first constructed for each group by K.S. Brown, and which is built out of the structure of the group itself. Here is an example of the use of simplicial complexes in group theory. We study the group G, consisting of invertible 3 by 3 matrices with entries in Z/2Z (the unique field of two elements). G has 168 elements, making it unwieldy to write down a multiplication table, though it is small from a group theorist's standpoint. We could consider the actions of this group on the space S=(Z/2Z)x(Z/2Z)x(Z/2Z), since G permutes the seven nonzero elements in this space. However, we need to look at something related to S but more involved to get results. We look at a particular simplicial complex, in this case a graph, called the "building" for G. To get this, consider the set of 1-dimensional subspaces of S. See Figure 1. This consists of seven lines through the seven nonzero points of G. >From this set of lines, we make the building, a graph with 14 vertices, one for each of the nonzero points and one for each of the lines. We join a vertex representing a point to a vertex representing a line on the graph if the point is on the line. G has the property that an element of G sends each line to another line. Since G permutes points in S while preserving lines, G acts on the building. See Figure 2. Considering a particular vertex in the building of G, the set of elements in G which leaves that vertex fixed is isomorphic to S(4), the group of permutations of four elements. Likewise, if two adjacent vertices are fixed, the edge between them is fixed. A given edge between adjacent points is fixed by D(8), the dihedral group with eight elements. See Figure 3. Using this information and a theorem due to Webb, we can get some results on the structure of G. The theorem is quite general; even as it applies to this group, it is too technical to state here. Thus, I only show the following very restricted case; denoting the abelianization of H by H/H', and [H]_2 for the 2-torsion subgroup of H (elements whose order is a power of 2), the theorem says that we have a short exact sequence 0 --> [G/G']_2 --> [S(4)/S(4)']_2 + [S(4)/S(4)']_2 --> [D(8)/D(8)']_2 --> 0 >From this we can deduce that [G/G']_2 is the identity (the unique one element group). The above example demonstrates that by studying the subgroups of a group which fix given parts of a simplicial complex and by considering the geometry of that complex, we are able to gain information about the structure of the group. To see that this is more efficient than writing down a multiplication table, compare of the number of elements in the three groups: Group Number of Elements G 168 S(4) 24 D(8) 8 Note that G is much bigger than the other two groups. We have reduced the complexity of the problem by using actions of G on a set, instead of looking at multiplication inside of G. Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/groups.and.symmetry directory. If you have Mosaic, you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum/figures.html (If you don't have access to the figures, make Figure 2, the "building": draw points in a circle. Make an edge from each point to its two nearest neighbors. From every other point, make an edge to the point five ahead of it. Thus there are three edges connecting at each point.) This article is based on an interview with Webb. The reference for Webb's original article is: P.J. Webb, "A split exact sequence of Mackey functors," Commentarii Mathematici Helvetici, 66(1991), 34-69. There is also a survey in: P.J. Webb, "Subgroup complexes," Proceedings of the Symposia in Pure Mathematics, 47(1987), 349-365. From sander@geom.umn.edu Wed Mar 9 14:55:47 CST 1994 Article: 470 of geometry.pre-college Xref: news1.cis.umn.edu geometry.pre-college:470 geometry.puzzles:149 Newsgroups: geometry.pre-college,geometry.puzzles Path: news1.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Moon Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 8 Mar 1994 18:00:34 GMT Lines: 5 Status: OR This is addressed to high school students, but perhaps others want to think about it. Do not post answers until Friday so everybody can think about it. I will post my answer then too: What is the shape of the inner edge of the moon's crescent? From sander@geom.umn.edu Thu Mar 31 15:19:33 CST 1994 Article: 475 of geometry.pre-college Xref: news3.cis.umn.edu geometry.puzzles:157 geometry.pre-college:475 Newsgroups: geometry.puzzles,geometry.pre-college Path: news3.cis.umn.edu!umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Moon Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities References: Date: Fri, 11 Mar 1994 21:38:03 GMT Lines: 50 Status: OR > What is the shape of the inner edge of the moon's crescent? The moon is a sphere that is lit on the half facing the sun. As the moon rotates around the earth, the lit part of the moon is at different angles to the earth. Thus mostly we can only see part of the lit side of the moon. Although we are looking at part of a sphere, there is no shadowing to show depth, so the moon looks like part of a circle. What is the shape of the inner edge of the moon's crescent? First, two easy cases; when the moon is between the sun and the earth, the sun shines on the side of the moon facing away from the earth. No crescent is visible. In other words, we have a new moon. If the earth is between the sun and the moon, we see only the lit part. The crescent is a circle. We call that a full moon. In general, the lit half of the moon faces neither directly towards us nor directly away from us, so we have to use some geometry to know the shape inner edge of the crescent. The boundary of the lit half of the moon is a great circle. It is this boundary circle which forms the inner edge of the crescent. What does it look like? The entire sphere looks flat, in a plane perpendicular to the line of sight. Therefore the inner edge of the crescent is the projection of the boundary circle onto a plane. I will compute what this projection looks like. Fix a coordinate system so that the moon is radius 1 with center at the point (0,0,0), and the line of sight is along the x-axis. This means the plane of projection is perpendicular to the x-axis. Since the boundary circle is in a plane through (0,0,0), we can choose the z-axis so it is in the plane containing the boundary circle. Now the coordinate system is fixed. We know that the boundary circle passes through a point (a,b,0) on the x,y-plane, where a^2+b^2=1. Using this point, here is an equation for the boundary circle: -bx+ay=0 (ax+by)^2 + z^2=1, which is the same as x=(a/b)y [(b+a^2/b)y]^2+z^2=1. Projecting this into a plane perpendicular to the x-axis is the same as looking at all pairs (y,z) so that (x,y,z) solves the above equations for some x value. Since we can always find an x that solves the first equation, we only need to look at the second equation. The second equation is the equation for an ellipse. Thus the inside edge of the moon's crescent is part of an ellipse. From sander@geom.umn.edu Fri Apr 15 19:54:19 CDT 1994 Article: 153 of geometry.college Xref: news3.cis.umn.edu geometry.college:153 geometry.research:176 Newsgroups: geometry.college,geometry.research Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Automatic Groups Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 15 Apr 1994 21:43:14 GMT Lines: 128 Status: OR Automatic Groups: A class of groups solving the word problem One of the first things a group theorist would like to know about a group element is whether it is the identity element. When the group is a finitely generated group given in terms of generators and relations, the question is known as the word problem. Namely, the word problem asks if there is an algorithmic method to decide whether an element given in terms of a product of generators is the identity. Though this seems like a simple request, it is not possible to solve the word problem in general. Thus it is difficult to study finitely generated groups, since the above result implies that in general one cannot tell whether two given products of generators represent the same element. In some recent research in computational group theory, J. Cannon, W. Thurston, and D. Epstein were able to define a large class of groups for which it is possible to solve the word problem. Groups of this class are called automatic groups. Not only is it possible to solve the word problem for automatic groups, but by representing an automatic group by a special automatic structure, it is possible to solve the word problem in an efficient manner. The concept of an automatic structure is not abstract; Epstein, S. Rees, and D. Holt have written software which inputs a set of generators and relations of a group and, when the group is automatic, gives back an automatic structure. Automatic groups are based on the computer scientists' concept of finite state automata. Briefly, a finite state automaton uses an algorithmic method to determine whether objects called "words" have particular properties by looking at their components, called "letters." The automaton reads a word one letter at a time. After reading a letter, of which there are only a finite number, the automaton records that it is in a certain one of a finite number of "states." At any time, the only information the automaton knows is what state it is in. This state and the next letter read entirely determine the state the automaton goes to next. The state may be an "accept" state. This means that the word that led to this state has the desired property. If the automaton does not reach an accept state before it finishes reading all the letters in a word, then the automaton rejects the word. This means the word does not have the desired property. Note that since there are only a finite number of states in the automaton, it only takes a finite amount of information to determine whether to accept a given word. Thus even though there might be an infinite number of words, an automaton guarantees that a for one word, a property is determined in a finite amount of time using a finite amount of information. In the context of groups, the words in a finite state automaton are products of generators of the group, and the generators themselves are the letters. Note that two words can represent the same group element. For example, assuming that all elements of a finite group are generators, for any x in the group, the identity word and the word (x)(x^(-1)) represent the same group element. One possible group automaton is one which accepts a word only when it is a representation of the identity element. Since an automaton rejects or accepts words in a finite amount of time, whenever such an automaton exists, it is possible to solve the word problem. An automatic group is one with three kinds of associated finite state automata. They are a word acceptor, which determines whether a representation is in reduced form, an equality recognizer, which decides if two words are equal, and a multiplier for each generator, which determines whether a word is equal to a second word times the generator. By the second type of automata, we know that automatic groups solve the word problem. Although the existence of these three kinds of automata sounds like a stringent condition, many important groups are automatic. Two examples of automatic groups are Euclidean groups, meaning discrete groups of isometries of Euclidean spaces, and word-hyperbolic groups. In order to define word-hyperbolic groups, we need to discuss Cayley graphs; the Cayley graph is a graph with a point for each element of the group and with a directed line from one point to a second point whenever the second element is the first element times one of the generators. For example, Figure 1 shows Cayley graphs for the two distinct four element groups. A path in a Cayley graph is a series of lines that match start to end; a path corresponds to multiplication by a series of generators. The length of a path is the number of lines making up the path; path length corresponds to the number of generators in the product. The distance between two points on the Cayley graph is the length of the shortest path between the two points. A word-hyperbolic group is one whose Cayley graph has the following property: given a triangle with edges of shortest possible length to still join the vertices, the distance from a point on one edge to the union of the other two edges is bounded by some constant. For example, any group for which the Cayley graph can be embedded nicely in the hyperbolic plane is a word-hyperbolic group. See Figure 2. Aside from the word problem, the automatic structure of a group allows people to quickly and efficiently draw a variety of geometric objects. For example, see Figure 3, figures from a recent paper by Greg McShane, John Parker, and Ian Redfern. Some of the scenes in the Geometry Center movie "Not Knot" were made using automatic groups. I asked Charlie Gunn, the movie's technical director, what he thought of them: "The use of automatic groups can provide significant acceleration when providing large lists of group elements. For example, the Geometry Center poster depicting the a frame from the fly-through of hyperbolic space from 'Not Knot' used a list of over 100,000 group elements; this list took only a few minutes to compute with the automatic structure. Without automatic groups, the same list would require hours of computation. Another advantage of automatic structure is extremely low memory requirements; without it the full list of group elements must be kept, as matrices, in memory throughout the computation." Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/automatic.groups directory. If you have Mosaic, you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum/figures.html Caption for Figure 3: The limit set of the group < z |-> z+2 , z |-> mu + 1/z > where mu = 0.06469 + 1.912 i. The cosets of the stabiliser of the real line are enumerated using an automaton. This enables us to draw each image of the real line under the group exactly once. (Figure 1a in paper below.) This article is based on interviews with David Epstein and Silvio Levy, technical director of the Center, as well as referring to their book, written at the Center: D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, "Word Processing in Groups," Jones and Bartlett Publishers, Boston, 1992. Greg McShane, John R. Parker and Ian Redfern, "Drawing limit sets of Kleinian groups using finite state automata" Warwick Preprint, number 24/1994. From sander@geom.umn.edu Fri Apr 15 19:54:48 CDT 1994 Article: 174 of geometry.research Xref: news3.cis.umn.edu geometry.research:174 Newsgroups: geometry.research,geometry.college Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Seifert Conjecture Overthrown Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 12 Apr 1994 22:49:11 GMT Lines: 113 Status: OR Seifert Conjecture Overthrown One of the long-standing problems of dynamical systems is to prove or disprove the following 1950 postulate of Seifert: Seifert Conjecture: Every non-vanishing vector field on the three-sphere has a periodic orbit. After many years, this conjecture is now known to be false. In 1974, P.A. Schweitzer constructed a C^1-smooth counterexample (i.e. the vector field in this counterexample is once continuously differentiable). In 1988, J. Harrison constructed a C^2 counterexample. Finally, just this past summer, K.M. Kuperberg solved the problem conclusively, when she constructed an infinitely continuously differentiable counterexample. This article gives some of the background of the problem as it arises in Hamiltonian systems, differential geometry, and topology. Part 2 gives a heuristic description of Kuperberg's counterexample. Parts 1 and 2 are both based on a recent seminar by Richard McGehee, University of Minnesota mathematics professor and interim director of the Geometry Center. Notation: for the rest of this article, I refer to the two-sphere as S^2 and the three-sphere as S^3. The Hamiltonian Case Consider the following Hamiltonian system of differential equations on R^4 (x and y are vectors in R^2): x'=y y'=-x Solutions of this differential equation stay within a fixed level set of its Hamiltonian H(x,y) = |x|^2+|y|^2. The level sets, {(x,y): H(x,y)=h, where h is constant}, are diffeomorphic to S^3 for positive h. Thus we can restrict to a vector field on S^3. The simplest kinds of solutions to understand are fixed points and periodic orbits. Thus we are interested in whether there are any periodic orbits. In other words, does the Seifert conjecture hold for this example? In fact in this particular example, all solutions are periodic. What happens if we perturb this system by a Hamiltonian perturbation? For small enough perturbation, Weinstein showed that the system will still have at least two periodic orbits. We do not have to restrict our attention to perturbations of this specific system. More generally, consider any Hamiltonian system in R^4. If a level set for the Hamiltonian of the system is diffeomorphic to S^3, is there a periodic solution on this level set? In other words, does the Seifert conjecture hold if the vector field comes from a Hamiltonian system? Rabinowitz showed that if the level set is star-like, in other words every ray from the origin intersects it in exactly one point, then the answer is yes. The answer is not known for a general level set. Riemannian Geometry In the case of Riemannian geometry, one version of the Seifert conjecture is a question about closed geodesics. Namely, for a given Riemannian metric on S^2, are there any closed geodesics? The unit tangent bundle of S^2 is diffeomorphic to S^3, which means that a geodesic flow on S^2 corresponds to a flow on S^3. A periodic orbit for this flow in S^3 corresponds to a closed geodesic. In this very restricted case, the Seifert conjecture does hold. Bangert and Franks proved the much stronger result that every Riemannian metric on S^2 has infinitely many closed geodesics. Topology In the topological context, we generalize the question: Generalized Seifert Question: Let M be a compact n-dimensional manifold with Euler characteristic zero. This last condition guarantees that there exist non-vanishing vector fields on M. Does every non-vanishing C^r vector field on M have a periodic orbit? In 1966, W. Wilson showed that for a manifold described above of dimension four or larger, the answer to this question is no for arbitrarily smooth vector fields. In other words, Wilson found an infinitely differentiable non-vanishing vector field on M which contains no periodic orbits. I stated the result for S^3, but in fact Kuperberg's example also shows that for any manifold as above of dimension three, the answer to the Seifert question is no. As the above discussion shows, the Seifert conjecture comes up in a variety of fields. It is an exciting result that Kuperberg has disproved it. In part two of this article, I will give an outline of the ideas used Kuperberg's infinitely differentiable counterexample. References: Krystyna M. Kuperberg, "A C^infinity counterexample to the Seifert conjecture in dimension three," preprint, 1993. Paul Rabinowitz, "Periodic Solutions of Hamiltonian Systems," Communications of Pure and Applied Math. 31(1978), 157-184. Alan Weinstein, "Symplectic V-Manifolds, Periodic Orbits of Hamiltonian Systems, and the Volume of Certain Riemannian Manifolds," Communications of Pure and Applied Math. 30(1977), 265-271. Wilson, "On the minimal sets of non-singular vector fields," Annals of Math. 84(1966), 529-536. The following two references are less technical than this article: Barry Cipra, "Collaboration Closes in on Closed Geodesics," What's Happening in the Mathematical Sciences, AMS, (1)1993, 27-30. Barry Cipra, "Smoothing Out the Seifert Conjecture," What's Happening in the Mathematical Sciences, AMS, (to appear). McGehee's lecture took place February 24, 1994, as part of the Dynamics and Mechanics seminars at the University of Minnesota. From sander@geom.umn.edu Mon Apr 18 20:54:28 CDT 1994 Article: 154 of geometry.college Xref: news3.cis.umn.edu geometry.research:177 geometry.college:154 Newsgroups: geometry.research,geometry.college Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Seifert Conjecture Overthrown Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Sat, 16 Apr 1994 01:54:12 GMT Lines: 141 Status: OR Seifert Conjecture Overthrown Part 2 Part 1 of this article discussed some of the background of the following conjecture, which K.M. Kuperberg has recently disproved: Seifert Conjecture: Every non-vanishing vector field on the three-sphere has a periodic orbit. In this article, I give an outline of Kuperberg's construction of an infinitely differentiable counterexample. Kuperberg's construction builds on Wilson's construction mentioned in part one. Namely, on every compact manifold of characteristic zero and dimension greater than three, Wilson constructs a non-vanishing infinitely differentiable vector field with no periodic orbit. Wilson's Construction Given a manifold as above of dimension n>3, Wilson starts with a non-vanishing vector field with a finite number of periodic orbits. He then modifies the vector field in a small strangely-shaped neighborhood of a point of a periodic orbit. This neighborhood is called a plug. The modified vector field has the property that it breaks a periodic orbit without vanishing or creating additional periodic orbits. After a finite number of modifications, the vector field on the manifold has no periodic orbits. Here is a description of Wilson's plug to modify a vector field; assume that the original vector field is constant in the (0,0,...,1) direction at each point in a cube in R^n. This is a reasonable assumption, since any non-vanishing vector field in a small neighborhood looks like this under a change of coordinates. The strange shaped plug mentioned above is an embedding of (T^2) x ([0,1]^(n-3)) x ([0,1]) into this cube. Thus we describe below a vector field for points (x,y,z) in (T^2) x ([0,1]^(n-3)) x ([0,1]). This vector field has some very important properties: 1. It matches smoothly with the original vector field on the boundary. Since (T^2) x ([0,1]^(n-3)) can be embedded in R^(n-1) for n>3, the z coordinate of the embedded plug corresponds to the last coordinate for the cube. Thus the condition amounts to having z'=1 on the boundary. 2. The vector field has no periodic orbits inside the plug. 3. Any orbit which flows into and out of the plug has the same x and y values on exit as on entrance. 4. There is an orbit which flows into but never out of the plug. Property 1 guarantees that the modified vector field is still smooth. 2 implies that locally there are no periodic orbits, and 3 implies that the new vector field does not create new periodic orbits globally for the manifold, since we chose a neighborhood on which there was only one periodic orbit. In order to do this, Wilson uses a vector field which has a mirror image property; for this vector field this means: for 00 everywhere else. On 1/2 Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 3 May 1994 20:50:07 GMT Lines: 11 Status: OR Web Page by a Sixth Grade Class Some teachers in K-12 are introducing their classes to the World Wide Web (WWW discussed in my article "World Wide Web and Mosaic," geometry.forum, January 27, 1994.). In particular, a few have had their classes create a web page, including a sixth grade class from Hillside Elementary School in Cottage Grove, Minnesota; see the document url hillside.coled.umn.edu . This page also has references to other schools' web pages. Perhaps looking at these pages will be interesting to teachers who are incorporating Internet use into their geometry classes. From sander@geom.umn.edu Tue May 3 19:41:17 CDT 1994 Article: 145 of geometry.announcements Xref: news1.cis.umn.edu geometry.research:181 geometry.announcements:145 Newsgroups: geometry.research,geometry.announcements Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Basic Issues in Computer-Aided Math Visualization: August Seminar Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 3 May 1994 20:54:34 GMT Lines: 216 Status: OR Basic Issues in Computer-Aided Math Visualization August 13-14, 1993 A two-day course at the Geometry Center, University of Minnesota. The Geometry Center is an NSF Science and Technology Research Center whose mission is to foster research in geometry and related fields, and the communication of geometric ideas among mathematicians and to the public, using modern computation and visualization tools. Computer visualization has become an important tool in several fields of mathematics, helping mathematical understanding and communication, the formulation of conjectures, and the development of proofs. "Basic Issues in Computer-Aided Math Visualization" is designed as an introduction to the subject for mathematicians. While standard references such as "Computer Graphics, Principles and Practice" [1,14] provide a detailed background to computer graphics in general, we will concentrate on issues likely to be of particular interest to mathematicians. Here are some sample questions that we will try to address: How can I visualize surface X? or map Y? or the solutions of differential equation Z? How should I spend my hardware grant of X thousand dollars? How can I get hard copy of this image? How much work is it to make a video? The focus will be on how to find out about, evaluate, and use existing software. A certain amount of basic material must be covered first, before we address these specific issues. Also, we will make no attempt to cover the broad subject of volume visualization. The course will include lectures, software demonstrations, a Q&A session, and supervised participant experimentation on graphics workstations. It will be taught by Stuart Levy, Tamara Munzner, and Mark Phillips, three of the creators of Geomview [2]. The instructors, who have extensive experience in mathematical programming and graphics development, will be assisted by a team of skilled undergraduates. Printed notes will be distributed, including a full bibliography. Registration Procedures The workshop will take place from 9:30 a.m. to 5:30 p.m. on Saturday and Sunday at the Geometry Center, 1300 South Second Street on the West Bank Campus. Enrollment is limited to 60 participants, on a first-come-first-served basis, due to the largely interactive nature of the course. For further information and an application form, see the April, 1994 issue of the Notices of the AMS. Preliminary Syllabus Note: In some cases, a selection of topics will be made depending on the audience's interest. Going from the Mathematics to Helpful (or Pretty) Pictures "I loved the picture on the cover of 'Scientific American' [12]. How can I make pictures like that?" Much depends on what you want a picture of, but there are some general principles. We distinguish two steps: going from the mathematical object to a low-level description in terms of "primitives", and generating an image from that description. We will use the words modeling and rendering for these two steps. Using a number of software systems, both locally developed and commercial [8,9,10], we will exemplify the modeling process for specific problems: parametrically defined surfaces implicitly defined surfaces curves and tubing maps from C to C and R^2 to R^2 vector fields (cf. [16]) solutions of differential equations higher-dimensional objects (cf. [11,12]) Basic 3D Graphics We can only give a brief outline of the main ideas relevant to three-dimensional computer graphics. Many of these ideas will be illustrated using Geomview [2], the locally developed interactive 3D viewer. modeling, animation, and rendering scene description: geometry, cameras, lighting, shading, shadows low-level geometric "primitives": points, lines, polygons, meshes, Bezier patches; 3D geometric data standards 4 by 4 matrix notation for 3D transformations; application to non-Euclidean geometries [3,4] interactive rendering versus photorealistic, offline rendering [7] three-component color theory; additive versus subtractive color; the color triangle [5] Graphics Hardware "What can I buy with my $3,000 grant?" (Or $10K, or $30K, or $100K.) general overview of existing hardware as we know it; graphics versus general-purpose workstations special-purpose hardware for graphics computation (hidden-surface elimination, projective transformations, shading) buffering, color depth, other desirable characteristics Still Pictures Mathematicians can now use computers to generate pictures that would be tedious or impossible to generate by hand. However, the process of approximating a picture seen on a computer monitor on a transparency or in the pages of a journal is nontrivial. 