Long before the advent of the WorldWide Web, Tom Banchoff was experimenting with ways of using electronic media to enhance mathematical research and aid in mathematical education. Banchoff helped install one of the first mathematics computer labs in the country, and continues to lead the development of innovative geometric software and curricula for undergraduate mathematics courses. He uses computer graphics as an integral part of his own research, and has used mathematical videos for the last 30 years as a means of disseminating his results.
When the WorldWide Web was being developed in the early 1990s, Banchoff immediately saw how it could be applied to mathematical education and research. On a trip to the Geometry Center in 1995, he began a number of webbased projects in an attempt to work out the new paradigms for using this medium effectively. (A number of these early attempts are linked below.)
Banchoff continues to use the Web as a fundamental part of his own work today. Many of his classes are held on line (using course software developed under his direction), and he routinely includes websites for the mathematical events in which he participates. This conference site is one such example, and several more are linked below.
Surfaces Beyond the Third Dimension This is an electronic art gallery that is based on an exhibition that took place in March of 1996 at the Providence Art Club in Providence, Rhode Island. Although the physical exhibit is long over, it lives on as a virtual experience on the web. The artworks all have their origins as computer graphics of surfaces with connections to fourdimensions. In this electronic version of the show (unlike the original), there are animations and virtual reality files that allow the visitor a dynamic view of the phenomena; there also are explanations of the mathematics involved and comments from the artist to help encourage the viewer to learn more about the underlying mathematics of the images. See his artwork page for more details.
Para Além da Terceira Dimensão This exhibit is based on the virtual gallery, "Surfaces Beyond the Third Dimension", described above. That web site was translated into Portuguese, and became a travelling exhibit, visiting over thirteen cities in Portugal and Brazil. It includes all the original artwork, plus two new images developed specially for this show, together with 13 new movie clips and associated descriptions. Unlike the original physical exhibit, this one incorporated computers where the visitors could interact with the various objects or view the movies. There was also an associated website, and a CDROM containing the artwork and complete web site so that viewers could take the show home with them. See the paper "A virtual reconstruction of a virtual exhibit" (5.8M PDF) for more details, and the artwork page for some photos from the traveling exhibit.
Math Awareness Month 2000 The MAM2000 poster had two components: a printed version distributed to mathematicians, high schools and colleges around the country, and an electronic version available over the Web. The theme of the month was "math spans all dimensions", and the poster revolved around a central cone image that progressed in dimension from 0dimensional points, to 1dimensional curves, to 2dimensional planes, to 3dimensional surfaces; the suggestion is, of course, that we can continue on to higher dimensions as well. Around the cone are pictures of people whose work relates to dimensions. The interactive poster includes links to information about the people shown, the various dimensions, and the central cone itself. There are animations to help you understand the central shape, and interactive areas where you can manipulate objects in the various dimensions. There are links to books and other web sites where you can find more information, and there are links to online articles about dimensions. See "Math Awareness Month 2000: an interactive experience" (3.8M PDF) for more information about the poster and its development process.
Computer Graphics in Mathematical Research: from ICM 1978 to ICMS 2002 In 1978, Banchoff gave a paper at the ICM conference in Helsinki that discussed five projects related to computer graphics. These developed into themes that he revisited over and over again, and 25 years later, in 2002, he gave a followup paper at the ICMS conference in Beijing that was a retrospective on these five topics. This poster illustrates the graphics available then and now, and gives an indication of the different mathematics that are suggested when the same object is viewed in two different ways. As with so much of Banchoff's recent work, there is an associated web site that explains the mathematics that appear in the poster. His paper, "Computer Graphics in Mathematical Research from ICM 1978 to ICMS 2002: A Personal Reflection" is available as a 2MB PDF file.
