To begin with, I encourage you to look at this paper. Even if you have little understanding of what most of the words in the title mean, it is very self-explanatory and worth diving into. This paper was NOT written like a regular research paper; that is, a paper sparsely written with only the barest of definitions in a linear format. Papers like these are the stuff of printed journals, which usually only experts on the subject can read through without a reference book on hand. Unlike those in traditional formats, this paper is non-linear and uses many of the advantages a hypertext format has to offer. First off, with almost every mathematical term, Davide created a link to a definition of that term. For instance if the reader follows the link for the phrase: "Euler Characteristic" the following definition comes up:

- Euler Characteristic
- The Euler characteristic of a closed surface is a topological
invariant that can be computed in several ways. Two important
ones are by counting critical points (the Euler characteristic
is the number of maxima and minima minus the number of saddles)
and by counting vertices, edges and faces of a polyhedral surface
(the Euler characteristic is the number of vertices and faces
minus the number of edges).
The Euler characteristic is a fundamental value: this number uniquely classifies closed surfaces up to orientability. That is, given the Euler characteristic and orientability of a surface, the topological type of the surface is determined. This makes the Euler characteristic a powerful computational tool.

For instance, in the section of the paper describing the history of the problem, one finds the sentence:

- Every non-orientable surface with Euler characteristic strictly less than -1 admits a tight immersion into three-space.

- Tight Non-Orientable Surfaces
- The non-orientable surfaces are divided into two families, one formed by adding handles to the Klein bottle, the other by adding handles to the real projective plane (just as all the orientable surfaces can be formed by adding handles to a sphere). The surfaces based on the Klein bottle have even Euler characteristic, and those based on the projective plane have odd Euler characteristic ...

Besides including references for the terms, Davide also includes historical references that give a better background of the original formulation of the problem and the work that has been done on it. Again, this gives the novice or curious reader a chance to gain further information on the subject without having to delve for further references in the library.

Information available at the fingertips is nice, but what else is there? Another thing Davide has included is movies and still images of the polyhedral surface and its level sets. Some web browsers may not be able to display these images, but by including a range of different formats, Davide has insured that a majority of people will be able to view at least one of them. These images are valuable because they enable the user to get a clearer understanding of the immersion by having many different pictures to view.

Reading through Davide's paper many potential problems came to mind about writing articles in a hypertext format and in particular the HTML format. For those of you unfamiliar with HTML (Hyper Text Markup Language) it is a text language used to write many of the documents on the Web. It is a simple language that allows the writer to insert hyperlinks to pictures and other documents with ease within the original document. It is a powerful tool since it does not take too long to learn, and anyone can write a work in hypertext after a short time. However there are some drawbacks. One of its problems involves mathematical formulae. In HTML it is not easy to include documents in mathematical formats such as Tex, LaTex, or AMS-Tex. To solve this problem, Davide kept mathematical formula to a minimum and made most of the formula he did use into pictures, which were included into the document so that they appear to be part of the text. Also it is difficult to combine formula with text within the same line. There is software available that allows one to include formula in text, but Davide chose not to use this because its results are not that good. This is a minor issue, but an issue that should be addressed and solved because computer technology should allow the easy manipulation of formula.

Another difficulty lies in writing up definitions to the terms. This was not a mentally difficult task, just time-consuming. Davide believes that in the future a database of glossary items will be available so that a writer will only need to create a link to the database, instead of having to write one up herself.

Some problems I was not aware of involved pictures. A user that changes the size of the type-say for easier readability-will find that the pictures next to this larger type do not change in size. The result can be to have some awkward looking pictures.

These format problems are minor; I doubt that many people will be distracted by these small problems. However these are issues people should be aware of if they desire to emulate Davide's work and write articles in a hypertext format.

With the hypertext format, there are some new issues that will need to be addressed, such as: How will the user know that an article is authentic? How will they be able to access it? How much (if any) should the access costs be? Should the traditional linear paper form be kept? Should the writer be allowed to make revisions to the article even after it is posted? Some of the problems of authenticity, access, and cost are issues that are carried over from the traditional bound journal format, and probably will be addressed in the same fashion as they are in the original format. The issues as to whether an article must remain in a static form, and its form in general are a matter of much debate. As for Davide's paper, he has made some revisions since the original posting, and has marked each page with a date so that the reader will know when it has been last updated.

As I mentioned at the beginning of this article, Davide's paper is a wonderful work and I encourage everyone to take a look at it. A hypertext format like this will probably become commonplace in the near future, so it is exciting to see it today and to watch this format grow and develop.

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Created: November 7 1994 ---
Last modified: Jun 18 1996