2D standards: bitmaps (TIFF, GIF, etc.) vs. page description (PostScript) preparing images for publication color printers Video Animation While video is less immersive than interactive software, it is much more portable. It can be prohibitively difficult or expensive to recreate the hardware and software environment necessary to show or use an interactive program, whereas arranging for a VCR and monitor is usually easy. The Center has produced many videos along a continuum ranging from multi-year efforts aimed at a broad audience and distributed widely [15] to "video overheads" intended for use during a talk by the author, made in a matter of hours. common pitfalls; predictable time sinks; useful caveats real-time vs. frame-by-frame recording video recording hardware References [1] James Foley, Andries van Dam, Steven Feiner, and John Hughes, "Computer Graphics, Principles and Practice", 2nd ed., Addison-Wesley,1990. [2] Mark Phillips, Silvio Levy, and Tamara Munzner, "Geomview: An Interactive Geometry Viewer", Notices of the Amer. Math. Soc. (1993), 985-988. [3] Charlie Gunn, "Discrete Groups and Visualization of Three-Dimensional Manifolds", SIGGRAPH 93 (Anaheim, CA, August 1-6, 1993), ACM (1993), 255-262. [4] Mark Phillips and Charlie Gunn, "Visualizing Hyperbolic Space: Unusual Uses of 4 by 4 Matrices", 1992 Symposium on Interactive 3D Graphics, ACM (1992), 209-214. [5] R. W. G. Hunt, "The reproduction of colour", 3rd ed., Wiley, New York, 1975. [6] Alan Watt and Mark Watt, "Advanced Animation and Rendering Techniques", Addison-Wesley, 1993. [7] Steve Upstill, The RenderMan Companion, Addison-Wesley, 1990. [8] Stephen Wolfram, "Mathematica, A System for Doing Mathematics by Computer", 2nd ed., Addison-Wesley, 1991. [9] Bruce W. Char, et al. "Maple V Language Reference Manual," Springer, Waterloo, Ont., 1991. [10] Kenneth A. Brakke, "The Surface Evolver", Experimental Mathematics, Vol. 1 (1992), 141-165. [11] A. J. Hanson and R. A. Cross, "Interactive Visualization Methods for Four Dimensions", Proceedings of Visualization '93, (San Jose, CA, Oct 25--29, 1993), IEEE Computer Society Press, 1993. [12] John Horgan, "The (Purported) Death of Proof", Scientific American, October, 1993. [13] John Sullivan, "Generating and Rendering Four-Dimensional Polytopes", The Mathematica Journal (Winter, 1991), 76-85. [14] David F. Rogers and J. Alan Adams, "Mathematical Elements for Computer Graphics", 2nd ed., McGraw-Hill, 1990. [15] Charlie Gunn and Delle Maxwell, "Not Knot (video and printed supplement)", A K Peters, 1991. [16] Brian Cabral and Leith Leedom, "Imaging Vector Fields Using Line Integral Convolution", SIGGRAPH Proceedings 1993, ACM, 263-270. This announcement originally appeared in the April, 1994 Notices of the American Mathematical Society. For an application form, please see the Notices. From sander@geom.umn.edu Tue May 10 14:17:25 CDT 1994 Article: 148 of geometry.announcements Xref: news3.cis.umn.edu geometry.announcements:148 Newsgroups: geometry.announcements Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Forum Articles in Print Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 10 May 1994 16:22:57 GMT Lines: 24 Status: OR Evelyn Sander's Forum Articles in Print A complete collection of my articles for the Geometry Forum is now available in book form, as part of the Geometry Center's preprint series. It is free and available to anyone who requests it (see below). The book includes a table of contents, numbered pages, a partial annotated list of articles, and all referenced figures. I have chosen to make the articles available in print for two reasons. First, as a convenience to people who have already read the articles electronically. Second, so that people who do not have access to the Internet have a chance to read the articles. For a copy, request Geometry Center Preprint Series GCG69. They are just binding them now, so it may take a few weeks before you receive anything. Write to: The Geometry Center 1300 South Second Street, Suite 500 Minneapolis, Minnesota 55454 or email admin@geom.umn.edu From sander@geom.umn.edu Fri May 20 11:21:30 CDT 1994 Article: 163 of geometry.college Xref: news1.cis.umn.edu geometry.pre-college:644 geometry.college:163 Newsgroups: geometry.pre-college,geometry.college Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Curvature of Lettuce Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 16 May 1994 19:24:01 GMT Lines: 79 Status: OR Curvature of Lettuce Jeff Weeks of the Geometry Center is has been touring the state of Minnesota this week as part of the Minnesota Mathematics Mobilization. This article describes his talk at the University of Minnesota focusing on math education. In accordance with Weeks educational values, "examples before theory," most of this article describes the illustration that Weeks gave of his teaching method. This illustration is a lecture on the Gauss-Bonnet theorem called "Curvature of lettuce." "My lecture is about curvature. The sphere is an example of a shape with positive curvature. The standard Euclidean plane is an example with zero curvature. A saddle has negative curvature. In order to give you an idea of how to calculate curvature, I have here three kinds of lettuce. I'm going to pass them out to everyone, and I want you to determine their curvature." At this point everyone received a piece of leaf lettuce, romaine lettuce, or cabbage and a pair of scissors. "Here is how you calculate curvature: cut a shape from your lettuce leaf. Now cut a narrow band of lettuce from the outside edge of your shape. The band will look like a fattened curve, but it is no longer closed since you need to slit it somewhere to cut it from your shape. Put the band on your desk and flatten it as much as possible. Measure the angle through which a tangent vector turns as it goes from one end of the curve to another. The curvature equals 2*pi - that angle." Everyone now cut out their lettuce and marked their results on a table labeled as follows: Cabbage Romaine Leaf Lettuce Curvature Area of the shape The results were that everyone measured cabbage with positive curvature, romaine around zero, and leaf lettuce negative. Further, Weeks pointed out a slight increase in the amount of curvature when the area increased. Although it was not a particularly noticeable difference, it provided motivation for the Weeks' next proposition, for which he then gave a justification; namely, the curvature of the union of two areas is the sum of their curvatures. Using this proposition, Weeks was able to talk about the curvatures of closed surfaces. "We define the total curvature of a closed surface by triangulating the surface and summing the curvatures for each element in the triangulation. What do I need to do to show that this is a good definition?" Someone in the audience suggested that we should check whether the definition gave the same answer for every triangulation. Weeks then proved this property. He allowed the audience to make suggestions on how to proceed with the proof. At this point, Weeks calculated the curvature of some basic examples: sphere, the torus, and a genus two surface. He then calculated the curvature of any genus g surface in two different ways; in one way he showed that for a triangulation of a genus g surface, the number of faces - number of edges + number of vertices is the total curvature/(2*pi). The quantity (the number of faces - number of edges + number of vertices) is called the Euler characteristic for a triangulation. Calculating curvature in another way, Weeks showed that the total curvature is 2(1-g). Putting these facts together, the Euler characteristic is 2(1-g) and is therefore a quantity independent of the triangulation. In addition, 2*pi times the Euler characteristic is the total curvature. This is a special case of the Gauss-Bonnet theorem. "This is the end of the lecture within the lecture. Now let me ask you something: did I give you a rigorous proof of the Gauss-Bonnet theorem?" The audience agreed that it was not a strict mathematically rigorous proof. In fact Weeks never even gave a precise definition of curvature. However, people felt they had a better understanding of the theorem than if he had just given formal definitions and proof. "Should we remove rigor altogether? My answer is no, we should not remove the formal proofs. After all, I had to work a bit to give you a non rigorous proof of a theorem that is true. If I worked harder, I could give such a proof for a theorem that wasn't true! Thus I believe that we should put the examples and ideas first. Then, when the students really understand the ideas, we can give them the formal proofs." Jeff Weeks' talk took place May 13, 1994. From sander@geom.umn.edu Fri May 27 16:54:12 CDT 1994 Article: 180 of geometry.puzzles Xref: news6.cis.umn.edu geometry.puzzles:180 Newsgroups: geometry.puzzles Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Photograph Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Thu, 26 May 1994 18:21:58 GMT Lines: 9 Status: OR Photograph Puzzle Here is part one of a two part puzzle, to get you warmed up: You have a photograph of a person in front of a sunset over the sea. Relative to the person in the picture, how tall is the photographer? Don't post an answer until Monday to give everyone a chance to think about it. From sander@geom.umn.edu Wed Jun 1 15:03:18 CDT 1994 Article: 664 of geometry.pre-college Xref: news1.cis.umn.edu geometry.puzzles:181 geometry.pre-college:664 Newsgroups: geometry.puzzles,geometry.pre-college Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Photograph Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Tue, 31 May 1994 15:00:01 GMT Lines: 31 Status: OR Photograph Answer and Puzzle Part 2 First puzzle: You have a photograph of a person in front of a sunset over the sea. Relative to the person in the picture, how tall is the photographer? The answer: Although it looks impossible at first, this problem does indeed have a solution. The important fact here is that the horizon is always at eye level. This is because we can assume the earth is flat. If you look up, you see the sky. If you look down, you see the ground. Thus to see the point transition between earth and sky, the horizon, you must look exactly horizontally. Assuming that the photographer holds the camera at eye level to take a picture, the placement of the horizon in the picture is the height of the photographer's eyes. Thus if the horizon is above the eyes of the person in the picture, the photographer is taller, whereas if the horizon is below the eyes of the person in the picture, the photographer is shorter. Second puzzle: Now that you have tried that one, here is a harder question. You have a photograph of a person standing on a long perfectly straight road, which you can see all the way to the horizon. In the photograph you can see two mile markers. How do you work out how far the person is from the first mile marker? From sander@geom.umn.edu Fri Jun 10 15:21:32 CDT 1994 Article: 149 of geometry.announcements Xref: news5.cis.umn.edu geometry.announcements:149 Newsgroups: geometry.announcements Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: New Forum Writer Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 10 Jun 1994 19:56:23 GMT Lines: 19 Status: OR As I have received a summer research fellowship, after next week I will not write for the Geometry Forum until fall. There are a number of articles that I hope to write up and post in the next week, having gotten bogged down in studying lately. (The studying was successful; I passed my oral qualifying exam for the PhD on Wednesday.) May each of you have a happy and productive summer. I very much enjoy writing for the Forum and will look forward to doing so again in three months. In the mean time, let me introduce Bob Hesse, a fellow University of Minnesota graduate student who will take my position for summer and split it with me next year. Below is a note from Bob introducing himself to the Forum. Evelyn Sander Hi, my name is Bob Hesse and I will be continuing Evelyn Sander's work this summer. I have just completed my third year as a mathematics graduate student at the University of Minnesota. I am excited about writing for the Forum, and I am looking forward to hearing from you. From sander@geom.umn.edu Mon Jun 13 11:36:16 CDT 1994 Article: 700 of geometry.pre-college Xref: news5.cis.umn.edu geometry.puzzles:188 geometry.pre-college:700 Newsgroups: geometry.puzzles,geometry.pre-college Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Photograph Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: julia.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: <9405311823.AA11542@ginger.princeton.edu> Distribution: inet Date: Fri, 10 Jun 1994 20:28:26 GMT Lines: 76 Status: OR I posted this a while ago, but it seems to have gotten lost on the way to the newsgroup. Photograph Answer and Puzzle Part 2 First puzzle: You have a photograph of a person in front of a sunset over the sea. Relative to the person in the picture, how tall is the photographer? The answer: Although it looks impossible at first, this problem does indeed have a solution. The important fact here is that the horizon is always at eye level. This is because we can assume the earth is flat. If you look up, you see the sky. If you look down, you see the ground. Thus to see the point transition between earth and sky, the horizon, you must look exactly horizontally. Assuming that the photographer holds the camera at eye level to take a picture, the placement of the horizon in the picture is the height of the photographer's eyes. Thus if the horizon is above the eyes of the person in the picture, the photographer is taller, whereas if the horizon is below the eyes of the person in the picture, the photographer is shorter. Second puzzle: Now that you have tried that one, here is a harder question. You have a photograph of a person standing on a long perfectly straight road, which you can see all the way to the horizon. In the photograph you can see two mile markers. How do you work out how far the person is from the first mile marker? Answer to the second puzzle: This answer has to do with the invariance of cross ratios. Thus I make a small digression to say what cross ratios are. This treatment is based on the beautiful treatment in "What is Mathematics" by Courant and Robbins. In the plane, a point p determines a map between two lines called the projection map. To project point x on line L1 to a point on line L2, draw a line through p and x and see where it hits L2. See Figure 1. What quantities are invariant for all projections from L1 using p? The cross ratio is such a quantity; given points A,B,C,D on a line, their cross ratio is (CA/CB)/(DA/DB). If A2,B2,C2,D2 are the projections of A,B,C,D respectively, then the cross ratio of these points is equal to the cross ratio of A,B,C,D. For a proof, see Figure 2. Back to the puzzle. The camera is projecting points along a straight road onto the straight image of the road on the film. We know the placement on the photograph of the person's image, mile marker 1's image, mile marker 2's image, and the horizon's image. We know the distance between the actual mile markers. We want to find the distance from the person to the first mile marker. Call the points on earth P (the person), M1 (mile marker 1), M2 (mile marker 2), infinity. Call the images of these points IP,IM1,IM2, and IH (the image of the horizon) respectively. See Figure 3. Now set up a cross ratio; in the special case D is a point at infinity, just take a limit and get the cross ratio of A,B,C,D to equal (CA/CB). Thus: [dist(M2,P)/dist(M2,M1)] = [dist(IM2,IP)/dist(IM2,IM1)]/[dist(IH,IP)/dist(IH,IM1)]. Everything on the right hand side is known: just measure the distances in the photograph. dist(M2,M1) is known since the mile markers are marked with distances. Thus we have a formula for dist(M2,P). So the distance from the person to the first mile marker is just dist(M2,P) - the distance between the two mile markers. So we are done! Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/photo_puzzle directory. If you have Mosaic, you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum.html Thank you to Oliver Goodman for his help on these puzzles. From sander@geom.umn.edu Fri Jun 17 12:31:55 CDT 1994 Article: 167 of geometry.college Xref: news5.cis.umn.edu geometry.pre-college:720 geometry.college:167 Newsgroups: geometry.pre-college,geometry.college Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Science Museum Math Exhibit Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: eudoxus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 17 Jun 1994 17:01:46 GMT Lines: 89 Status: OR Science Museum Math Exhibit Geometry Center staff collaborated with the Science Museum of Minnesota to produce a museum exhibit on triangle tilings. Starting with a module for Geomview written by Charlie Gunn, staff members Tamara Munzner and Stuart Levy, with assistance from Olaf Holt, worked with exhibit developers at the museum to make software and explanations which are accessible and interesting to the general public. This is an especially difficult task at a museum; the average length of stay at the exhibit is only about five minutes. Despite this, the Geometry Center and museum collaborators managed to create an exhibit which contains sophisticated concepts such as tilings of the sphere and the relationship between tilings and the Platonic and Archimedean solids. Here is a brief description of the exhibit. The sum of the angles of a planar triangle is always 180 degrees. Repeated reflection across the edges of a 30,60,90 degree triangle gives a tiling of the plane, since each angle is an integral fraction of 180 degrees. Figure 1 shows the exhibit's visualization of these ideas. What about a triangle whose angles add up to more than 180 degrees? Such triangles exist on the sphere. Whenever the angles of such a spherical triangle are integral fractions of 180 degrees, repeated reflections across the edges give a tiling of the sphere. The exhibit shows this for triangles with the first two angles always 30 and 60 degrees, and the third angle selected as 45, 36, or 60 degrees. See figure 2. A spherical triangle which tiles and a point of the triangle, called the bending point, uniquely determine an associated polyhedron as follows. Repeated reflection through the edges of the triangle gives a tiling of the sphere, each tile of which contains a reflected version of the bending point. These bending point reflections are the vertices of the associated polyhedron. The edges are chords joining each bending point and its mirror images. The faces are planes spanning the edges. Associated to the spherical triangle which tiles the sphere, there is a flattened triangle which tiles the polyhedron. The bending point is the only point of the flattened triangle which is still on the sphere. See figure 3. Also compare the spherical tiling with marked bending point in figure 2 with the associated polyhedron in figure 4. Tilings of the sphere and polyhedra visually demonstrate the idea of a symmetry group. Each choice of angles for the base triangle selects a different symmetry group. Reflections across the edges of the base triangle are the generators of the group. In the language of group theory, the vertices are images of the bending point under the action of the group. The choices of group and bending point completely determine the polyhedron. The exhibit software allows the viewer to move the bending point to see how the resulting polyhedron changes. For particular choices of bending point, the resulting polyhedra are Platonic and Archimedean solids. See figure 4. The software allows viewers to see the relationship between these polyhedra more easily than would a set of models. Using the mouse, viewers can watch the polyhedron change as they drag the bend point. Thus they can begin to understand the idea of duality of Platonic solids, as well as the idea of truncation to form Archimedean solids. The triangle tiling exhibit is currently on view at the Science Museum of Minnesota. In addition to the software, the exhibit contains books and posters explaining the software, toys for constructing Platonic and Archimedean solids, and other gadgets useful for understanding the ideas of tilings of the sphere. For example, the exhibit includes a set of mirrored triangular tubes, each of which contains a spherical or flattened triangle. The mirrored walls make it appear as though inside each tube there is a sphere or a polyhedron. This gives a physical demonstration that repeated reflections of some spherical triangles tile the sphere, and repeated reflections of certain flattened triangles result in the Platonic and Archimedean solids. The triangle tiling exhibit is successful with museum visitors; around 2500 people use it each week. In addition, the exhibit has been accepted for display at the annual meeting of the computer graphics organization SIGGRAPH. It will be part of graphics display called The Edge. (For more about SIGGRAPH, see Evelyn Sander, "SIGGRAPH Meeting," geometry.college, 17 August, 1993.) This article is based on an interview with Tamara Munzner and a visit to the Science Museum of Minnesota. Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/museum.exhibit directory. If you have Mosaic,you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum.html. If you are interested in trying the software, which only works on an SGI, a version is available by anonymous ftp from geom.umn.edu as priv/munzner/tritile.tar.Z . The exhibit can easily be duplicated at other science museums. If interested, contact Munzner (munzner@geom.umn.edu). From sander@geom.umn.edu Fri Jun 17 12:32:08 CDT 1994 Article: 168 of geometry.college Xref: news5.cis.umn.edu geometry.college:168 Newsgroups: geometry.college Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Energies of Hopf Links Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: eudoxus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 17 Jun 1994 17:22:13 GMT Lines: 68 Status: OR Energies of Hopf Links Rob Kusner of the University of Massachusetts and John Sullivan of the University of Minnesota have been at the Geometry Center for the last month to work on energy minimization in the study of knots and links. Basically, minimal energy configurations are those for which parts of the knot or link stay far apart. For example, an unknotted loop has as its minimal energy a round circle. They hope that looking at energies will give a way to distinguish between different knots and links. A particularly nice class of links is the class Hopf links. These are links composed of particular great circles on the three-sphere which come from the Hopf fibration, defined as follows. The three-sphere is embedded in four-space. Regarding four-space as complex two-space, Hopf circles are the intersection of the sphere and the complex linear subspaces (two-dimensional real subspaces). Equivalently, they are points on the three-sphere such that the ratio of the two complex coordinates is constant. Hopf circles turn out to be great circles of the three-sphere. In fact, each point on the three-sphere is contained in exactly one Hopf circle. This division into Hopf circles is called the Hopf fibration. Any two Hopf circles link exactly once. (Their linking number is +1, giving the circles the same orientation as the complex linear subspaces in which they lie.) Thus a collection of Hopf circles form a link. For Hopf links, the associated energy depends only on the distance between loops in four-space, since each loop in the link is itself a round circle. The energy Kusner and Sullivan use is proportional to one over the square of the distance. However, it is not necessary to work with the three-sphere in four-space to compute this quantity; for each Hopf circle, there is a uniquely defined point on the two-sphere. This is the point on the Riemann sphere given by the constant ratio of the two complex coordinates for points on a Hopf circle. Minimizing the associated energy for configurations of points on the two-sphere is equivalent to looking at the original energy function for Hopf links on the three-sphere. The associated energy function for points on the two-sphere turns out to be the same as the classical Coulomb's law energy between charged particles, in other words, inversely proportional to the distances between points in three-space. Some classical physics papers discuss the energy minimization problem of point charges on a sphere. However, work on the problem stopped around 1912 due to the discovery of quantum mechanics. Here are the first four particle minimal energy configurations; for two particles, antipodal points have minimal energy. Assuming that they are at the north and south poles, the corresponding link consists of loops which are the intersections of the three-sphere with the (z,0)- and the (0,w)-complex lines. For three points, the minimum energy occurs when the three are evenly spaced on a great circle of the sphere. Four points have minimum energy when placed in a regular tetrahedron, but there is another critical configuration; can you find it? The minimum energy configurations are known for up to six particles. For larger numbers of points, little is known about critical configurations and the number of stable configurations (local minima). Last summer Kusner and Sullivan discovered, using Ken Brakke's program Surface Evolver, that there are two different stable configurations of sixteen particles. One appears to be the global minimum; the other has slightly more energy. Kusner and Sullivan are currently using Morse theory to try to classify all the critical configurations of charges on the sphere up through sixteen particles. This article is based on an interview with Rob Kusner. Kusner and Sullivan's paper about knot and link energies is available on the University of Massachusetts World-Wide Web using Mosaic. Look in the url: www.gang.umass.edu . From sander@geom.umn.edu Tue Nov 1 10:57:38 CST 1994 Article: 157 of geometry.announcements Xref: news6.cis.umn.edu geometry.announcements:157 Newsgroups: geometry.announcements Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: John Nash shares Nobel prize Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Wed, 26 Oct 1994 22:11:32 GMT Lines: 34 Status: OR Here is a forwarded message from Al Marden: [An old timer told of `Nash`s paradox` which was discussed around Princeton in the 1950s. In a world of two people, one had a house, and the other who wanted the house, had a bag of dynamite. In math, Nash`s main fame is for the Nash embedding theorem. Perhaps someone can expand on the story of John Nash.] TIDBITS #22 21 OCTOBER 1994 MATHEMATICAL SCIENTIST SHARES 1994 NOBEL PRIZE IN ECONOMICS John Nash, Visiting Research Collaborator in the Mathematics Department at Princeton University, was awarded the 1994 Nobel Prize in Economic Sciences last week, sharing the prize with John Harsanyi of Berkeley and Reinhard Selten of the University of Bonn. They won "for their pioneering analysis of equilibria in the theory of non-cooperative games." Nash put forth his key idea--the Nash equilibrium--in his 1950 PhD thesis, which defined a new concept of equilibrium and used methods from topology to prove the existence of an equilibrium point for n-person, finite, non-cooperative games. Nash's proof has had a major impact on economic theory, making it possible in some cases to predict the strategies that economic actors will adopt in the long run, at the Nash equilibrium point at which no player can change to improve his or her outcome. Nash was a Westinghouse scholar, receiving undergraduate and Masters degrees in three years, 1945-1948, at the Carnegie Institute of Technology (now Carnegie-Mellon University). He received his PhD in mathematics from Princeton two years later. From sander@geom.umn.edu Tue Nov 1 10:57:21 CST 1994 Article: 156 of geometry.announcements Xref: news6.cis.umn.edu geometry.announcements:156 geometry.research:274 Newsgroups: geometry.announcements,geometry.research Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Fermat's Last Theorem Finally Proved Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Wed, 26 Oct 1994 22:03:23 GMT Lines: 44 Status: OR I hope that I am not repeating what has already appeared in the Forum. I recieved the following message: >From rubin@math.harvard.edu Tue Oct 25 09:58 CDT 1994 Date: Tue, 25 Oct 94 10:24:46 EDT From: rubin@math.harvard.edu (Karl Rubin) Subject: update on Fermat's Last Theorem As of this morning, two manuscripts have been released Modular elliptic curves and Fermat's Last Theorem, by Andrew Wiles Ring theoretic properties of certain Hecke algebras, by Richard Taylor and Andrew Wiles. The first one (long) announces a proof of, among other things, Fermat's Last Theorem, relying on the second one (short) for one crucial step. As most of you know, the argument described by Wiles in his Cambridge lectures turned out to have a serious gap, namely the construction of an Euler system. After trying unsuccessfully to repair that construction, Wiles went back to a different approach, which he had tried earlier but abandoned in favor of the Euler system idea. He was able to complete his proof, under the hypothesis that certain Hecke algebras are local complete intersections. This and the rest of the ideas described in Wiles' Cambridge lectures are written up in the first manuscript. Jointly, Taylor and Wiles establish the necessary property of the Hecke algebras in the second paper. The overall outline of the argument is similar to the one Wiles described in Cambridge. The new approach turns out to be significantly simpler and shorter than the original one, because of the removal of the Euler system. (In fact, after seeing these manuscripts Faltings has apparently come up with a further significant simplification of that part of the argument.) Versions of these manuscripts have been in the hands of a small number of people for (in some cases) a few weeks. While it is wise to be cautious for a little while longer, there is certainly reason for optimism. Karl Rubin From sander@geom.umn.edu Tue Nov 15 13:56:01 CST 1994 Article: 229 of geometry.puzzles Xref: news3.cis.umn.edu geometry.pre-college:1052 geometry.puzzles:229 Newsgroups: geometry.pre-college,geometry.puzzles Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Geometry Games Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: archimedes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 14 Nov 1994 20:37:58 GMT Lines: 137 Status: OR Geometry Games The following article describes software written for the Macintosh by Jeff Weeks. It also includes puzzlers which can be solved without the software. All of the programs are available free using anonymous ftp. See the end of the article for details. "Explaining simple closed manifolds such as the two-torus, the Klein bottle, and three-torus becomes easier," says Jeff Weeks, "if students get a good gut-level understanding by playing games in them. The students experience the manifolds first, and only later discuss them theoretically." To this end, Weeks has written programs to allow students to get an intuition for some two- and three-dimensional manifolds. He has used the programs with high school and college students and says they make a good introduction to simple two- and three-dimensional manifolds. Here is a brief description of three of them, including some puzzlers for torus and Klein bottle chess. 1. Flight Simulator This is a program to learn about three-dimensional manifolds. It shows what you see when you travel in the three-sphere, projective three-space, or many other three-manifolds. You can travel any direction you want using the mouse. For example, Figure 1 shows a cube as viewed in the three-torus. (See the program's About Box, the first item in the apple menu, for flying instructions.) 2. Hyperbolic MacDraw This program allows the user to draw lines, draw shapes, rotate, and translate, in the same way as any draw program, except that rather than Euclidean space, the drawings appear in the user's choice of Poincare, Klein, or upper half plane model of the hyperbolic plane. The user can discover the many oddities of hyperbolic space. For example, the user immediately notices that in some models of the hyperbolic plane, lines appear straight, whereas in other models, they appear curved. Figure 2 shows two regular polygons drawn using Hyperbolic MacDraw in the Poincare model. The sides are equal length and straight in the hyperbolic plane. When students first wrote Hyperbolic MacDraw, as a term project in Weeks' undergraduate Geometry class, they came to Weeks and said, "The translation seems to work, except that when you translate an object around a loop in the plane, when it gets back to where it started, it seems to have rotated. We can't find the bug in our program." The program was working correctly. The students had discovered one of the odd properties of curved spaces. For example, in Figure 3b, the object in Figure 3a has been translated around a loop. In the hyperbolic plane, when a translated object returns to its starting point, it has rotated by the amount of area enclosed by its path. This is true on a sphere as well, except that on the sphere the object rotates in the same direction that you transversed the loop, whereas in hyperbolic plane, the rotation is in the opposite direction. 3. Torus and Klein bottle chess One way to represent the torus or Klein bottle is as a square with opposite sides identified as in Figure 4. This chess program presents a chess board with sides identified in this fashion. The players can move a chess piece off one side of the board, at which point it appears on the other side of the board. There are two alterations to the rules for standard chess. One, the initial position is as shown in Figure 5. It is different from the starting position for standard chess, since with the usual starting point on the torus or Klein bottle, the game would begin with massive slaughter. Second, in this game pawns can move one square in a straight line in any direction and capture diagonally in any direction. Figures 6a and 6b show a possible rook move on a Klein bottle board, to give you an idea of how the play looks. Figure 6c shows the same board as in 6b, except that the board has been scrolled. Here are some torus and Klein bottle chess puzzlers: (1) In Klein bottle chess, starting from the initial position in Figure 5, can the white bishop capture the black rook in one move? (2) What happens when you move a piece straight into a corner in torus chess? In Klein bottle chess? (3) Starting from the initial position in Figure 5, how can a black knight capture a white rook in one move? Note that that one of the white knights in Figure 6c appears to be mirror-reversed >from the way it was in Figure 6b. Similarly, with the board in its initial position, the black knight appears to get mirror-reversed as it captures the white rook. But if you first scroll the board down a few rows, then the black knight does not appear to get reversed as it captures the white rook. Does it get reversed or doesn't it? Here are some more involved questions for chess experts: (A) In traditional chess, a pawn is worth 1, a knight or bishop 3, a rook 5 and a queen 9. How do these relative values change on a torus? On a Klein bottle? (B) Experiment with different starting positions for the pieces. Send particularly good starting positions to weeks@geom.umn.edu. To learn more about the manifolds in the flight simulator and torus chess, see Jeff Weeks, "The Shape of Space," Marcel Dekker, Inc, New York, 1985. For more on hyperbolic space, see two previous articles on the Forum, Evelyn Sander, "Hyperbolic.m: A general-purpose hyperbolic geometry package," geometry.college, February 1, 1994 and "Models of the Hyperbolic Plane," geometry.college, February 11, 1994. All puzzlers are by Jeff Weeks. Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/weeks_software directory. If you have Mosaic, you can also read the http version of this article with figures included. It is a separate posting or it is available in the url http://www.geom.umn.edu/docs/forum/weeks_software/weeks_software.html All software is available by anonymous ftp from geom.umn.edu in the file geometry_games.sea.hqx in the pub/software directory. From sander@geom.umn.edu Wed Nov 30 10:03:06 CST 1994 Article: 225 of geometry.college Xref: news5.cis.umn.edu geometry.college:225 geometry.puzzles:231 Newsgroups: geometry.college,geometry.puzzles Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Circle Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Mon, 28 Nov 1994 20:29:00 GMT Lines: 27 Status: OR Circle Puzzle Can you find a way to cover all of R^3 with disjoint geometric circles? In other words, can you place disjoint geometric circles in R^3 so that every point in R^3 is on one and only one of the geometric circles? A geometric circle is the set of points in a fixed plane that lie a fixed positive distance from a center point that lies in the same plane. (Notice that circle refers to the curve, and not to the disk it bounds.) To give the idea, here is a way to fill all of R^3 except the vertical axis: label the vertical axis the z-axis. At each fixed height (z = constant), put a circle of every positive radius centered at the z-axis. Since circles of radius of zero are not allowed, this set of circles does not cover the z-axis. In this example, nearby points in R^3 (off the z-axis) are contained in circles with nearby centers and nearby radii -- in other words this example shows a continuous method of filling R^3 minus the z-axis with disjoint geometric circles. However, for this puzzle, it is not necessary to fill R^3 in a continuous manner. In fact, the solution that I give will have some discontinuities. A solution to follow in a few days. From sander@geom.umn.edu Thu Dec 1 11:03:34 CST 1994 Article: 229 of geometry.college Xref: news6.cis.umn.edu geometry.college:229 geometry.puzzles:233 Newsgroups: geometry.college,geometry.puzzles Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Circle Puzzle: Solution Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Wed, 30 Nov 1994 16:20:23 GMT Lines: 71 Status: OR (Previously Posted) Circle Puzzle Can you find a way to cover all of R^3 with disjoint geometric circles? In other words, can you place disjoint geometric circles in R^3 so that every point in R^3 is on one and only one of the geometric circles? A geometric circle is the set of points in a fixed plane that lie a fixed positive distance from a center point that lies in the same plane. (Notice that circle refers to the curve, and not to the disk it bounds.) To give the idea, here is a way to fill all of R^3 except the vertical axis: label the vertical axis the z-axis. At each fixed height (z = constant), put a circle of every positive radius centered at the z-axis. Since circles of radius of zero are not allowed, this set of circles does not cover the z-axis. In this example, nearby points in R^3 (off the z-axis) are contained in circles with nearby centers and nearby radii -- in other words this example shows a continuous method of filling R^3 minus the z-axis with disjoint geometric circles. However, for this puzzle, it is not necessary to fill R^3 in a continuous manner. In fact, the solution that I give will have some discontinuities. SOLUTION Here is one solution to the circle puzzle. If you have others, please post them. In the entire solution, when I say circle, I mean geometric circle. With center (x,y,z) = (1/2,0,0), place circles in the xy-plane of radii .5, 1.5, 2.5, ..., i.e. at radius n+0.5, where n is nonnegative integer. Notice that a sphere of any radius centered at (0,0,0) either intersects one of the above circles at two points, or has a point of tangency to each of two of the above circles. Remove from each sphere centered at (0,0,0) the two points that are already on circles. These punctured spheres need to be covered with additional circles. Here is how to cover each of the above twice-punctured spheres with disjoint circles. Place tangent planes to the two removed points. If the planes are parallel, the two points are like the north and south poles of a globe, and you can cover the rest of the sphere with circles of latitude. If the tangent planes are not parallel, they intersect in a line. Call this line the "hinge line." The intersection of the sphere with any plane containing this hinge line is a circle (as long as it is non-empty). In addition, since a line and a point not on the line determine a plane, every point on the sphere is on a unique plane containing the hinge line. Thus this covers each twice-punctured sphere by disjoint circles. The disjoint union of the circles on the xy-plane with center (1/2,0,0) and radius n+0.5 along with the above twice-punctured spheres is all of R^3. Thus all of R^3 is covered with disjoint geometric circles. Thanks to Dan Asimov for telling me about this puzzle and solution. It is discussed in the following book: Halmos, Paul R., "Problems for Mathematicians, Young and Old," Mathematical Association of America, 1991. From sander@geom.umn.edu Sat Dec 3 17:46:34 CST 1994 Article: 231 of geometry.college Xref: news6.cis.umn.edu geometry.college:231 geometry.puzzles:234 Newsgroups: geometry.college,geometry.puzzles Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Circle Puzzle: Solution 2 Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Fri, 2 Dec 1994 17:38:22 GMT Lines: 11 Status: OR Oliver Goodman (oag@mundoe.maths.mu.oz.au) sent me another solution: Take an open spherical ball. Remove a circle of half the radius going through the center and one pole. The intersection of a plane perpendicular to the removed circle and going through the pole is an open disk minus a point. Open disk minus a point can be made out of circles. Since all these open disks minus a point are disjoint we can make the open ball minus a circle out of circles. Putting back the circle we get an open ball plus a point. Join copies of these pole to pole along a line like a string of beads. So now we have exactly covered the union of a line and a collection of touching open balls. Fill the remaining space with circles centered on and perpendicular to the line. From sander@geom.umn.edu Sun Jan 8 17:13:56 CST 1995 Article: 246 of geometry.puzzles Xref: news6.cis.umn.edu geometry.puzzles:246 geometry.research:298 Newsgroups: geometry.puzzles,geometry.research Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Jig Saw Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 3 Jan 1995 20:24:01 GMT Lines: 35 Status: OR Jig Saw Puzzle Bolt a metal rod to the end of the blade of a jig saw so that the rod can pivot in one plane. Here is a sketch of the object: ------------- | ^^^^^^^^^^_|_______________________ | _________________________| | ___________| -------------- saw handle blade metal rod moves <---> pivots in the plane of the picture In the picture the rod and blade are facing to the right. However, notice that with the motor off, since the rod can pivot, if it is not held in this position, it will fall and point downward. What would you expect to happen if with the motor running you point the object so that the rod and the blade face directly upward? Will the rod fall to point downward? Will it continue to point upward direction? Or will it instead point at some angle from the vertical? Consider the case in which the length of the rod is much larger than the gravitational acceleration over the square of the frequency of the jigsaw blade. This puzzle is by no means easy. However, since it is not too hard to visualize what is happening physically, I wanted to give people a chance to at least think about the situation before I say what happens and why. I will post an answer in a few days. Please do not post an answer until then. From sander@geom.umn.edu Fri Jan 13 15:39:20 CST 1995 Article: 253 of geometry.puzzles Xref: news1.cis.umn.edu geometry.puzzles:253 geometry.research:301 Newsgroups: geometry.puzzles,geometry.research Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Jig Saw Puzzle: Solution Message-ID: Summary: Solution and Second Puzzle Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: archimedes.geom.umn.edu Organization: The Geometry Center, University of Minnesota References: Date: Mon, 9 Jan 1995 23:02:07 GMT Lines: 123 Status: OR Solution to the Jig Saw Puzzle and a Second Puzzle Answer: For a sufficiently long rod, the rod will continue to point upward. Why: This is a famous physics problem, but the explanation I give here is mathematical. The equation for the angle x of the rod is: L x'' + (-g + a(t)) x = 0 where L is the length of the rod (if we assume that the mass of the rod is concentrated at the end), g is the gravitational acceleration, and a(t) is the acceleration of the jigsaw blade. We want to show the stability of the equilibrium point where the rod points upward with no angular velocity. Call this equilibrium point E. Note that this corresponds to a point in phase space, the space angular velocity vectors of the form (x,x')=(angle, angular velocity). Since the saw blade moves very quickly up and down, the acceleration of the blade is much larger than g. Therefore we can treat g as a small perturbation. In fact, we show stability in the case of g=0 and moreover, that stability cannot be destroyed by small perturbations, which means stability persists when g is present. Assume that a(t) is constant through each half-period. The gravity-free equation is x" + w^2 x = 0 when accelerated downward, and x" - w^2 x = 0 when accelerated upward. In the second case, this rod behaves like the standard swinging pendulum, in which the equilibrium point at the top of the swing is unstable. Thus the rod accelerates away from the vertical. Since the length of the rod is assumed large compared to the time that the saw blade is accelerated downward, the rod does not move very far during a single half-period of the saw blade. For the first equation, the rod's acceleration is restoring, i.e. it points toward the vertical. Thus the restoring accleration during the down phase competes with the destabilizing acceleration during the up phase. Interestingly, the stabilization wins over destabilization. Here is a geometrical reason behind this phenomenon. The points (x, x') move in ellipses for one half period; we stretch these ellipses into circles by setting Y=x'/w (and keeping X=x). The equations become: X'= w Y *** Y'= -/+ w X. So for half the period now we have the usual rotation with angular velocity w. For the other half period a hyperbolic rotation with the same rate w. That is, the flow moves towards the origin along the line x=y, moves away from the origin along the line x=-y, and all other points move on hyperbolas. How do we combine the information about the two half-periods to get information about the total system? Rather than looking at the entire flow for stability, we can look at only the angular velocity vector of the rod when the saw blade is at its highest point. This turns out to give us enough information to determine stability; let P be the two by two matrix corresponding to the linearized system at the equilibrium point. It gives the angular velocity vector of the rod after the blade goes one period based on the rod's current position. In other words, if T is the period of the blade, P is the matrix such that for the linearized system: (x(T),x'(T))= P (x(0), x'(0)). It turns out that the equilibrium point is stable exactly when all iterates of the matrix P are bounded. P has determinant one, since it must be area preserving since the system is conservative. The key observation is that the composition of the rotation in equation *** and the "equal strength" hyperbolic rotation in equation *** turns every point by a nonzero angle. This is because the angular velocity of the vectors under the rotational flow is w, while the angular velocity of the vectors under the hyperbolic flow is strictly less (except on the x- and y- axes, which are passed intantaneously). This is clear geometrically since the speed of both vector fields is the same, while one is perpenducular to radius vector and the other is not, and thus "wastes" itself on lengthening or shortening rather than just rotating. Thus the composition of the two turns every initial condition by a nonzero amount (less than pi, if the frequency of the jigsaw is high enough). In other words, matrix P for angular velocity of the system after time T has no real eigenvectors and thus must diagonalize to diag(lambda_1,lambda_2) with lambdas on the unit circle. Thus P has all its iterates bounded. Finally, the above property of P is not destroyed by a small perturbation, and thus adding g does not affect stability. Thus E is a stable equilibrium. This article is based on a seminar and interview with RPI professor Mark Levi during his recent visit to the Geometry Center. References to follow with Solution to Puzzle 2. ------------------------------------------------------------------- Part 2 of this answer will address another "puzzle" which appears similar but is actually much more difficult. If you are not an expert in this kind of thing, you may want to just wait for the answer. The answer will in fact only give a general explanation rather than all the specific details. Jigsaw Puzzle 2 What happens as the length of the rod becomes arbitrarily small? Answer to follow. -------------------------------------------------------------------- From sander@geom.umn.edu Tue Jan 17 09:51:12 CST 1995 Article: 259 of geometry.puzzles Xref: news6.cis.umn.edu geometry.puzzles:259 geometry.research:304 Newsgroups: geometry.puzzles,geometry.research Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Re: Jig Saw Puzzle 2: Solution Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities References: Date: Fri, 13 Jan 1995 21:35:37 GMT Lines: 50 Status: OR > Jigsaw Puzzle 2 > > What happens as the length of the rod becomes arbitrarily