OnLine Classroom Tom Banchoff now runs nearly all of his classes on line using coursemanagement software developed under his direction by his student assistants. This software allows him to assign problems to which all the students reply, and at a certain point, they can see each others' responses and can comment on them. Banchoff also responds to each student, pointing out the strengths and weaknesses of his or her solution. Many courses include interactive graphics modules using a Java applet (developed by Banchoff's students) that allows the instructor to set up demonstrations where the student can move a slider or change a formula and see the results on screen immediately. Students can modify the demonstrations themselves, and can save the results and post them as part of their solutions to the problems posed in the course. Banchoff's current and past courses are available from this link, together with sample pages and demos.
Midpoint Polygons Revisited on the Internet In the Spring semester of 1999, Professor Thomas Banchoff of Brown University presented the midpoint polygons problem to the students in his Fundamental Problems of Geometry course, Math 104. He mentioned Rose Mary Zbiek's article, and challenged his students to go beyond the earlier investigations. The students discussed the midpoint problem on the interactive course webpage, as well as in class, and came up with a number of new results, including an explicit general formula for the area ratio. This paper is based on the results of that discussion.
Twice as Old — Again! One summer, Banchoff's older daughter, Ann, observed that he was at that point twice as old as her younger sister, Mary Lynn, again! He was already 30 at the time of the birth of ML (her preferred name at that time), and he made a comment about being twice as old as she was when she turned 30. Then Banchoff turned 61 and was no longer (exactly) twice as old. He thought that was the end of it. But Ann's observation was correct, sure enough. Of course, Banchoff used this as a jumping off point for an interactive discussion in his honors multivariable calculus course, documented in this report.
Beyond the Third Dimension This link is to chapters 1 & 2, "Introducing Dimensions" and "Scaling and Measurement", of the nine chapters in the book Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, by Thomas F. Banchoff. It appeared as part of the Scientific American Library in hardcover in 1990, and then in softcover in 1996, and is distributed by
W. H. Freeman and Company. Translations into several foreign languages are now in print. These two chapters have been augmented by some questions for study and suggestions for projects, and are reproduced here with permission of the publisher as part of the MAM2000 project described above.On the Shoulders of Giants This is chapter 2, "Dimension" from the book On the Shoulders of Giants: New Approaches to Numeracy, edited by Lynn Arthur Steen for the Mathematical Sciences Education Board of the National Research Council. It was published by the National Academy Press in 1990, and includes other chapters on pattern, quantity, uncertainty, shape, and change. This chapter presents the pedagogy behind the material in Beyond the Third Dimension above, and is reproduced here with permission of the publisher.
The Abbott Project Banchoff has an impressive collection of information about Edwin Abbott Abbott, the author of the Victorian novel Flatland. This project begins to document some of that material, and includes letters and images of Abbott, a bibliography, essays, and even a virtual tour of an exhibit created by Banchoff for Brown's Rockefeller library in 1999. The complete text of Flatland also is included, along with auxilliary material for the book. For the past three years, the Abbott project has been a collaboration with William Lindgren, professor of mathematics at Slippery Rock University.
The following are earlier, experimental works from Banchoff's stay at the Geometry Center. His visit to the Geometry Center affected him very strongly, and it was there that he began to integrate the interactive mathematics he had been doing for years with the new webbased hypertext medium. The examples below represent some of his earliest efforts in this area. While the graphics and movies are crude by today's standards, the technology was just being developed for this when these pages were written, and they represent the cutting edge of the time.Communications in Visual Mathematics This is the prototype volume for an electronic journal in mathematics developed in 1998. Unlike most contemporaneous electronic journals, which were designed to get papers to you faster and cheaper and included only a minimum of hypertext and alternatemedia components, the CVM was intended to explore the possibilities presented by the new hypertext technologies. The articles are not traditional, linearflow papers, but are constructed so that the structure of the papers corresponds more closely to the structure of the information involved. In addition, many include animations or interactive demonstrations that are central to the development of the ideas they present. The journal was a bit before its time, and the slow emergence of the MathML standards for mathematical markup on the web, and the even slower development of browsers that could handle the standard, hampered its success. Unfortunately, the journal did not progress past its initial volume.
