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 Eng