small? > Answer to follow. > > -------------------------------------------------------------------- Solution to Jig Saw Puzzle 2 What happens as the length of the rod becomes arbitrarily small? As the rod's length vanishes, there are infinitely many regions in which the equilibrium point is stable. Because of this relationship between the differential equation and the the space of two by two matrices with determinant one, the proof of stability relies on the geometry of this space of two by two matrices with determinant one. Every two by two matrix with determinant one can be written as the product of a symmetric matrix and a rotation. Each symmetric matrix is uniquely determined by the expansion direction and the amount of expansion, where the expansion direction is irrelevant if the amount of expansion is one. Thus the set of symmetric matrices is homeomorphic to the open disk. Each rotation matrix is uniquely determined by an angle of rotation. The set of rotation matrices is homeomorphic to the circle. Combining the last two paragraphs, the set of two by two matrices with determinant one is a solid torus. Also notice that if M is any symmetric matrix times a rotation of pi/2, then M to the fourth power is equal to M. Thus at the angle pi/2, every matrix is stable. In fact an open neighborhood of the set of these matrices consists of stable matrices. This means that every time you go around the solid torus, you hit a region of stability. It turns out that if the length goes to zero, the matrix goes around the torus infinitely many times. More precisely, as the length shortens so that the number of vertical moments (i.e. moments during the jigsaw's period when the pendulum becomes vertical) changes by 5 or more, the matrix goes at least once around the torus. Thus as the length of the rod decreases to zero, there are infinitely many regions of stability. As with the previous part, this was based on a seminar, interview, and subsequent correspondence with Mark Levi. References: Jearl Walker's "The flying circus of Physics" discusses the inverted pendulum and has other references. From sander@geom.umn.edu Wed Jan 25 17:34:46 CST 1995 Article: 182 of geometry.announcements Xref: news1.cis.umn.edu geometry.announcements:182 geometry.pre-college:1303 geometry.software.dynamic:31 Newsgroups: geometry.announcements,geometry.pre-college,geometry.software.dynamic Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Torus Chess Updated: 1.1 Available Message-ID: Keywords: Torus Chess Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: descartes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 23 Jan 1995 20:46:58 GMT Lines: 13 Status: OR Torus Chess Updated: Version 1.1 Now Available Jeff Weeks has just made available Torus Chess 1.1. (For more information about Torus Chess, see my previous article "Geometry Games," geometry.puzzles, geometry.pre-college, Mon, 14 Nov 1994.) This version of Torus Chess fixes a problem moving the cursor from one side of the (torus) window to the other. It should work fine with all current Macintosh hardware and system software. It is available for anonymous ftp as geom.umn.edu//pub/software/geometry_games/TorusChess-1-1.hqx From sander@geom.umn.edu Mon Feb 13 09:47:16 CST 1995 Article: 185 of geometry.announcements Xref: news6.cis.umn.edu geometry.announcements:185 geometry.pre-college:1427 Newsgroups: geometry.announcements,geometry.pre-college Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Mac Version of Science Museum Software Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Sat, 4 Feb 1995 04:53:40 GMT Lines: 107 Status: OR Macintosh Version of Science Museum Software This is an announcement that Jeff Weeks has written a Macintosh version of the science museum math exhibit developed at the Geometry Center last year. The program is called KaleidoTile, version 1.0. It is available for anonymous ftp from ftp://geom.umn.edu/pub/software/KaleidoTile I have included a copy of the article I wrote last June about the museum exhibit. Although the SGI version described below is not precisely the same as the Macintosh version, it will give you an idea of what KaleidoTile does. Science Museum Math Exhibit geometry.college, geometry.pre-college, Fri, 17 Jun 1994 Geometry Center staff collaborated with the Science Museum of Minnesota to produce a museum exhibit on triangle tilings. Starting with a module for Geomview written by Charlie Gunn, staff members Tamara Munzner and Stuart Levy, with assistance from Olaf Holt, worked with exhibit developers at the museum to make software and explanations which are accessible and interesting to the general public. This is an especially difficult task at a museum; the average length of stay at the exhibit is only about five minutes. Despite this, the Geometry Center and museum collaborators managed to create an exhibit which contains sophisticated concepts such as tilings of the sphere and the relationship between tilings and the Platonic and Archimedean solids. Here is a brief description of the exhibit. The sum of the angles of a planar triangle is always 180 degrees. Repeated reflection across the edges of a 30,60,90 degree triangle gives a tiling of the plane, since each angle is an integral fraction of 180 degrees. Figure 1 shows the exhibit's visualization of these ideas. What about a triangle whose angles add up to more than 180 degrees? Such triangles exist on the sphere. Whenever the angles of such a spherical triangle are integral fractions of 180 degrees, repeated reflections across the edges give a tiling of the sphere. The exhibit shows this for triangles with the first two angles always 30 and 60 degrees, and the third angle selected as 45, 36, or 60 degrees. See figure 2. A spherical triangle which tiles and a point of the triangle, called the bending point, uniquely determine an associated polyhedron as follows. Repeated reflection through the edges of the triangle gives a tiling of the sphere, each tile of which contains a reflected version of the bending point. These bending point reflections are the vertices of the associated polyhedron. The edges are chords joining each bending point and its mirror images. The faces are planes spanning the edges. Associated to the spherical triangle which tiles the sphere, there is a flattened triangle which tiles the polyhedron. The bending point is the only point of the flattened triangle which is still on the sphere. See figure 3. Also compare the spherical tiling with marked bending point in figure 2 with the associated polyhedron in figure 4. Tilings of the sphere and polyhedra visually demonstrate the idea of a symmetry group. Each choice of angles for the base triangle selects a different symmetry group. Reflections across the edges of the base triangle are the generators of the group. In the language of group theory, the vertices are images of the bending point under the action of the group. The choices of group and bending point completely determine the polyhedron. The exhibit software allows the viewer to move the bending point to see how the resulting polyhedron changes. For particular choices of bending point, the resulting polyhedra are Platonic and Archimedean solids. See figure 4. The software allows viewers to see the relationship between these polyhedra more easily than would a set of models. Using the mouse, viewers can watch the polyhedron change as they drag the bend point. Thus they can begin to understand the idea of duality of Platonic solids, as well as the idea of truncation to form Archimedean solids. The triangle tiling exhibit is currently on view at the Science Museum of Minnesota. In addition to the software, the exhibit contains books and posters explaining the software, toys for constructing Platonic and Archimedean solids, and other gadgets useful for understanding the ideas of tilings of the sphere. For example, the exhibit includes a set of mirrored triangular tubes, each of which contains a spherical or flattened triangle. The mirrored walls make it appear as though inside each tube there is a sphere or a polyhedron. This gives a physical demonstration that repeated reflections of some spherical triangles tile the sphere, and repeated reflections of certain flattened triangles result in the Platonic and Archimedean solids. The triangle tiling exhibit is successful with museum visitors; around 2500 people use it each week. In addition, the exhibit has been accepted for display at the annual meeting of the computer graphics organization SIGGRAPH. It will be part of graphics display called The Edge. (For more about SIGGRAPH, see Evelyn Sander, "SIGGRAPH Meeting," geometry.college, 17 August, 1993.) This article is based on an interview with Tamara Munzner and a visit to the Science Museum of Minnesota. Figures are available by anonymous ftp from forum.swarthmore.edu in the /pictures/articles/museum.exhibit directory. If you have Mosaic,you can also read this article with figures included in the WWW document http://www.geom.umn.edu/docs/forum/forum.html. If you are interested in trying the software, which only works on an SGI, a version is available by anonymous ftp from geom.umn.edu as priv/munzner/tritile.tar.Z . The exhibit can easily be duplicated at other science museums. If interested, contact Munzner (munzner@geom.umn.edu). From sander@geom.umn.edu Tue Feb 14 13:50:49 CST 1995 Article: 247 of geometry.college Xref: news3.cis.umn.edu geometry.college:247 geometry.research:309 Newsgroups: geometry.college,geometry.research Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Implicit Surface Algorithm Message-ID: Summary: Description of Algorithm and Fluid Dynamics Example Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Mon, 13 Feb 1995 15:48:21 GMT Lines: 97 Status: OR Mike Henderson's Implicit Surface Algorithm A standard objective in computational mathematics is to graph an implicitly defined surface. Here is a geometric description of a practical but conceptually elegant continuation algorithm written by Mike Henderson of IBM. Since this is a description of a graphing algorithm, it is more interesting when accompanied by figures. Either read the World-Wide Web version of this article in url: http://www.geom.umn.edu/docs/forum/henderson/henderson.html or ftp the figures from ftp: forum.swarthmore.edu/pictures/articles/henderson Given a point of an implicitly defined surface with full rank Jacobian, the implicit function theorem guarantees that there is some neighborhood in the tangent space that maps onto the surface in a bijective manner. In Riemannian geometry, this map is called the exponential map. Starting with a point on the surface, Henderson's algorithm computes the exponential map in a small elliptic region in the tangent space using many applications of Newton's method. Figure 1 shows a (light yellow) region of the surface x^2+y^2-z=0 computed starting from the (dark blue) tangent plane shown in Figure 2. Starting with the computed region on the surface, the algorithm picks a point on the boundary. It computes the exponential map to the surface mapping into a small neighborhood of this point. Since the point is on the boundary of the computed region, the exponential map always gives new information on the surface. Figure 3 shows two overlapping elliptic regions on the surface from Figure 1. We want the algorithm to keep repeating the process in the previous paragraph until it has computed the entire surface. (intersected with a compact region) However, in order to get new information each time, the algorithm must identify the overlap between the boundary of the new region and the boundary of the previously computed region. This overlap is interior to the entire computed region, namely the union of the old and new regions. To avoid unnecessary calculation, the algorithm must remove this overlap from consideration before finding a new boundary point. Finding and removing the overlap is the most computationally difficult part of using Henderson's method. After this, the algorithm repeats the process in the previous paragraph. This entire procedure continues to repeat as often as necessary until the algorithm finds no more boundary points, at which time it has finished computing the surface. Figure 4 shows the computed part of the surface in Figure 1 after repeating the entire procedure several times. There are many applications for such an algorithm. Here is a description of a problem from fluid dynamics called the Taylor-Couette convection problem. Start with two cylinders, one inside the other; put fluid between them. Keep the outside cylinder fixed; spin the inside cylinder. When the rate of spin is sufficiently slow, the fluid just spins slowly around; this is called Couette flow. More precisely, Couette flow is characterized by saying that the fluid velocity along the axis of the cylinders is zero and the fluid velocity in the radial direction is also zero. Since the outside cylinder does not move, fluid moves more slowly at the outside edge. As the spin rate increases, centrifugal force throws the faster moving fluid at the inside cylinder outwards. This results in a rolling or convection of the fluid in the radial direction. At the bifurcation point at which the fluid ceases to move in Couette flow, two possible kinds of motion can occur together or separately; one possibility is that the fluid moves around two torus-shaped regions around the cylinder; that is, a cross section of fluid moves around two circles in the radial direction. This is what is called a two-convection cell. Another possible fluid motion is a four-convection cell; in other words, the fluid moves radially in four circles in the cross section. Near the bifurcation point from Couette flow, two- and four-cells coexist with different amplitudes, depending on the height and the spin rate of the inside cylinder. To study the bifurcation point, it is informative to plot the amplitudes of the two- and four-cells as a function of the height of the cylinders and the velocity of the inner cylinder. This information is given implicitly using the Navier-Stokes equations for fluid flow. This is an example of a 2-dimensional surface in 4-dimensional space. Figure 5 is a picture of a projection of this surface as computed by Henderson's algorithm. He considers the specific situation of infinitely long cylinders such that the flow periodic in the direction along the axis. The aspect ratio is the ratio of the height of the cylinders to the distance between the two cylinders. The Reynolds number is a non-dimensional way of measuring the speed of the inner cylinder. Namely, R= (angular velocity of inner cylinder)*(radius of inner cylinder)* (distance between cylinders)/viscosity This article is based on Henderson's talk in the University of Minnesota Dynamics and Mechanics seminar on February 2, 1995. I would like to thank him for providing additional advice, as well as preparing the figures for this article. From sander@geom.umn.edu Wed Apr 19 12:22:45 CDT 1995 Article: 367 of geometry.puzzles Xref: news5.cis.umn.edu geometry.puzzles:367 Newsgroups: geometry.puzzles Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Ladder Puzzle Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: dehn.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 18 Apr 1995 20:27:49 GMT Lines: 6 Ladder Puzzle A person is standing 8 feet up on a 12 foot ladder. The ladder slides down the wall. What is the path that the person follows while falling? Evelyn Sander From sander@geom.umn.edu Wed May 3 14:36:48 CDT 1995 Article: 337 of geometry.research Xref: news6.cis.umn.edu geometry.research:337 geometry.college:258 Newsgroups: geometry.research,geometry.college Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Lotka-Volterra Equations Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: dehn.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 2 May 1995 20:58:50 GMT Lines: 95 Lotka-Voterra Equations Through Computer Visualization The simplest model of population growth says that population increase is proportional to the current population. However, this does not take into account that there are limited resources. In addition, there may be many species competing more or less successfully for these same resources; some species may hunt others. To begin to take these factors into account, ecologists came up with the nonlinear model given by the Lotka-Volterra equations. This model consists of a system of ordinary differential equations for the rate of population growth of n interacting species depending on the current population. Using the Lotka-Volterra model, what happens to each species after a long time? Does one species beat out the others and become the only species left? Do the populations of the species remain constant? Mary Lou Zeeman, mathematics professor at the University of Texas, San Antonio and Drew LaMar, undergraduate at U.T. San Antonio, are looking at this question of long term behavior using dynamical systems theory. They recently visited the Geometry Center to use computer visualization to gain some intuition. Here is some of the theory of the Lotka-Volterra differential equations and a description of Zeeman and LaMar's visualizations with accompanying figures. To see the figures, either read the World-Wide Web version of this article, in http://www.geom.umn.edu/docs/forum/lotka/lotka.html, or ftp the figure >from ftp:forum.swarthmore.edu/pictures/articles/lotka We look at the populations of n species as a point in n-dimensional space. The question of eventual behavior becomes a mathematical question; starting at a given point in n-dimensional space, what is the eventual behavior of the solution of the Lotka-Voterra differential equations? This is an n-dimensional system with n^2+n parameters. We wish to know how the dynamics change as the parameters vary. By practical considerations, the origin must be fixed by the Lotka-Volterra flow, since zero population can never result in any positive population; we restrict attention to the positive cone in n-space. When species are strictly competitive, then work of M.W. Hirsch shows that other than the origin, every orbit is asymptotic to an invariant set called the carrying simplex. This carrying simplex is a hypersurface homeomorphic to the standard unit simplex in n-space via radial projection. Thus, since the carrying simplex is globally attracting, understanding the dynamics on it will show the eventual behavior of every initial point. There is at most one fixed point with all coordinates positive. Call this fixed point P. It lies on the carrying simplex. In some cases it is known that there are no other recurrences except on the axes. Mary Lou Zeeman and Christopher Zeeman have proved that any time that the carrying simplex is convex or weakly convex (i.e. the carrying simplex lies all to one side of it's the tangent plane at the fixed point), there can be no periodic orbits, and the fixed point P is the only recurrence off of the axes. In this case, if P is an unstable fixed point, then ecologically, one species eventually dominates. In some non-convex cases, it is known that there is a periodic orbit on the carrying simplex. For example, M.L. Zeeman has proved that for certain paths in parameter space, there is a Hopf bifurcation from the fixed point P. Zeeman and LaMar are investigating the transition from no periodic orbit to a periodic orbit through a Hopf bifurcation. What happens to the shape of the carrying simplex at the bifurcation? Does the Hopf bifurcation occur immediately as the carrying simplex loses convexity, or does the loss of convexity precede the periodic orbit? To gain intuition, they have come up with a method for computing the carrying simplex. They are animating the carrying simplex through a Hopf bifurcation. Here are a series of pictures of the carrying simplex for the three species case. The figure shows the carrying simplex as the periodic point goes through the Hopf bifurcation. Note that in the top picture, the carrying simplex is concave, in the middle picture, the carrying simplex is equal to the unit simplex, in the bottom picture, the simplex is convex, and thus there are no periodic orbits. Through a combination of dynamical systems theory and computer visualization, Zeeman and LaMar are gaining intuition for the dynamics of solutions to the Lotka-Volterra equations. Though the equations are still very simplistic, understanding them gives insight into what happens in a more complex situation. This article is based on Zeeman's talk in the University of Minnesota Dynamics and Mechanics Seminar on February 23, 1995, as well as an additional interview with Zeeman and LaMar. Thank you to them for their time taken for the interview, as well as preparing the figure for this article. From sander@geom.umn.edu Thu May 18 13:45:41 CDT 1995 Article: 390 of geometry.puzzles Xref: news3.cis.umn.edu geometry.puzzles:390 Newsgroups: geometry.puzzles Path: umn.edu!sander From: sander@geom.umn.edu (Evelyn Sander) Subject: Simple Puzzles with Mathematical Ideas Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: riemann.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Fri, 12 May 1995 21:13:19 GMT Lines: 50 Simple Puzzles The following is Steve Gray's response to my posing of the chess board puzzle. They are puzzles which require the solver to use fundamental mathematical ideas. Does anyone else have any such puzzles? Perhaps we could compile such puzzles in the Forum archives. I have separated the answer from the puzzle by a few lines: The following are a few simple math problems which require only elementary knowledge and which illustrate certain basic observations or principles. Thus these are more elegant and informative problems than most, as is the chessboard covering problem. (The first is obviously not geometry.) 1. A prime pair is two integers each of which is prime and whose difference is two, such as (17,19). Show that the number in between the prime pair is always divisible by 6 (excluding prime pairs lower than (5,7). Answer: The proof follows from the obvious but oft-overlooked fact that of every two consecutive integers, one must be divisible by two, and of every three consecutive integers, one must be divisible by 3. (This is not a well-known problem. Some people try algebra to solve it, which approach is fruitless.) For the more sophisticated, it illustrates the difference between algebra and purely number problems.) 2. You have a wood cube 3 inches on a side. You want to cut this cube into 27 one-inch cubes. This can obviously be done with six straight saw cuts. Can it be done in fewer by possibly rearranging pieces already cut so that further saw cuts accomplish more separating? Answer: Because of the general principle that the center piece has six faces which must be freed from surrounding wood, and there is no way to free any two of these faces in the same cut. Thus the answer is no. (This IS a well-known problem. It can be generalized to some extent to shapes other than cubes. For kids you could apply it to cutting along the lines of a tic-tac-toe diagram. Can 9 pieces be separated in fewer than 4 cuts?) -- Steve Gray Computers, Geometry, Inventions Santa Monica CA From sander@geom.umn.edu Thu May 18 13:46:15 CDT 1995 Article: 208 of geometry.announcements Xref: news3.cis.umn.edu geometry.announcements:208 geometry.pre-college:2017 geometry.forum:274 Newsgroups: geometry.announcements,geometry.pre-college,geometry.forum Path: umn.edu!news From: sander@geom.umn.edu (Evelyn Sander) Subject: Kali for the Macintosh Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: dehn.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 8 May 1995 17:02:30 GMT Lines: 30 Announcement: Kali for the Macintosh The Geometry Center is pleased to announce Kali for Macintosh. Kali lets you draw symmetrical patterns based on any of the 17 wallpaper groups. It was written for a broad audience, with particular attention to the needs of the following two groups: (1) Young children. Even the youngest children enjoy Kali. In particular, Kali does not assume the user knows how to read. (2) High school and college geometry students. Kali lets students systematically explore the wallpaper, frieze and rosette groups. The Show Singularities command and the Conway notation provide support for a theoretical analysis. Kali for Macintosh is based on Unix Kali by Nina Amenta. Optimized versions are provided for PowerMacs, 680x0 Macs, and 680x0 Macs with FPUs. All are freely available for anonymous ftp from the directory ftp://geom.umn.edu/pub/software/Kali/ A ReadMe file provides guidance in selecting the best version for your Mac. The Geometry Center is planning to develop a curriculum module using Kali. If you would like more information about it, of if you have other questions, problems or suggestions, please contact Jeff Weeks at weeks@geom.umn.edu. From sander@geom.umn.edu Thu Jun 8 11:31:19 CDT 1995 Article: 213 of geometry.announcements Xref: news6.cis.umn.edu geometry.pre-college:2073 geometry.announcements:213 Newsgroups: geometry.pre-college,geometry.announcements Path: umn.edu!sander From: sander@geom.umn.edu (sander) Subject: Shape of Space: Geometry Center's New Video Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: bers.geom.umn.edu Organization: The Geometry Center, University of Minnesota Date: Tue, 6 Jun 1995 19:49:41 GMT Lines: 85 Shape of Space: Geometry Center's New Video Evelyn Sander What is the shape of the universe? It is possible that it is Euclidean three-space: infinite volume, with no straight path returning to meet itself. However, the universe might also be finite in volume. You might be able to travel on a straight path and return to your starting point. How is this possible? This is the question addressed in the Geometry Center's new movie The Shape of Space. This article describes the movie, the making of the movie, and describes some future materials development for it. Although the plans for distribution are not yet decided, I wanted to write this article now to announce the good news that the movie has just been accepted to the SIGGRAPH 95 Computer Animation Festival/ Electronic Theatre and also into this year's SIGGRAPH Video Review. The Shape of Space is an eleven minute introduction to two- and three-manifolds intended for students grades 6-12. It describes the sphere, torus, and Mobius strip, and related three-dimensional objects. Although the ideas are quite sophisticated, the movie manages to develop them clearly enough that a junior high and high school audience can understand. It is designed to stand alone without any need for preparatory lectures or exercises before the video. Here is a brief illustrated description of the plot; how can a space ship go in a straight line and return to where it started? The movie describes this possibility in two-dimensional objects such as the sphere, the torus, and the Mobius strip. Figure 1 shows a space ship returning to its starting point in the two-dimensional torus. It shows both the standard torus and the square with sides identified. Moving on to three-dimensions, the movie describes a series of three-manifolds created by identifying sides of a box. This is the three-dimensional analogue of identifying sides of a square. For example, Figure 2 shows a view from inside a three-torus. Figure 3 shows the view from inside the three-dimensional analogue of a Klein bottle. In both figures, although there are only two stars and one space ship, the identification of sides makes the space appear to go on forever and be filled with stars and ships. Jeff Weeks has been lecturing on the shape of space for the last ten years and has written a book on the subject. The Geometry Center movie is based on Weeks' book. In 1992, Celeste Fowler created a draft of the video at the Center. However, it was not until December that a proposal for a production quality video went through. Technical production started in mid-January. Stuart Levy, Tamara Munzner, and Lori Thomson worked on the video at the Center, with remote help from Charlie Gunn, Delle Maxwell, and Jeff Weeks. On April 25th, 1995 with rejoicing and popcorn, the movie had its debut at the Geometry Center. An under four month production time makes this by far the shortest production time video from the Center. This is especially remarkable, since it also had the smallest production staff. They were able to speed production time so enormously by starting with a carefully followed script. In addition, in March, a draft went to three test sites of students and a test site of teachers. The feedback from these groups was very important in knowing what the students did not understand and how to fix it. This summer, teachers and students in the Center summer programs will work as a team to create class materials to use in conjunction with The Shape of Space. In addition to these related class materials, Jeff Weeks intends to improve his related Macintosh geometry software and make it into arcade style video games. (For more information about the current versions of Weeks' software, see Evelyn Sander, "Geometry Games for the Macintosh," geometry.pre-college, geometry.puzzles, 14 Nov 1994, or WWW version http://www.geom.umn.edu/docs/forum/weeks_software/) This article is based on an interview with Lori Thomson, a member of the production staff of The Shape of Space. Figures are stills from the movie and are available inline the WWW version of the article. For the WWW version of this article containing figures as well as mpegs >from the movie, see http://www.geom.umn.edu/docs/forum/sos/. Jeff Weeks' book is Jeff Weeks, "The Shape of Space," Marcel Dekker, Inc, New York, 1985. From news3.cis.umn.edu!umn.edu!hesse Fri Apr 14 17:37:20 GMT 1995 Newsgroups: geometry.pre-college,geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@siegel.geom.umn.edu (Bob Hesse) Subject: Angle Trisection Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: siegel.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 14 Apr 1995 17:37:20 GMT Lines: 133 Xref: forum.swarthmore.edu geometry.pre-college:2052 geometry.college:277 Angle Trisection Note: For those people with World Wide Web Capabilities, this article can be found at the URL address http://www.geom.umn.edu/docs/forum/angtri at the Geometry Center. Otherwise, figures mentioned in this article can be retrieved via anonymous ftp at ftp://forum.swarthmore.edu/pictures/articles/angtri/ at the Geometry Forum. Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a circle among other constructions. A few weeks ago I explained my job to a group of professors visiting the Geometry Center . I mentioned that I wrote articles on a newsgroup about geometry and that sometimes people write to me with geometry questions. For instance one person wrote asking whether it was possible to divide a line segment into any ratio, and also whether it was possible to trisect an angle. In response to the first question I explained how to find two-thirds of a line segment. I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle. One professor I told this story to replied by saying, "Bob it is possible to trisect an angle." Before I was able to respond to this shocking statement he added, "You just needed to use a MARKED straightedge and a compass." The professor was referring to Archimedes' construction for trisecting an angle with a marked straightedge and compass. When someone mentions angle trisection I immediately think of trying to trisect an angle via a compass and straightedge. Because this is impossible I rule out any serious discussion of the manner. Maybe I'm the only one with this flaw in thinking, but I believe many mathematicians make this same serious mistake. Why tell people it is impossible to trisect an angle via straightedge and compass? Instead we could say it is possible to trisect an angle, just not with a straightedge and a compass. When told that it is impossible to trisect an angle with a straightedge and compass people then often believe it is impossible to trisect an angle. I think this is a mistake and to rectify my previous error I will now give two methods for trisecting an angle. For both methods pictures are included that will hopefully illuminate the construction. The first method, Archimedes' trisection of an angle using a marked straightedge has been described on the Geometry Forum before by John Conway. The first picture, fig1, shows the angle to be trisected, angle ABC, and a line parallel to BC at point A. Next use the compass to create a circle of radius AB centered at A, as shown in fig2. Now comes the part where the marked straightedge is used. Mark on the straightedge the length between A and B. Take the straightedge and line it up so that one edge is fixed at the point B. Let D be the point of intersection between the line from A parallel to BC. Let E be the point on the newly named line BD that intersects with the circle. Move the marked straightedge until the line BD satisfies the condition AB = ED, that is adjust the marked straightedge until point E and point D coincide with the marks made on the straightedge as shown in fig3. Now that BD is found, the angle is trisected, that is 1/3*ANGLE ABC = ANGLE DBE. To see this is true let angle DBC = a. First of all since AD and BC are parallel, angle ADB = angle DBC = a. Since AE = DE, angle EAD = a, and so angle AED = Pi-2a. So angle AEB = 2a, and since AB = AE, angle ABE = 2a. Since angle ABE + angle DBC = angle ABC, and angle ABE = 2a, angle DBC = a; in fig4 the angles and line segments are marked to show these relations. Thus angle ABC is trisected. The next method does not use a marked ruler, but instead uses a curve called the Quadratrix of Hippias. This method not only allows one to trisect an angle, but enables one to partition an angle into any fraction desired by using the Quadratrix. This curve can be made using a computer or graphing calculator and the idea for its construction is clever. Let A be an angle varying from 0 to Pi/2 and y=2*A/Pi. For instance when A = Pi/2, y=1, and when A=0, y=0. Plot the horizontal line y = 2*A/Pi and the angle A on the same graph. Then we will get an intersection point for each value of A from 0 to Pi/2. This idea is illustrated in fig 5. This collection of intersection points is our curve, the Quadratrix of Hippias. We will now trisect the angle AOB. First find the point where the line AO intersects with the Quadratrix. The vertical coordinate of this point is our y value. Now compute y/3 (via a compass and straightedge construction if desired). Next draw a horizontal line height y/3 on our graph, which gives us the point C. As one can view by fig6, drawing a line from C to O gives us the angle COB, an angle one third the size of angle AOB. As I mentioned before this curve can be computed and plotted via a computer. The formula to find points on the curve is defined as x = y*cot(Pi*y/2). Yes here the vertical variable, y, is the independent variable, and the horizontal variable, x, is the dependent variable. So once the table of values is found, the coordinates will need to be flipped to correctly plot the Quadratrix. Now a justification for the formula. In fig7, B =(x,y) is a point on the Quadratrix of Hippias. Let BO be a line segment from the origin to B and BOC be our angle A. If we draw in a unit circle, and drop a vertical line from the intersection of the angle we get similar triangles and see that sin(A)/cos(A) = y/x, or tan(A) = y/x. But earlier we defined a point (x,y) on the Quadratrix to satisfy y=2*A/Pi. So we get tan(Pi*y/2)=y/x or equivalently x = y*cot(Pi*y/2). So we now have two different ways of trisecting an angle. I learned about the construction of the second method in Underwood Dudley's book: "A Budget of Trisections". In the book Dudley describes several other legitimate methods for trisecting an angle as well as compass and straightedge constructions that people have claimed trisect an angle. The book also contains entertaining excerpts of letters from these "angle trisectors". Besides stating it is impossible to trisect an angle, I think other problems occur in discussing angle trisection. One difficulty is in explaining what it means for something to be impossible in a mathematical sense. I definitely remember in high school being told that it was impossible to trisect an angle. But I think at the time it meant the same thing to me as that it was impossible for me to drive a car. I was only 14 years old and I could not get a license to drive a car for another two years so it was just not possible AT THAT TIME. I do not remember being told that when something is impossible in mathematics, it was not possible five million years ago, it is not possible now, and it will never be possible in the future. Granted I may not have been ready for an explanation of mathematical logic and proof, but a statement like, "It is impossible to trisect an angle with a straightedge and a compass. This means it is no more possible to trisect an angle with those tools, then it is to add 1 and 2 and get 4" would have been much more powerful. (I am assuming here that we all count the same way: 1, 2, 3, 4, ...) I did not intend to attack my high school teacher; I learned an incredible amount of mathematics from her as well as a deep love for the subject. Maybe my teacher did explain what the words "mathematically impossible" meant, and I just do not remember her comments. Regardless, I think a discussion of impossible in the mathematical sense would be an interesting and valuable topic to discover in high school. Are there any teachers out there who have spent time talking about mathematical impossibilities? From news3.cis.umn.edu!umn.edu!hesse 23 May 1995 20:13:25 GMT 1995 Path: forum.swarthmore.edu!news From: Bob Hesse Newsgroups: geometry.college,geometry.precollege Subject: Banchoff at the Geometry Center Date: 23 May 1995 20:13:25 GMT Organization: The Geometry Forum Lines: 85 Distribution: inet Message-ID: <3ptfl5$kbl@forum.swarthmore.edu> NNTP-Posting-Host: bers.geom.umn.edu Banchoff at the Geometry Center Thomas Banchoff, a professor currently on leave from Brown University, has been visiting the Geometry Center. Professor Banchoff reminded me of a commercial Paul Wellstone, U.S. Senator from Minnesota, ran in his 1990 campaign. The short commercials were clips of Mr. Wellstone furiously running from one part of the state to another; the idea being that he did not have enough time or money to spend on a lengthy commercial. Professor Banchoff reminded me of this by the way he has been moving around at the Geometry Center. He seems to be everywhere at the Center learning everything, from working on web applications to using Geometer's Sketchpad. He intends to use the technology available and applications discovered to communicate research. One of the first things Professor Banchoff did was to learn the HyperText Markup Language, HTML, which is used to create documents for the World Wide Web. He has created a homepage at the URL address http://www.geom.umn.edu/~banchoff/ of the Geometry Center. From this homepage, he has made links to other documents that show more of his work. One of these links goes to a page that lists all of his publications. Professor Banchoff would like to include his articles, as well as links to articles that reference him. By doing this researchers can see the introduction to a problem, where it has lead, and future areas of work. Besides having these links, he is considering adding pictures and movies to illustrate his work. Another link on Professor Banchoff's homepage goes to a projects page. As mentioned earlier, he has been working not only with learning HTML, but other current applications. Professor Banchoff has created a storyboard for a film he had made twenty years ago titled, COMPLEX FUNCTION GRAPHS. The storyboard contains a description of how to view complex graphs such as w=z^2 and w = log z, and then shows some beautiful pictures of these graphs. If one has an mpeg player, it is also possible to view movies of these graphs as the projections vary. Professor Banchoff has already made some movies on self-linking curves in the 3-sphere. Like all of the other work he has done in HTML, these movies are accessible from his homepage and anyone who has an mpeg viewer can see these movies by clicking on the appropriate link in his document. Professor Banchoff not only created links to pictures and movies but also has made links to programs from his documents. In particular, he has written a HTML document titled Monge's Theorem and Desargues' Theorem Identified, which contains work from Geometer's Sketchpad. Any one viewing this document with a machine containing a version of Geometer's Sketchpad program, can view and manipulate the sketches he has made. Using the Sketchpad program, he gives visual arguments for both theorems. For those curious here is Monge Theorem: "For three disjoint circles of unequal radii, with no one contained in any other, the pairs of external tangents meet in three points that are collinear." A version of Desargues' Theorem can be stated as "If two triangles have their corresponding vertices on three parallel lines, then the intersections of the lines containing the corresponding sides will be collinear." Professor Banchoff has also made pictures of the three-dimensional generalization of Monge's Theorem, as well as explanations. Not only has he learned many new techniques for communicating mathematics, but he has also been working on ways to help other researchers learn them. In particular he has been working with Davide Cervone of the Geometry Center creating software for making movies such as the ones mentioned above. This software will be a more effective way of working with Geomview to create such movies. As mentioned at the beginning of this paper, Professor Banchoff has constantly been moving around at the Center like the ads remembered from years ago. However, unlike the candidate that was too busy to appear in a commercial, he has been able to take time and explain his work to others. Maybe some bright politician will try to incorporate that idea into some future campaign. From news3.cis.umn.edu!umn.edu!hesse Thu Jun 30 14:04:14 GMT 1994 Newsgroups: geometry.college,geometry.precollege Path: forum.swarthmore.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: 1994 Summer Course for Teachers: CHANCE Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Thu, 30 Jun 1994 14:04:14 GMT Lines: 66 Each summer, the Geometry Center hosts a two-week course for teachers. This year's course, titled CHANCE, is a probability and statistics college class taught by Laurie Snell at Dartmouth College. Instead of teaching the theory and then giving the students examples, Laurie has students read current news items that involve probability and statistics and are asked questions pertaining to the articles. Some of the recent examples used in class were whether major-league baseballs are juiced or not, the reliability of DNA testing, and grade inflation at colleges over the past thirty years. The goal of this class is to "make students better able to make informed and critical judgements about news reports of chance issues that affect their daily lives." (CHANCE Pamphlet) Because the class topics depend on current events, there is no course outline. CHANCE instructors also use perennially popular issues such as polls, trials, and drug testing. In the summer course one morning, the CHANCE team (Peter Doyle, Laurie Snell, Joan Garfield, Mark Foskey and Linda Green) had the participants read an article about a study on the effects of taking daily supplements of vitamin E or beta carotene or both. Along with the article the participants were given a question sheet. They were asked to interpret a percentage the article claimed, to use statistics to verify a claim, and to tell the instructors whether they should continue taking doses of beta carotene and vitamin E. The students worked in groups of four or five discussing possible answers to the solution. After about twenty minutes of the discussion the instructors called on the groups and had them respond to the questions. Many of the participants and the instructors went beyond the original questions and debated different approaches to the problems. In one of the more interesting discussions, the instructors compared a point in the beta carotene article with the tossing of a fair coin. The related coin-tossing question was this: What is the approximate probability of getting 474 or more heads when a coin is tossed 876 times? Several of the groups approached the problem differently. The instructors showed a way of answering the problem using a computer; they had a laptop run a thousand simulations of flipping a fair coin 876 times. In those thousand simulations, only four times did 474 or more heads appear. They also used statistical theory to show that such an incidence is unlikely. The combination of using technology (which gave one a strong feeling of what was going on) along with statistical theory (which confirmed the feeling) sparked interest in the material and aided in appreciating the power of statistics in everyday life. The instructors approach to teaching probability was fast-paced and gripping. Both the participants and the instructors were caught up in the discussion. Although the participants were all practicing mathematicians, I believe that this method can work for many students. Laurie believes that even if one could not teach a class in this format the entire time, it would be a good supplement to an already existing probability and statistics class. A second component to the summer CHANCE course is the use of computers. Participants use programs to analyze data and run simulations. They also learn how to access and use the Internet system so that they could retrieve articles and databases useful for projects. One of the more useful files participants are able to access via Internet is the CHANCE database. This database contains articles from newspapers, raw data to use for projects, and eventually some probability simulation programs. For example besides containing the newspaper articles used in the summer course, the database contains statistics on the 1993 SAT scores. There is also a BiWeekly chance newsletter that abstracts current news that involves statistical and probability concepts. This newsletter can be obtained by e-mail by sending a request to dart.chance@dartmouth.edu or can be read on the Chance Mosaic found on the Geometry Center Mosaic (http://www.geom.umn.edu/) in their Online Document Library. From news3.cis.umn.edu!umn.edu!hesse Mon Mar 13 18:02:02 GMT 1995 Newsgroups: geometry.college,geometry.pre-college Path: forum.swarthmore.edu!umn.edu!news From: hesse@siegel.geom.umn.edu (Bob Hesse) Subject: University of Minnesota Curriculum Initiative Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: siegel.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 13 Mar 1995 18:02:02 GMT Lines: 74 Xref: forum.swarthmore.edu geometry.college:273 geometry.pre-college:1859 University of Minnesota Curriculum Initiative Next fall, one hundred University of Minnesota Institute of Technology, IT, students will have the opportunity to participate in a pilot calculus course. This five quarter class will heavily stress utilize technology such as graphing calculators and computer algebra systems such as Maple. Students will also be required to attend workshops where they will work on interdisciplinary modules containing problems from the sciences and engineering. If this pilot project is a success, it will become a standard calculus sequence for two-thirds of all IT students. The class will be structured in a different way then the usual lecture/recitation format at the University. Currently a student attends a professor's lecture three times a week for one hour with at least a hundred other students. The student also attends one-hour recitations twice a week held by a Teaching Assistant with about thirty other students. In the pilot class, students will attend lecture only twice a week. The lecture will be team taught by two professors, one a senior faculty member and the other a postdoc. Twice a week students will attend one-and-a-half hour workshops containing 25 students. At the workshops not only will the TA's be present to aid the students, but once a week a faculty member will be assisting students as well. Next year one of the Teaching Assistants will be an instructor from a local community college or high school. The University hopes to spread its program format to other schools because they think it is an exciting way for students to learn calculus, and so that incoming students taking calculus elsewhere will be familiar with this format. As mentioned above students will work on interdisciplinary modules in the workshops. Some of these modules will be written in hypertext format as well as require the use of programs such as maple. These modules are similar to those written by Davide Cervone and Rick Wicklin for the UMTYMP Multivariable Calculus class. (For more information on these modules see the article "UMTYMP Multivariable Calculus at the Geometry Center.") This core sequence will last five quarters. In the first year, the first three quarters, students will cover the traditional areas of calculus. First quarter will be single-variable differential calculus and second quarter will be integral calculus. In the third quarter, students will cover some differential equations and linear algebra related to differential equations. They will also cover power series, parametrization of curves in two and three dimensions, tangent vectors, unit normals, curvature, and invariance under parametrization. Students will be using calculus reform texts which differs from the regular university calculus classes that are using the more traditional texts. The curriculum planning committee for this course included engineering and science department faculty. The committee advocated the creation of modules which both the mathematics and other departments could build upon in higher level classes. Eventually the committee plans to include one instructor from the sciences and engineering departments as well as one mathematics instructor for the team instruction. The goals for this new sequence are simple. students will not only leave the course with a strong background in calculus, but also be adept with using technology to aid in solving problems, be proficient in a computer algebra system such as Mathematica or Maple, and be able to build upon models from previous classes. Further information about the initiative can be obtained by contacting Tracy Bibelnieks (tracyb@math.umn.edu) or Harvey Keynes (keynes@math.umn.edu) of the Mathematics Special Projects Office, University of Minnesota. From news3.cis.umn.edu!umn.edu!hesse Wed Aug 3 16:57:56 GMT 1994 Newsgroups: geometry.pre-college Path: forum.swarthmore.edu!concert!news.duke.edu!godot.cc.duq.edu!newsfeed.pitt.edu!uunet!spool.mu.edu!umn.edu!news From: hesse@siegel.geom.umn.edu (Bob Hesse) Subject: Minesota Mathematics Mobilization Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: siegel.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Wed, 3 Aug 1994 16:57:56 GMT Lines: 52 In a previous article I wrote about a Calculus Consortium sponsored by the Minnesota Mathematics Mobilization held at the Geometry Center. This is the long-awaited sequel to that article explaining what M^3 is and what its goals are. Tracy Bibelnieks, an administrator of Minnesota Mathematics Mobilization, gave me a brief history of the organization. M^3 was founded in 1986 as an attempt to bring together Minnesota instructors, government, and business leaders to improve mathematics education. Over the past eight years membership has swelled to over 6000 members, including educators in the elementary, secondary, and post-secondary levels, as well as leaders in industry and government. The goals of the Mobilization are to: Advocate state policies that support national mathematics education goals. Encourage implementation of locally-developed mathematics education programs that are consistent with national goals. Generate consensus among Minnesota's corporate, education, and public policy leaders regarding goals, programs, and progress in mathematics education. Develop timely responses to state mathematics education needs. Provide state agencies, institutions, and organizations with reliable information regarding state activities to improve mathematics education. (mobilization pamphlet) M^3 achieves some of these goals through seminars, a bi-monthly newsletter and information sent through electronic mail. Some seminars such as the Calculus Consortium are a medium for instructors to get together and exchange ideas of teaching. Other seminars are a place where different promoters of mathematics, such as educators and business leaders, discuss their needs and what is available. The newsletter is a forum for promoting various workshops, groups, and other general information. It exposes those in M^3 to different opportunities and information that they possibly would not have seen. For instance educators may be informed of bills going through the state legislature that may pertain to mathematics. Members of M^3 are also periodically sent electronic mail from Danna Elling at St. Olaf College. Danna gathers mathematics-related news together, sets up a table of contents, and distributes the news to all members of M^3 with an e-mail address. For further information about M^3, please contact Tracy Bibelnieks at the following address: Tracy Bibelnieks University of Minnesota 115 Vincent Hall 206 Church Street S.E. Minneapolis, MN 55455 (612) 625-2861 From news3.cis.umn.edu!umn.edu!hesse Tue Oct 4 13:40:54 GMT 1994 Newsgroups: geometry.pre-college Path: forum.swarthmore.edu!uunet!spool.mu.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: Making Math Fun Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 4 Oct 1994 13:40:54 GMT Lines: 74 Have you ever gone to a party where someone asks you what your occupation is and upon hearing that you are a mathematician or that you teach math, their face becomes white and they mumble, "I hated math in school."? How does one alleviate the fear and dread of mathematics that people have in our society? Obviously one way to remove this fear is by making math fun. How does this work? Susan Addington of California State University, San Bernardino, has come to the Geometry Center to look at ways to make math fun. In particular she is writing a proposal to create a traveling math show that visits shopping malls, community centers, middle and junior-high schools, and libraries. By having the exhibit at these places rather than a museum, Susan believes she can reach out to a wider audience, especially to those people who hate math. Currently, Susan is creating a pilot exhibit that would involve the ideas of symmetry. She is hoping to acquire a Silicon Graphics Iris computer that would run a Triangle Tiling program created at the Geometry Center. (For further information about the tiling program please look at the article by Evelyn Sander entitled, "Science Museum Math Exhibit", posted June 17 on the geometry.college newsgroup). Besides the SGI, there would also be wallpaper tiling programs on MACs, and an area in the exhibit where people tile a space by using different geometric blocks. Not only is Susan planning on making a more accessible exhibit than those at museums, but she is also trying to create an exhibit that contains more depth than those at museums. Susan plans on doing this by including seating areas in the exhibit where people could read or listen to the broader ideas that are being discussed. Another possibility is to include a store where people can purchase math puzzles and books related to the exhibit. A third idea is to include modules for teachers so that they can instruct their students before they attend the exhibit, as well as after. Another project Susan has been working on for a different future museum exhibit is the Ames Room on a computer. An Ames room appears to be a typical rectangular room from one perspective, but in reality is a room of a very non-rectangular shape. One can construct such a room with the aid of projective geometry. One takes a standard rectangular room and then multiplies the vectors in the room by a 4 by 4 invertible matrix that satisfies certain constraints. Some of the constraints are the following: 1) Fix a position for the viewers eye. The projection of the Ames Room from the viewers position must be the same as the projection of an ordinary room from that position. 2) Vertical lines should be preserved 3) The family of horizontal lines in the back wall should be sent to concurrent lines, with the point of concurrency directly to the right of the eye of the viewer. Adelbert Ames, a lawyer by profession, is attributed as the inventor of this room. With the aid of Stuart Levy, Geometry Center Senior Technical Staff, Susan has constructed an Ames Room module that runs on Geomview as well as one for Mathematica. In the Geomview version, the user can bring up the room as well as two identically sized objects, say two chess bishops, and view their placement in the room from several perspectives. One perspective shows the placement of the bishops from directly overhead, another shows the entire room from a distance, and a third window shows the room and bishops from the perspective which tricks the eye. In this third window, the user sees the two bishops in an apparently rectangular room where one bishop is larger then the other. When looking at the overhead view, one sees that this affect is created by having one bishop further back in the room than the other. Moreover one can adjust the parameters of the room so as to distort the appearance even more (or less) to get an idea of how the projection works. From news3.cis.umn.edu!umn.edu!hesse Fri Jun 17 13:43:37 GMT 1994 Newsgroups: geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@diophantus.geom.umn.edu (Bob Hesse) Subject: 1993 Summer Institute Video Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 17 Jun 1994 13:43:37 GMT Lines: 34 The Summer Institute is a ten-week research experience for undergraduates and high school students hosted by the Geometry Center. A video has been made that captures highlights from the 1993 Summer Institute. The sixty minute film displays students' computer projects which range from simulating natural objects such as neural nets, clouds and the human heart, to studying mathematical objects such as the Weierstrass p-function and the Complex Henon Map. I'm impressed by the students' projects. They applied mathematical theory to natural and theoretical problems and with the aid of state-of-the-art computer hardware and software, as well as export support from researchers and programmers, produced visually stunning results. I encourage mathematics undergraduates to watch this video and see how much fun research can be. Media Magic sells the video for $15 plus shipping. If interested, write to the following address: Media Magic P.O. Box 598 Nicasio, CA 94946 415/662-2426 For a free book which comprehensively details research done by the 1993 Summer Institute (a nice book to put on one's coffee table!) send a request to the Geometry Center for Preprint Series GCG59 at: The Geometry Center 1300 South Second Street, Suite 500 Minneapolis, Minnesota 55454 or email: admin@geom.umn.edu From news3.cis.umn.edu!umn.edu!hesse Mon Jan 9 20:16:49 GMT 1995 Newsgroups: geometry.research,geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@siegel.geom.umn.edu (Bob Hesse) Subject: Systems Administation Workshop Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: siegel.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 9 Jan 1995 20:16:49 GMT Lines: 148 Xref: forum.swarthmore.edu geometry.research:341 geometry.college:265 Systems Administration Workshop Between December 15-17 I attended parts of the Systems Administration Workshop or SAMP, held at the Geometry Center. SAMP is a workshop for computer systems administrators of mathematics departments. Over 60 people attended the workshop from as far away as California, Florida and England, ranging from large research Universities like Berkeley to small liberal arts colleges like Gustavus Adolphus College in St. Peter Minnesota. Participants in SAMP not only covered the daily technical issues, but also discussed future projects and developments relevant to anyone whose mathematical work involves computers. Although I have no desire to become a System Administrator, I decided to sit in on the SAMP workshop for three reasons: 1) I needed a subject for an article. 2) There was free food provided. 3) I was curious as to what these people would talk about for three days. Surprisingly I learned that there was a lot of issues these administrators wanted and apparently needed to discuss while attending the workshops. The first talk was given by David Oliver of GANG (Geometry, Analysis, Numerics Group) the University of Massachusetts at Amherst. David titled his talk "A long-term approach to mathematical computing". He began his talk by mentioning the problems administrators have. One problem he mentioned was that only a fraction of the software and hardware needed by academia is available via the trade magazines. Many of the tools needed are often found by talking with other systems administrators and searching the internet. A second problem involves the changes in hardware. Previously a systems administrator was in charge of mainframe systems and the user needed to customize their work towards the system. Now systems administrators oversee a series of individual workstations, most of them customized for a particular user. One surprising problem David mentioned was how data on certain mediums has and is becoming obsolete. System Administrators had held the common fear that data stored on magnetic tape would not last for a long period of time. David believes what will more likely happen is that people will not be able to recover data from tapes because they don't have a machine or software that can access the tapes! David continued his speech by discussing the role of a systems administrator. One part of the systems administrator's job should be to introduce new software and hardware to the faculty instead of the faculty searching for the products they need. By finding the software for them, the faculty have more time to what they were hired for: research, instructing, and service. System administrators also need a good technical staff. Often departments cover this cost by having graduate students or faculty members to these jobs, instead of hiring a support staff. Although hiring a support staff is more visible cost in the record books, David insists that this is money well spent. Instead of constantly replacing graduate students who continue their studies and graduate or faculty who return to their research, a professional staff provides continuity. One main theme David mentioned throughout his talks was keeping future plans, projects, and proposals "informal". Being informal is necessary due to the constantly changing nature of computers where companies like Silicon Graphics and Hewitt Packard produce a new top-of-the-line machine every nine months. Because of this planned obsolescence, administrators need to be less formal in their purchasing strategies to stretch their budgets. For instance instead of purchasing the latest new model from Silicon Graphics thereby spending the entire 5 year hardware budget, David suggests leasing it orhaving some sort of payment plan with the intent of exchanging it for an upgrade before it is obsolete, and you are stuck with a system unable to run the latest programs. David also suggests being informal towards goals. Instead of focusing strictly and laboriously on a formal proposal for a grant, one needs to move through faster, more informal channels. He mentioned his mistake in applying for a grant of 150 hours of supercomputer time a few years back. Originally there was a need for this CPU time. However as this proposal was pushed through the proper channels, two years passed and in the end only 15 minutes of Cray time was ever actually used. This is where being more informal comes into play. The systems administrator needs to find solutions to problems in a short amount of time. Spending a lot of time focussed on a particular problem such as the one above wastes time, since the initial problem often has been changed or is no longer there. David also suggests share resources with different departments as well as working on more joint ventures. The second talk given by Russ Ruby of Oregon State University was titled distributed computation. Ruby uses parallel computing for factoring large integers. He has created a program to access all of the computers hooked up on his system, and has each of them work on a part of the particular problem. Russ' solution is not a unique approach, what is different is the availability of resources. He does not have the luxury of a system of identical computers that are stable and whose CPUs can be strictly devoted to working on the problem. This is often a necessary condition for parallel processing. Instead Ruby needed to create a program that would be able to send jobs to a hodge-podge of different computers, be able to continue working on the problem if one of the computers crashes,and be able to transfer a job to another computer or distribute the job on a given computer if a user want to use it. He has created and is still working on this generalized code. The third talk title "What's new in computer algebra systems" was given by Paulo Ney de Souza of UC Berkeley. Paulo gave a brief history and future of the CAS programs available. He placed them in to three groups: the four big "M" programs out on the market; Mathematica, Maple, Macsyma and axioM, two programs that run on every type of system: Reduce and Derive, and four new small, lean and fast programs: GAP, Magma, Peri, and Macaulay. Paulo first mentioned that all of the big "M" programs now do computations more-or-less with the same skill, i.e. now one particular program does overall better computations then the others. However each of these programs are expanding in newer ways. Mathematica has packages for solving combinatorial problems as well as tensor analysis. One of the outgrowths of Macsyma includes a PDE solver. The Maple kernel is inside other programs like MathCad, MatLab, and even a word processor! As for the future developments, the Axiom program will incorporate NAG routines (a Fortran library) within its program, so that the user within Axiom is able to compile codes. For difficult calculations, this ability to compile could drastically increase computation speed. The Maple program has a test version that allows the user within Maple to send problems or parts of problems to any other machines using Maple, i.e. parallel computing of Maple calculations. Paulo mentioned the four newer programs as being small, lean and fast. They each have small kernels that allow the user to solve specific problems in certain mathematical areas. For instance Peri is stronger in solving number theory problems. It can solve problems commonly much faster then the four larger programs can due to its specialization. Also the writers of these programs are much quicker in updating their codes to include better algorithms. Ney de Souza gave the following example. To compute n!, many programs implement the standard natural code: n! = n*(n-1)*...*1. However, notice that for n even, n!=(n)*((n-1)*2)*((n-2)*3)*((n-3)*4)*...*((n-n/2+1)*n/2). This approach can be defined as a product of a recursively defined sequence which reduces the number of multiplications needed to compute n!, consequently speeding up the process. All of the big "M" companies as well as the four lean companies learned about this algorithm at the same time. None of the big four had modified their code, three of the lean four have. What I found as interesting as the talks were the differences that these system operators had. For instance, Paulo felt that most of the big "M" CAS programs should be available on one computer system. However many of the systems administrators have to choose one of the big "M" programs as their package, because that is all that their University can afford. Paulo Ney de Souza's options varied greatly from those of say Russ Ruby, whose system consists of donations by companies and the most affordable models. However they both share the desire to maximize the resources they have. In conclusion I noticed that these administrators all came to the workshop with the same set of problems: lack of equipment, lack of staff, trouble keeping up with constantly new, bigger and faster systems, as well as software upgrades. But I also observed some of the potential uses of computer systems as well as how to maximize a system's potential power. From news3.cis.umn.edu!umn.edu!hesse Wed Feb 22 20:03:50 GMT 1995 Newsgroups: geometry.pre-college,geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@archimedes.geom.umn.edu (Bob Hesse) Subject: UMTYMP Multivariable Calculus at the Geometry Center Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: archimedes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Wed, 22 Feb 1995 20:03:50 GMT Lines: 104 Xref: forum.swarthmore.edu geometry.pre-college:1673 geometry.college:272 UMTYMP Multivariable Calculus at the Geometry Center This year two of the Geometry Center's Postdoctoral Fellows, Rick Wicklin and Davide Cervone, are teaching the UMTYMP Calculus III Class. UMTYMP stands for the University of Minnesota Talented Youth Math Program. Students are discovering multivariable calculus and differential equations at the Geometry Center by a combination of techniques including computer-aided labs and "mastery problems". Mastery problems are non-routine, sometimes open-ended homework problems. UMTYMP is a mathematics program for gifted secondary students throughout the State of Minnesota. These students are given the opportunity to take calculus classes for college credit, often as early as their freshman or sophomore year of high school. The Calculus I program consists of single variable differential and integral calculus. Calculus II involves linear algebra, sequences and series, and Calculus III covers multivariable calculus and differential equations. Rick and Davide are extensively using the computers at the Geometry Center to teach the students. These labs are written as hypertext files that the students view using a reader such as Mosaic or Netscape. By writing the documents in hypertext, the labs may be linked to previous labs, to pages with general information, or to programs students need to use for the lab, for instance Maple. Some of the labs can be found at the URL address http://www.geom.umn.edu/~fjw/Calc3Labs.html of the Geometry Center. (For legal reasons, the hypertext links to software such as Maple will not work outside of the Geometry Center. Also a few links will be unaccessible to readers on PCs or Macintosh computers.) Half of each class period is spent working on the computer-aided labs. Rick mentioned that the half-lecture half-lab format has many benefits. Besides giving students the chance to develop group work skills, students usually work on the computers in groups of three, the labs also give the instructors another way to evaluate students. Some of the students who do poorly on exams and homework do quite well on leading the group in labs. Also, all the students are excited about using the computers to do mathematics, and look forward to doing the labs. In one lab, students are asked to find the area and volume of a domed stadium, the HHH Metrodome in Minneapolis, Minnesota. They are first given a crude approximation to the base of the dome (a rectangle) to familiarize themselves with how the Maple program plots domains of integration, and can compute iterated integrals. Next, students are given better models for the shape of the Metrodome's walls (first an ellipse, and later a fourth degree polynomial in two variables) and are asked to set up and solve double integrals representing the area enclosed by the Metrodome. They are also asked to compare their models with reality, suggest a better model for the Metrodome's walls, and describe the important features for modeling the building. Similarly after being given a formula modeling the height of the dome roof, students are asked to set up and solve the double integral for the volume enclosed by the roof. They are again asked open questions as to whether the "roof function" is a good model for the actual roof, and of course the students need to justify their answers. Another lab introduces differential equations by looking at Kepler's Problem which is a special case of the 2-body problem from celestial mechanics. The equations of an n-body problem describe the motions of n different bodies or planets. In Kepler's Problem, one body is assumed to be fixed at the origin due to it having a tremendous mass, e.g. the sun. The differential equation describes the motion of the other body, e.g. the earth. In this lab students use the program DsTool, a software package for studying the behavior of differential equations or iterated mappings By using DsTool, students can compute and visualize trajectories, i.e., the solution, to the differential equation corresponding to the initial conditions for the velocity and position they choose. Through DsTool, students discover that the trajectories of Kepler's problem in the plane are ellipses, parabolas, and hyperbolas. Students also vary the initial velocity of the particle in order to discover the escape velocity needed to break free of the sun's attraction. Besides discovering mathematical concepts on the labs, Davide and Rick give students "mastery" homework problems each week. These problems often carry over the concepts discussed in the labs or in lecture and introduce new concepts. For instance after becoming familiar with differential equations, students studied the gradient differential equation, dx/dt = -Grad(F(x)) where F(x)=x^4-2x^2. This differential equation could be thought of as modeling the dynamics of a particle moving along a line. Students were asked to connect the geometry of the graph of F to physical phenomena such as the location of equilibria. By generalizing the results of this problem, students discovered some of the fundamental concepts of dynamical systems, and in doing so have learned another method to use to understand differential equations. Davide and Rick even have students discover new mathematics on the exams. Students were asked to derive the arc-length formulas for graphs of functions, parametric functions, and functions written in polar coordinates. Students were also asked to analyize a system of differential equations similar to a predator-prey model designed to simulate two tribes of cannibals! (This problem was adapted from a article by J.M McDill and B. Felsager, The ligher side of differential equations, The College Math Journal, vol 25 no. 5, November 1994, pp 448--452.) Although Rick and Davide are teaching Calculus III to high school students, the lab format and mastery problems appear to be a useful method for teaching multivariable calculus and differential equations to any group of students. By having them work through these problems and labs, students discover mathematics on their own, which is the best way to learn and understand any discipline--especially mathematics. From news3.cis.umn.edu!umn.edu!hesse Wed Jun 21 18:16:00 GMT 1995 Path: forum.swarthmore.edu!news From: Bob Hesse Newsgroups: geometry.pre-college,geometry.college Subject: UMTYMP Calculus III Math Fair at the Geometry Center Date: 21 Jun 1995 18:16:00 GMT Organization: The Geometry Forum Lines: 123 Distribution: inet Message-ID: <3s9nl0$6re@forum.swarthmore.edu> NNTP-Posting-Host: picard.geom.umn.edu Mime-Version: 1.0 X-Mailer: Mozilla 1.1S (X11; I; IRIX 5.3 IP22) Xref: forum.swarthmore.edu geometry.pre-college:2272 geometry.college:287 X-URL: news:geometry.pre-college Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii UMTYMP Calculus III Math Fair at the Geometry Center On May 23, third year UMTYMP calculus students studying at the Geometry Center held a math fair. Two of the Center's postdocs, Rick Wicklin and Davide Cervone are teaching differential equations and multivariable calculus to the class and the fair is part of the curriculum. (For more information on the class and the UMTYMP program see the article UMTYMP Multivariable Calculus and the Geometry Center.) The Math Fair was structured as a science fair event with students presenting their projects at stations. Students worked in groups of three or four, either trying to model a physical phenomena or delving further into a mathematical concept they had studied. Their work ranged from models of heartbeats, plagues and roller-coasters to methods of visualizing mathematical concepts like involutes and evolutes. Some groups made their projects into modules that may be used in following years' classes. Modeling was a popular topic for the Math Fair. One group of students, Martin Almlof, Tierre Christen, Sarah Fellows and Apurv Kamath decided to model the beating of a heart using differential equations. For the model to be credible, it needs to contain two states, the diastole state (the state when the heart relaxes and fills with blood) and the systole state (the state when the heart contracts and pumps the blood out). They researched and found an ordinary differential equation that models the change in electrochemical activity and fiber length over time given by the following system of differential equations: e*x' = -(x^3-T*x+b) b' = x - x0 + u*(x0-x1) where u = 1, if b0 <= b <= b1 and (x^3-T*x+b) > 0 or if b > b1 u = 0, otherwise. The two variables b and x represent electrochemical activity and muscle fiber length respectively. This model has two equilibrium states, one at (b0,x0), the diastolic state, and the other at (b1,x1), the systolic state. Students plotted phase portraits of this system and showed how the dynamics vary as parameters are changed. Students also plotted the electrochemical activity versus time, which produces the famous Electrocardiogram or EKG graph. Students created a lab in hypertext form. In this lab one could alter the above parameters to the differential equation and see by producing phase portraits and EKG graphs, how that would change the model. The lab contains questions about what happens in the model when various parameters are changed, what happens to the EKG graph when various values are changed, and what do some of the parameters represent. Two groups did work on modeling epidemics. They too researched and found differential equations to model outbreaks. They started with the following model for the spread of a disease: S' = -a*S*I I' = a*S*I - b*I R' = b*I Where S represents the number of people susceptible to the disease, I the number of people infected, and R the number of people recovering or immune to the disease. The parameter a represents the infectiveness of the disease, and b the rate of removal from the infected population. One group of students Mike Arvold, Bjorn Soderlund, and Jeremy Tremblay, created a hypertext lab. In the lab they explained and or showed mathematical phenomena such as equilibria states and bifurcations. They plotted phase portraits of the above differential equations and included plots of the rate of infection as a function of time. They also made the model more intricate by altering the differential equation to include seasonal changes, as well as allowing births and non-disease deaths. Another group of students, Anastacia Rohrman and Kelly Plummer, compared their mathematical model to data from smallpox statistics in Mexico from 1922-1951. They incorporated vaccination into an Ordinary Differential Equation in order to model the effects of the actual vaccination efforts occurring in Mexico during that time. How does one make a safe roller coaster ride? This was a question behind two groups' work. One approach to answer this question comes from a roller coaster designer, Schwarzkopf. A result from his work is that safe roller coaster loops are ones where the radius of curvature of the loop decreases at a constant rate. The curvature K of a curve at a point is the magnitude of the rate of change of the direction of the curve with respect to arclength. Sandy Choi, Laura Wenzel, and Kris Keller then created via Maple, a mathematical software program, various parametrized curves. They had Maple color segments of the curve according to the degree of its curvature. They also plotted the curvature as a function to illustrate which curves would be safe for roller coaster rides, and which would not. What is the involute of a curve? What is an evolute? What happens when you take an evolute of an evolute, or an evolute of an involute? Chris Wyman, Erik Streed, and Tim McMurray tried to answer this in their project. Roughly speaking the evolute of a curve x(t) at t0 is another curve y(t) for which a circle of radius |y(t0) - x(t0)| centered at y(t0) best fits the curve at x(t). The involute can be thought of as a curve w(t) created by unwrapping a taught string from another curve z(t). The students plotted various curves as well as the involutes and evolutes of them, and made some interesting discoveries such as the evolute of a cardiod is a cardiod, in fact one that is similar to the original cardiod. They also found some curves where the involute and evolutes are inverses of one another. Overall the fair was an entertaining and enjoyable experience. Students demonstrated not only their mathematical acumen, but also were able to communicate their knowledge using several different media in several different ways. My only disappointment is that I wasn't able to see all of the exhibits. Fortunately, some of the students' hypertext documents are accessible on the Web at the URL address http://www.geom.umn.edu/docs/education/student/ of the Geometry Center. From news3.cis.umn.edu!umn.edu!hesse Fri Dec 2 14:07:54 GMT 1994 Newsgroups: geometry.research,geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@archimedes.geom.umn.edu (Bob Hesse) Subject: Some Applications of Virtual Reality Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: archimedes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 2 Dec 1994 14:07:54 GMT Lines: 101 Xref: forum.swarthmore.edu geometry.research:329 geometry.college:253 Imagine the following academic fiction: Eighteen professors from five departments decide to work together and submit a request for a virtual reality system. Suppose further that the administration actually believes that this is a wonderful idea and approves the proposal, provided that the virtual reality system is put to use in the classroom. The faculty eagerly agree to this condition, and to their amazement they acquire the funds to purchase an SGI Onyx 2 Reality Engine and 10 SGI Indigos. The above scenario is not some introduction to a John Grisham suspense novel, but a real story at Clemson University. Recently Steve (D.E.) Stevenson from the Department of Computer Science at Clemson University came to the Geometry Center and talked about applications of Geometry with computers. Steve mentioned briefly how various departments had been using the virtual reality system they acquired, and showed specific examples of what they had done with them. The departments using the system range from those which traditionally might use virtual reality, such as the Computer Science department, the Mechanical Engineering department and the Architecture department, to fields not generally associated with the technology such as the Biomedical Engineering department and the Performing Arts department. All these disciplines' projects use the technology in ways that create images and objects that otherwise would take a long time to construct, or not be feasible to construct at all. In particular, software is currently under development for Mechanical Engineering students that extends CAD/CAE software to virtual reality. Instead of clicking keystrokes to try to alter perspective views, a user is able to wear a helmet and by moving their head around are able to view an object as if it were before them. Moreover one is able to look through different layers of an object to view how the device is operating internally. Although these are all things that CAD/CAE software allows, the virtual reality system gives a user a more natural way to view an object, which accordingly allows one to easier ask the question, "what if?" Some of the other projects involving engineering are simulation-based design, multipurpose design optimization and visualization in High Performance Computing-Computer Formulated Design structures. Lastly one professor dreams of creating a simulation of the famous Tacoma Narrows bridge collapsing so that Civil and Mechanical Engineers can fully appreciate the consequences of their errors. In the Biomedical Engineering department some of the projects mentioned are use of virtual reality for viewing of X-RAY's and MRI's, using stereolithography to make prototypes of joints, and even having students perform test surgery. In the Computer Science department some of the projects range from creating a toolkit for non-computer science designers, rendering and 3-D lighting, viewing non-euclidean geometries, and modeling for resource management. Projects in the Architecture department include creating a virtual reality model of campus, and a laboratory on building design. People in the Performing Arts department use virtual reality for Stage Lighting and Stage Design Courses. Of the above projects, two of the more interesting applications common to both Mechanical Engineering and Biomedical Engineering, involve stereolithography or 3D printing. One is able to design or input given data about an object and actually create a prototype made out of polymers of the object viewed in the virtual reality. One interesting example is that of an image of a Pelvis taken from an MRI, piped into the virtual reality software so that one is able to view it, and then a model of the bone is manufactured using the polymer machine. Figure_1 is the virtual reality image of this pelvis. Similarly, a model of a "ship in a bottle" (Figure_2) was created using CAD/CAE software viewed through the virtual reality software, and then made. The virtual reality machines nicely compliment the polymer machine. One is able to thoroughly view an object before making a prototype, thus saving on the production costs of making a prototype. The Computer Science department has also created some interesting programs. Two software programs are titled Steve's Room and Oliver's Room. Steve's Room is a program which allows the user via the helmet to look around a room, turn on lights, and place objects by voice or mouse commands. Oliver's Room also is a high resolution room. In this room, one can see in high resolution, an Impressionist painting on the wall, a tiled floor, and a window with a view of mountains. Figure_3 shows a view of Oliver's Room. As with Steve's Room, the user is able via voice commands to move about the room. Figure_4 is an image of what one might see through the helmet after a request to move has been made. The visual results from these projects are amazing, both in a practical sense and in a pure aesthetic sense. The images created are useful in understanding the structure of an object, as well as being suitable for framing. However, what is equally impressive is that various departments were able to get together and pool their resources so that this system could be acquired. By doing this, they have provided themselves, and more importantly, their students, an opportunity to use computer systems today that will no doubt be commonplace in the future. Credit is given to Darren Crane at Clemson for creating the pictures for this article. The pictures may be acquired via anonymous ftp from the Geometry Forum in the directory /pictures/articles/virtual.reality. One can also read a version of this article with pictures on the World Wide Web at the URL address http://www.geom.umn.edu/docs/forum/vr/vr.html. From news3.cis.umn.edu!umn.edu!hesse Fri Jun 22 14:54:13 GMT 1994 Newsgroups: geometry.college Path: forum.swarthmore.edu!news.cc.swarthmore.edu!netnews.upenn.edu!newsserver.jvnc.net!howland.reston.ans.net!gatech!news-feed-1.peachnet.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: Calculus Consortium at the Geometry Center Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 22 Jul 1994 14:54:13 GMT Lines: 103 Last Monday the Minnesota Mathematics Mobilization sponsored a calculus seminar at the Geometry Center. I along with sixty secondary and post-secondary educators attended. The seminar was broken into a series of three workshops which were titled: "Selecting a Reformed Text, Developing Syllabi, and Designing Evaluation Methods"; "Precalculus/Calculus Cooperative Learning Methods"; and "Designing Class Materials and Exams to Incorporate Technology". Each attendee could register for two of the above offerings, I personally chose "Selecting a Reformed Text, Developing Syllabi, and Designing Evaluation Methods" and "Designing Class Materials and Exams to Incorporate Technology" In the "Selecting a Reformed Text, Developing Syllabi, and Designing Evaluation Methods" workshop the moderator, Paul Froeschl, broke the class into three groups of five and gave us the following scenario: You are the members of the Mathematics Department [of a small state college of 5000 students, mostly undergraduates with modest computing facilities] and you will "reform" your calculus course. Your Academic Dean has been at the job for three years. Coming out of the History Department, the Dean's goal is to be a college president some day. Seeing the job as that of a mediator and a resolver of conflicts, the Dean does not take academic leadership. The College President was hired to "strengthen" the university. Coming out of business--a Ross Perot type, the President is not really comfortable around academics. With this scenario in hand we were asked to develop a profile of the typical student coming out of the first semester of calculus, a final examination for the course in light of the student profile and other types of assessment modes during the course. During the second half of "Selecting a Reformed Text, ..." we were asked to select a calculus textbook, to develop a syllabus, and to discuss ways to evaluate the course. As we worked on the assignment I discovered that our group had a diverse background. Besides another graduate student from the University of Minnesota, there were two instructors from small liberal arts colleges (under 2000) and an instructor from a large university. In developing the student profile, we were deciding what a student should know after a semester of calculus. Because of our backgrounds, this job was a difficult task, and one which we spent most of the allotted time discussing. Those individuals from the smaller schools wanted to stress ideas of calculus, while those from the large schools wanted to emphasize techniques. Also, group members that had more experience with graphing calculators wanted to evaluate students in ways that incorporate this technology. When all the groups got together and reported on their work, we found that there was debate within all of the groups. The other groups also struggled with what were the important ideas of calculus, and what it meant to reform a calculus class. However, the discussion was healthy and I got a glimpse of how diverse groups of people could approach changing a course. The second workshop I attended involved designing class materials and exams to incorporate technology. Tracy Bibelnieks lead the workshop by presenting some problems from calculus texts of old. Here are a sample of those problems: Approximate (25)^(1/3). Find the dimensions of an "open top" box with largest volume created from a square sheet of cardboard 12 inches long by cutting off squares x inches long from the corners and folding up the sides. Find the number of real roots of the polynomial x^3 + x + 1. With the advent of the graphing calculator as well as programs like Mathematica, Maple and Derive, the above problems need to be rethought and rephrased, or maybe not asked at all. As in the previous workshop, we split into groups of five and looked at a particular area of calculus. Our task here was to create or rewrite problems to use the new technology. One group came up with the idea of designing a waterslide shaped like a backwards J. Three points on the slide were fixed: the top of the slide, the bottom of the slide before the curve, and the point on the slide after the curve. Students would be asked to piece together a function that would have the values at the fixed points, and be smooth so that someone going down the slide would not have a bumpy ride. In this problem, students would discover how to piece functions together so that the functions are continuous at the seams as well as having the first and maybe even the second derivatives match. Although this workshop consisted of post-secondary instructors, there was a brief discussion of the change in the 1995 AP exam permitting (in fact requiring!) the use of graphing calculators. Later, Tracy mentioned that in the morning workshop (consisting of high-school teachers) there was a great deal of discussion about the changes in the AP Exam, and especially on how to fit instruction of the technology into the classroom under the time constraints. After a few days of reflection, I believe the workshop was a good experience although not in ways I had expected. I expected and did indeed learn some concrete approaches to problems addressed in the workshops. But I learned much more through the informal discussions of the diverse crowd during the breaks. Sharing experiences with educators in both the secondary and post-secondary levels gave me new ideas and recharged me. As mentioned above, this workshop was sponsored by the Minnesota Mathematics Mobilization. I intend shortly to post a sequel to this article explaining what the organization is about and some of its goals. From nes3.cis.umn.edu!umn.edu!hesse Fri Nov 4 21:04:07 GMT 1994 Newsgroups: geometry.college,geometry.research Path: forum.swarthmore.edu!umn.edu!news From: hesse@diophantus.geom.umn.edu (Bob Hesse) Subject: Research Paper on the Web!? Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 4 Nov 1994 21:04:47 GMT Lines: 140 Xref: forum.swarthmore.edu geometry.college:243 geometry.research:317 Recently, I talked with Geometry Center postdoc Davide Cervone about his multimedia research paper, "A Tight Polyhedral Immersion of the Real Projective Plane with one Handle." which is located on the web at the address http://www.geom.umn.edu/docs/dpvc/RP2.html. The paper is easy to read, and I couldn't do any better explaining it than the author does himself. The purpose of this article is to discuss the format of Davide's paper, the problems associated with writing a paper in this form, and the general issues of having research papers available in this form. To begin with, I encourage you to look at this paper. Even if you have little understanding of what most of the words in the title mean, it is very self-explanatory and worth diving into. This paper was NOT written like a regular research paper; that is, a paper sparsely written with only the barest of definitions in a linear format. Papers like these are the stuff of printed journals, which usually only experts on the subject can read through without a reference book on hand. Unlike those in traditional formats, this paper is non-linear and uses many of the advantages a hypertext format has to offer. First off, with almost every mathematical term, Davide created a link to a definition of that term. For instance if the reader follows the link for the phrase: "Euler Characteristic" the following definition comes up: Euler Characteristic: The Euler characteristic of a closed surface is a topological invariant that can be computed in several ways. Two important ones are by counting critical points (the Euler characteristic is the number of maxima and minima minus the number of saddles) and by counting vertices, edges and faces of a polyhedral surface (the Euler characteristic is the number of vertices and faces minus the number of edges). The Euler characteristic is a fundamental value: this number uniquely classifies closed surfaces up to orientability. That is, given the Euler characteristic and orientability of a surface, the topological type of the surface is determined. This makes the Euler characteristic a powerful computational tool. Besides including links to definitions of words, Davide also has links tied to entire sentences. If a reader wants further explanation of a particular sentence, they need only to follow the link at the end of a sentence. For instance, in the section of the paper describing the history of the problem, one finds the sentence: "Every non-orientable surface with Euler characteristic strictly less than -1 admits a tight immersion into three-space." Following the link at the end of the sentence brings up a page containing this explanation: Tight Non-Orientable Surfaces: The non-orientable surfaces are divided into two families, one formed by adding handles to the Klein bottle, the other by adding handles to the real projective plane (just as all the orientable surfaces can be formed by adding handles to a sphere). The surfaces based on the Klein bottle have even Euler characteristic, and those based on the projective plane have odd Euler characteristic ... One great advantage of a paper like this is that it can be read by people at many different levels of understanding the subject. Experts in the field can go through the many points of the article without having to read through definitions and theorems they are familiar with, while a novice has the opportunity to go though the paper without needing additional reference books to understand the document. Besides including references for the terms, Davide also includes historical references that give a better background of the original formulation of the problem and the work that has been done on it. Again, this gives the novice or curious reader a chance to gain further information on the subject without having to delve for further references in the library. Information available at the fingertips is nice, but what else is there? Another thing Davide has included is movies and still images of the polyhedral surface and its level sets. Some web browsers may not be able to display these images, but by including a range of different formats, Davide has insured that a majority of people will be able to view at least one of them. These images are valuable because they enable the user to get a clearer understanding of the immersion by having many different pictures to view. Reading through Davide's paper many potential problems came to mind about writing articles in a hypertext format and in particular the HTML format. For those of you unfamiliar with HTML (Hyper Text Markup Language) it is a text language used to write many of the documents on the Web. It is a simple language that allows the writer to insert hyperlinks to pictures and other documents with ease within the original document. It is a powerful tool since it does not take too long to learn, and anyone can write a work in hypertext after a short time. However there are some drawbacks. One of its problems involves mathematical formulae. In HTML it is not easy to include documents in mathematical formats such as Tex, LaTex, or AMS-Tex. To solve this problem, Davide kept mathematical formula to a minimum and made most of the formula he did use into pictures, which were included into the document so that they appear to be part of the text. Also it is difficult to combine formula with text within the same line. There is software available that allows one to include formula in text, but Davide chose not to use this because its results are not that good. This is a minor issue, but an issue that should be addressed and solved because computer technology should allow the easy manipulation of formula. Another difficulty lies in writing up definitions to the terms. This was not a mentally difficult task, just time-consuming. Davide believes that in the future a database of glossary items will be available so that a writer will only need to create a link to the database, instead of having to write one up herself. Some problems I was not aware of involved pictures. A user that changes the size of the type-say for easier readability-will find that the pictures next to this larger type do not change in size. The result can be to have some awkward looking pictures. These format problems are minor; I doubt that many people will be distracted by these small problems. However these are issues people should be aware of if they desire to emulate Davide's work and write articles in a hypertext format. With the hypertext format, there are some new issues that will need to be addressed, such as: How will the user know that an article is authentic? How will they be able to access it? How much (if any) should the access costs be? Should the traditional linear paper form be kept? Should the writer be allowed to make revisions to the article even after it is posted? Some of the problems of authenticity, access, and cost are issues that are carried over from the traditional bound journal format, and probably will be addressed in the same fashion as they are in the original format. The issues as to whether an article must remain in a static form, and its form in general are a matter of much debate. As for Davide's paper, he has made some revisions since the original posting, and has marked each page with a date so that the reader will know when it has been last updated. As I mentioned at the beginning of this article, Davide's paper is a wonderful work and I encourage everyone to take a look at it. A hypertext format like this will probably become commonplace in the near future, so it is exciting to see it today and to watch this format grow and develop. From news3.cis.umn.edu!umn.edu!hesse Tue Sep 6 15:54:12 GMT 1994 Newsgroups: geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: Viewing Four-dimensional Objects in Three Dimensions Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Tue, 6 Sep 1994 15:54:12 GMT Lines: 133 Viewing four-dimensional objects in three dimensions Given that humans only visualize three dimensions, how is it possible to visualize four dimensional, or higher, objects? This question is the underlying idea of a short novel written over a hundred years ago by Edwin A. Abbot called FLATLAND. FLATLAND is a story about two-dimensional creatures--triangles, squares, circles and other polygons--that live on a plane. The story contains a section where one of the squares is visited by a three-dimensional object, a sphere. The sphere explains to the square the existence of higher dimensional objects like itself, and ways in which the square can understand the form of such objects. The method the sphere gives to the square can be generalized so that the form of four-dimensional objects can be seen in three dimensions. This method of viewing higher dimensional objects as well as others is one way people can understand the shape of higher dimensional space. Before attempting to view four-dimensional objects in three-dimensional space, let us consider viewing a three dimensional object in two-dimensional space. In FLATLAND, the method in which the sphere showed its form to square was by raising its body through the Flatland surface. The square saw at first a point that quickly grew to a circle, which continued increasing in size, and then started decreasing in size until it became a point, and then it disappeared. So the square perceived the sphere to be an infinite collection of circles pieced together. Figure 1 is a series of pictures that show the sphere as it rises through the plane as the square saw it: The flatlander square just as easily could have seen what a cube looked like by the following rising of the cube through space in figure 2. Before continuing further, it should be mentioned that for simplicity's sake and for aesthetic purposes, the forms which we will consider viewing are polytopes, the generalized term for polyhedra and polygons. A second way to view three dimensional polytopes in two dimensions is by means of a projection. Projection is a popular method for Cartographers to create maps of the world from a globe. For instance the United Nations flag is created by a projection of the globe about the south pole. One especially useful type of projection in mathematics is called stereographic projection. Stereographic projection takes a sphere and maps it over the entire plane in the following manner. If one lays a sphere on a plane, the point of the sphere touching the plane stays fixed while the point directly opposite it, i.e. "the North Pole" gets sent to infinity. Any other point on the sphere is sent to the unique point on the plane found by intersecting the plane with a line made from the point at the north pole and the point on the sphere. Figure 3 is an example of a cube which is contained in the sphere, stereographically projected onto the plane. Such a picture is also called a Schlegel diagram. Note that instead of projecting the Cube in the manor shown above, the cube could have been rotated so that its faces were not parallel and perpendicular to the plane, but rather at different angles which would result in a different projection. A third way to view polyhedra in two-dimensions is through a method defined by Barbara Hausmann and Hans-Peter Seidel as "Cut-Throughs" and "Fold-Downs". Since polyhedra have as faces regular polygons, one could cut a polyhedra on the edges and fold it in a way so that all the faces are lying on the plane. Figure 4 is an example of the cube after it has been cut-through and folded-down. As you may have already surmised, all of the above methods can be used to visualize four dimensional polytope in three dimensions. But before showing these different ways of viewing polytopes, an explanation of how these polytopes are constructed is in order. As regular polyhedra are constructed from regular polygons, so are regular 4-dimensional polytopes constructed from regular polyhedra. Recall that there are only five regular polyhedra: 1. The tetrahedron, constructed from four equilateral triangles. 2. The cube, constructed from six squares. 3. The octahedron, constructed from eight equilateral triangles. 4. The dodecahedron, constructed from 12 regular pentagons. 5. The icosahedron, constructed from twenty equilateral triangles. There are only six four-dimensional polytopes. They are the following: 1. The 4-simplex, constructed from five tetrahedra, three tetrahedra meeting at an edge. 2. The hypercube, constructed from eight cubes meeting three per edge. 3. The 16-cell, constructed from sixteen tetrahedra, with four tetrahedra meeting at an edge. 4. The 120-cell, constructed from 120 dodecahedra, with three dodecahedra meeting per edge. 4. The monstrous 600-cell, constructed from 600 tetrahedra, with five tetrahedra meeting at an edge. Since the above examples of viewing three-dimensional polytopes in two dimensions all contain the cube, let us continue viewing in the fourth dimension by looking at the hypercube. First, let us look at some projections of the hypercube. Figure 5 is a series of diagrams created by rotating the hypercube about a plane in four dimensions, or a combination of plane rotations. Figure 6 is a sequence of pictures of a slicing of the hypercube into three dimensions. Note that as the hypercube passes through our three-dimensional space, it is growing and then shrinking from various polyhedra shapes. This is analogous to the slicing of the cube in the plane shown earlier. Figure 7 is a set of pictures showing the step-by-step approach of the hypercube being cut-through and folded-down. Note that in each stage of the process a cube pops out of the hypercube, which is analogous to a square coming out of a cube as one dissects a cube. In FLATLAND, the square discovered that the sphere was an infinite collection of circles. However the square was unable ever to actually view the sphere in the same way we three-dimensional beings are able. Similarly, we can discover what some four-dimensional objects look like by viewing aspects of them in three dimensions. But like the square, we are limited in understanding the whole nature of these objects. This article is based on an interview and a seminar given by Barbara Hausmann at the Geometry Center. Figures are available via anonymous ftp from forum.swarthmore.edu in the /pictures/articles/polytopes directory. If Mosaic is available, the following article with pictures included is in the WWW document http://www.geom.umn.edu/docs/forum.html. From news3.cis.um.edu!umn.edu!hesse Wed Jul 1 18:53:10 GMT 1994 Newsgroups: geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: Geomview Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Fri, 1 Jul 1994 18:53:10 GMT Lines: 94 Here is a question for Geometry Center trivia buffs. What is the most requested software program created at the Geometry Center? The answer (as could be concluded by the title of this article) is Geomview. Over seven thousand copies of the program have been distributed, five thousand more copies than any other program. So what is Geomview? Quoting the Geomview Manual (Preprint GCG62), Geomview, pronounced ge-om-view, is an interactive program for viewing and manipulating geometric objects, written by staff members of the Geometry Center. It can be used as a stand alone viewer for static objects or as a display engine for other programs which produce dynamically changing geometry. It runs on Silicon Graphics (SGI) IRIS workstations and NeXT workstations. (page 1) After seeing the large interest in Geomview, I thought I should take a run with it to find out the reason for its popularity. For my first session with Geomview I followed the Tutorial section in the Geomview Manual. I loaded in the program and opened two files: tetra and dodec. When the program came up, three panels appeared on the screen, two control panels and the Camera Window. Inside the Camera Window I saw a picture of a dodecahedron (a twelve-sided polygon with pentagular faces) and part of a tetrahedron (a four sided polygon with triangular faces) whose view was obscured by the dodecahedron. The control panels along with the mouse enabled me to view objects in the Camera Window. I could rotate, translate, and even zoom in on the objects. Within a half hour I was able to move the dodecahedron and the tetrahedron apart, cause the tetrahedron to spin about its center, and have the dodecahedron revolve about the tetrahedron. In addition to the two control panels that appear when starting the program, there are panels that alter the environment. Among other things, one can adjust the appearance of an object, alter the way an object moves, and change the lighting on an object. Working with these other panels was no harder than using the initial panels. Being able to study more than one object at a time is one of the many benefits of Geomview. But the program does more than that; it allows several objects to interact with each other. After a couple of hours playing with the dodecahedron and tetrahedron and loading other pre-existing objects, I felt fairly comfortable using the program, and was ready for a bigger challenge. For my second session with Geomview, I decided to see how the software worked with other programs. I tried it with Mathematica. What I wanted to do was construct three-dimensional objects using Mathematica, and then view them with Geomview. There are two ways to do this. One way is to plot the image with Mathematica and then save the image in a file that Geomview can read. The second method is to use Geomview to display any three-dimensional output created by Mathematica. Being interested in seeing immediately what the functions looked like, I choose that latter method. The first function I choose to view was an old nemesis of mine from multivariable calculus, the hyperbolic paraboloid, commonly called a saddle. In multivariable calculus, sketching a saddle filled me with dread, since my drawings had no resemblance to those beautiful pictures I had seen in my textbook. I plugged in the function f(x,y) = x^2 - y^2 for Mathematica to plot. However instead of having Mathematica's graphics output appear, the Geomview program appeared with the picture of the saddle in the Camera Window. With Geomview, viewing the saddle from several different perspectives was simple and instantaneous. I was able to gradually turn the saddle around and see how the surface varied, something impossible with Mathematica's standard graphics output. I also viewed the function f(x,y) = x^2 + y^2 and again was amazed with the ease in which Geomview gave a detailed clear view of the object. Mathematica is a wonderful program and gives great insights into the form of three-dimensional objects. I would compare the difference between viewing an object with Mathematica's standard graphics and with Geomview like the difference between seeing a picture of the Grand Canyon and viewing the Canyon from the rim. Although the routines were already set up for me to use Geomview with Mathematica, it is possible to take graphic output from any program and render it with Geomview. In addition, if a program outputs graphic information constantly, it is possible to run Geomview concurrently with that program. As I mentioned earlier, my intent was to discover the appeal of Geomview. Although my foray was by no means comprehensive I did learn a great deal and was amazed at how easy it was to clearly view three dimensional objects using this software. There are however, two more interesting things Geomview can do. Geomview allows the user to view objects in non-euclidean spaces and view objects in higher dimensions. I have not had the opportunity to try either of these yet, but because Geomview is such an easy and fun program to use, I'm sure that when I enter into those frontiers my journey will be safe and enjoyable. For those interested in acquiring Geomview and the Geomview Manual (they are both free) do an anonymous ftp on the Internet from host `software@geom.umn.edu'. From news3.cis.umn.edu!umn.edu!hesse Mon Sep 12 18:06:15 GMT 1994 Newsgroups: geometry.research Path: forum.swarthmore.edu!umn.edu!news From: hesse@archimedes.geom.umn.edu (Bob Hesse) Subject: A Second Look at Geomview Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: archimedes.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 12 Sep 1994 18:06:15 GMT Lines: 121 This article is a followup to an earlier one on Geomview. It contains information from an interview with one of the writers of Geomview, Tamara Munzner. In my first article about Geomview, I gave a brief introduction of the program by telling of my experiences with it. This article expands on the first by answering some questions about the program and mentioning other applications. To refresh one's memory, as quoted previously from the Geomview Manual (Preprint GCG62), Geomview, pronounced ge-om-view, is an interactive program for viewing and manipulating geometric objects, written by staff members of the Geometry Center. It can be used as a stand alone viewer for static objects or as a display engine for other programs which produce dynamically changing geometry. It runs on Silicon Graphics (SGI) IRIS workstations and NeXT workstations. (page 1) It should be mentioned that there is also an Xwindows version for Geomview. Currently there is only a beta release available, but a polished version should be out soon. A couple questions asked about Geomview were the following: Why was Geomview created, and what are the future plans for the program. The motivation for Geomview was twofold. First, Geomview was designed to be a tool by which mathematicians could easily view and manipulate geometric objects. However since it is distributed for free through the Internet, Geomview's audience has expanded to people in many disciplines. Second, Geomview was created to be a program that displays graphics output from other programs. With the exception of the X-Windows version, Geomview is not currently under major development. Since Geomview has satisfied its goals in the two areas mentioned above, the program is complete. Because Geomview is stable, the programmers have focused their efforts on creating new applications for it. One enjoyable application is a connection between Geomview and World Wide Web called WebOOGL. (OOGL is the name of the 3D data files that Geomview can read.) It is accessible on the web at http://www.geom.umn.edu/docs/weboogl/weboogl.html. The basic idea of WebOOGL is to extend the Web beyond 2D hypertext to 3D "hyperscenes". In traditional Web browsers like Mosaic, hyperlinks can be attached to text, images, or movies and are followed by simply clicking on them with a mouse. If the link points to hypertext, the new page appears in the Mosaic window. Links to images start up an image viewer, links to movie start up a movie player, and so on. In WebOOGL use of the hyperlinks has been extended one step further. The user is able to view a 3D scene and access hyperlinks within that scene. For instance, in one part of the WebOOGL page, the user is invited to click on a picture of a pear and a dodecahedron placed on a table. After doing so, Geomview starts up on the user's own machine and one discovers that the still life of pear and dodecahedron is actually a three-dimensional object which one can rotate, zoom-in on and all the other actions that Geomview allows. However, WebOOGL is much more then just Geomview on the Web, which has been around for a while. When one clicks on the right mouse button while the pointer is over an object in Geomview, the module follows the hyperlink through the Web. If the link points to another 3D world, the scene in Geomview is replaced with the new world. If the link points to text, a new page comes up in the Mosaic window. For instance, a page with a bad joke about parity appears in the Mosaic window when the pear is clicked on (ugh!). As briefly mentioned in the first article, an interesting aspect of Geomview is its ability to view and move through non-Euclidean space. Two modules for Geomview, "Hinge" and "Interactive Hyperbolic Flythrough" highlight this ability and give the user a powerful hands-on chance to learn about non-Euclidean space. "Hinge" is a module written by Mark Phillips that allows the user to hinge multiple copies of a polyhedron around the edge of an original polyhedron, in both Euclidean and non-Eulidean space. If the polyhedra fit together perfectly with no gaps and no overlaps, then the hinging creates a tessellation of space. For instance, the user can click on one edge of a cube in Euclidean space. Then clicking on an adjacent face creates a new cube attached to the original cube at the hinged edge. The position of new cube is a rotation of the old cube by a preset number of degrees. One can change the number of degrees: for instance, one can stack one cube on top of another in Euclidean space by setting the angle to 90 degrees. In Euclidean space one cannot tile space perfectly with dodecahedra, whose interior angles are 117 degrees. In hyperbolic space one discovers that it is indeed possible to hinge dodecahedra so that the space can be tiled with them! In hyperbolic space, one can have a dodecahedron with interior angles of 90 degrees, so that four fit together perfectly at each edge. What is equally fascinating is attempting to tile hyperbolic space with cubes and being unsuccessful. "Interactive Hyperbolic Flythrough" explores further the tiling of hyperbolic space by right-angled dodecahedra, as seen in the movie "Not Knot". In this Geomview module the user can fly around inside the tessellation of space by dodecahedra. Only the edges of the dodecahedra are drawn, since from inside one of the dodecahedra the user's view would be blocked by the faces. The edges are colored either white, red, blue or green. Each face has exactly one edge colored non-white, so that face could be colored by the non-white edge. This coloring, which corresponds to the Borromean rings, is explained in the video. The user can either fly freely through the tiled space with the Geomview motion controls or or travel on pre-set paths. There are three pre-set paths, all of which have as a starting point and ending point of what would be a green face of the dodecahedron. All three paths give visually stunning results and are incredibly useful in understanding how the space is tiled. Both "Hinge" and "Interactive Hyperbolic Flythrough" are modules that come with Geomview. After becoming familiar with Geomview both modules are straightforward, user-friendly, and give the user an idea of the shape of hyperbolic space, i.e. how objects change in unfamiliar ways as one travels about them. In the first article, I incorrectly stated how one could access the Geomview program. One can download Geomview by an anonymous ftp from geom.umn.edu in the pub/software/geomview directory. Geomview comes with a manual, which you can download separately or read on the Web at http://www.geom.umn.edu/docs/software/viz/geomview/geomview_toc.html. There's also a brief technical introduction to Geomview on the Web at http://www.geom.umn.edu/docs/software/viz/geomview/geomview.html. Also, if you have Mosaic, you can also download Geomview directly from these Web pages, rather than going through the FTP program. From news3.cis.umn.edu!umn.edu!hesse Mon Aug 8 16:26:01 GMT 1994 Newsgroups: geometry.college Path: forum.swarthmore.edu!concert!news-feed-1.peachnet.edu!umn.edu!news From: hesse@diophantus.geom.umn.edu (Bob Hesse) Subject: World's Largest Icosahedron! (Probably) Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: diophantus.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 8 Aug 1994 16:26:01 GMT Lines: 60 Recently the Geometry Center was the construction site of the world's largest icosahedron. For those unfamiliar with an icosahedron, it is a twenty-sided polygon whose sides are equilateral triangles. This particular icosahedron was constructed with triangles whose sides are approximately 37 inches long. Sixth through eighth graders in a summer program run by the Special Projects Office of the University of Minnesota decorated the triangles. Students worked in pairs decorating each of the triangles. They colored a two-inch border around the triangle and then put different mathematical objects and symbols in the interior. Some students sketched platonic solids and archimedian solids, others made their triangles into Sierpinski triangles, and still others wrote equations like Euler's formula (vertices + faces - edges = 2) or the first few rows of Pascal's triangle. There were five different colors chosen for the borders of the triangle. Since each vertex of an icosahedron is created by the intersection of five triangles, each vertex was surrounded by five colors. With this color scheme, every vertex was symmetric to every other vertex by a translation and a rotation. Rick Wicklin, a postdoc at the Geometry Center, calculated that the polygon would be about 70 inches tall from vertex to opposite vertex. He came up with this number after discovering that the length from vertex to opposing vertex of an icosahedron is equal to 5^(1/4)*w^(1/2)*s where w is the golden ratio and s is the length of the side of a triangle. When I asked him for some justification for this formula, Rick scrambled into his office and pulled out a sheet of paper on which he had done his work. On the paper he had left a note that said, "I have a very elegant proof of this result, but am unable to fit it in the margin." :-) If anyone knows of the proof that Rick discovered, maybe they should post a reply to this article. An easier puzzle is the following: Rick cut the equilateral triangles out of foam board that were 32 by 40 inches long. Why did he end up with triangles with side length of 37 inches? Rick and Tracy Bibelnieks, an administrator in the Special Projects Office, got the idea for the project after reading an article in February's issue of FOCUS. The article mentioned that 4th and 5th graders in Maryland constructed the world's largest Rhombicosidodecahedron. In that polygon each side had a length of 13 inches. At the end of FOCUS article, there was a challenge daring others to try and better their record. Instead of simply constructing a larger one, Rick and Tracy choose to make a icosahedron of the same size. Since constructing a icosahedron as big as the rhombicosidodecahedron would require making sides of much greater length, they thought this would meet the posted challenge. The icosahedron also happens to be a symbol on the FOCUS magazine and there is talk of having the icosahedron near the MAA's booth at the Minnesota Mathfest. Rick Wicklin also mentioned that an icosahedron with 12 inch triangular sides is a good classroom model. A diagram showing the order in which the icosahedron was put together and the color scheme mentioned earlier can be found in postscript format at ftp://forum.swarthmore.edu/pictures/articles/icosahedron/icofig1.ps or for gif people the diagram can be found at ftp://forum.swarthmore.edu/pictures/articles/icosahedron/icofig1.gif From news3.cis.umn.edu!umn.edu!hesse Mon Oct 3 14:27:18 GMT 1994 Newsgroups: geometry.pre-college,geometry.college Path: forum.swarthmore.edu!umn.edu!news From: hesse@markov.geom.umn.edu (Bob Hesse) Subject: A Pandora's Box? Message-ID: Sender: news@news.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: markov.geom.umn.edu Organization: University of Minnesota, Twin Cities Date: Mon, 3 Oct 1994 14:27:18 GMT Lines: 102 Xref: forum.swarthmore.edu geometry.pre-college:912 geometry.college:215 Dear High School Math Teachers, Have you ever wondered how your former students were placed into a college math class? This letter is an opportunity for you to find out. This is also an attempt to generate discussion about students' placement in college math classes. In particular, I am wondering how other colleges and universities place students, and if any of you have feedback from former students regarding their placement, and its appropriateness. As you can tell by the title, I am probably opening a pandora's box with this article, but curiosity and a sincere interest in mathematics education have compelled me to write. My motivation for discussing math placement is the following: For the past few years one of my jobs as a graduate student at the University of Minnesota has been Math Advising Specialist for the College of Liberal Arts. In this position I advise incoming students who are both recent high-school graduates and non-traditional students. Advising a student often has two parts. First, one has to determine the student's level of mathematical ability. Second, one has to determine whether the student needs business and "professional" math classes versus science or engineering math classes. This second part of advising is often the easier one. Many students have a strong feeling where their interests lie, if not a particular major already set. Explaining the class options and various consequences usually is no problem. The first part of advising is where there can be difficulty. At the University, there is a system to determine proper math placement according to level of comprehension. The first statistic used is a "math index". This index is computed by the students high school math grades and their ACT math scores. For instance, students who have had high ACT scores as well as four years of math with grades of B or better, have a high math index number, while students with only two to three years of math with average or below average grades score on the lower end of the range. Students are also encouraged to take a "Math Readiness Exam". There are two types: the Calculus Readiness Exam and the College Algebra Readiness Exam. Students take one or the other depending on whether they plan to take calculus at some time, or if they just need a non-calculus math class for their major. After gathering the above information about the student, a table is referenced which indicates what level of math classes are appropriate. For instance, if only the math index number is available, there is a row on the table which indicates the appropriate math level. If there is a both an index and a readiness exam score, there is another row on the table showing the combined information for placement. This placement system seems to work the best when there is as much information about the student as possible, i.e. both the math index and the readiness exam. At orientation, a student receives a sheet with their math index and readiness exam score (if taken) along with a recommendation for what level of math class to take. Most of the time students are comfortable about the recommendation, and don't come to see me about placement. Often when students do come to talk to me, their only concern is what course would be an appropriate math class for their intended field of study. A problem can occur when the student disagrees with the placement rating and believes they are capable of a much higher math class then the rating recommends. A common scenario is a student has taken four years of high school math with Calculus senior year, has gotten some C's, and the Calculus Readiness Exam and the ACTs scores are low. According to the placement table, the recommendation is that the student take a precalculus class. Some students strongly disagree with this recommendation, since they had Calculus in high school, feel quite capable of taking Calculus again, and definitely not any class lower than that. Another common scenario is that the student has had three years of math, gotten some D's in high school and scored very low on their placement exam. A student with a background like this would be told to take a math class below college level before starting a college level math class. These students often times disagree, stating that they have already taken an algebra class in high school, why should they do so again for no college credit? Many of these students tell me that they had a bad day when taking the ACT or math readiness exam, or that they are not good test takers, or that they didn't get along with their high school instructors so those grades shouldn't be considered. I do take these comments into consideration. If all the other indicators in the student's file suggest that the above explanations are the case, I may give them the benefit of the doubt and wave the suggested recommendation. As I mentioned at the beginning of this letter, I am interested in hearing from both college educators about placement as well as high school teachers. In particular, I am very interested in hearing from high school teachers about feedback they have received from their former students about math placement.