Understanding Complex Function Graphs This is a story board for a remake of a "classic" film that investigates a new method of understanding the graph of a complex function of one complex variable as a surface in (real) fourdimensional space. The paper develops a mechanism of navigating the various views of the surface as projections from four dimensions to three, and maps these views onto a tetrahedron, where various paths on the tetrahedron correspond to rotations of the object in fourspace. In this way, the traditional views of the real and imaginary parts of the graph are linked by a series of intermediate steps. New insight into the nature of a complex function can be gained through the study of these sequences. A more polished version was under development for use in the CVM, but the project was never completed.
The Flat Torus in the ThreeSphere This project is a remake of some scenes from The Flat Torus in the ThreeSphere, a film originally created in 1968. One of Banchoff's favorite surfaces, this torus is featured in the animation on the home page of the TFBCON, on the cover of Beyond the Third Dimension, and in the art show Surfaces Beyond the Third Dimension. This surface is described in detail in the page for In and Outside the Torus, one of Tom's artworks from that show.
SelfLinking on the ThreeSphere This is a research project that investigates a means of measuring a linking number for curves lying in the threesphere in fourspace called the curve's selflinking number. This paper looks at the selflinking of various torus knots, and their projections into threespace. Some familiar knots in threespace can be viewed as images of the same knot in fourspace, and so have the same selflinking. The paper includes a copy of a letter from Banchoff to Nicolaas Kuiper from 1991 that discusses these results. More modern movies in a number of formats are available on this page without commentary.
Monge and Desargues, Identified This is a discussion of the relationship between Monge's Theorem and Desargues' Theorem that was inspired by a demonstration of Geometer's Sketchpad that Banchoff attended while at the Geometry Center. As a result, he learned to use Sketchpad and developed several demonstrations that are linked to this page to illustrate the twodimensional theorems. The connection between the theorems is made in three dimensions, and Banchoff includes images of an interactive demonstration that he developed in a 3D graphics package called Fnord.
Spherical Geometer's Sketchpad Inspired by Geometer's Sketchpad, an interactive tool for investigating twodimensional Euclidean geometry, Banchoff investigated elliptical geometry on the sphere using demonstrations he developed in Fnord. This 3Dgraphics program was created for him by students at Brown University, and though he had been using this program for years, these sophisticated demonstrations were actually the first ones he ever wrote himself. In this paper, he demonstrates how to construct a quadrilateral with two right angles and having the same area as a given triangle.
Linear Least Squares This short treatment of the geometry of least squares uses computergenerated standard illustrations in traditional textbook mode. There is also a prototype interactive demonstration using Fnord. This was the very first HTML document written by Banchoff.
The Volume of an nDimensional Ball When Banchoff sat in on a session for highschool students at the Geometry Center, he was asked by one student about how to compute the volume of a hypersphere. He gave a brief response at the time, but afterward, wrote this more detailed explanation for the students. It is a traditional nonhypertext paper in PDF form, but illustrates Banchoff's investigative mode of operation.
The Klein Bottle in 4D One can not build the Klein bottle in threespace without introducing selfintersection, but that can be done in four dimensions. Our familiar threedimensional views of the Klein bottle can be thought of as projections of embedded Klein bottles in fourspace. This paper presents some animations showing various views of Klein bottles as they rotate in fourspace.
Tight Embeddings of Surfaces Tight embeddings of surfaces has been the subject of a longstanding and ongoing investigation by Tom Banchoff. This is a short survey article, written with Wolfgang Kühnel, describing the main ideas and results of the field.
The Spherical TwoPiece Property Tight embeddings of surfaces can be characterized geometrically by the twopiece property: a surface is tight provided every plane cuts the surface into at most two pieces. This paper investigates the related spherical twopiece property (where the cutting is done by spheres rather than planes) and the cyclides of Dupin, which have this property.
From Flatland to Hypergraphics Another longstanding interest of Banchoff's is Edwin Abbott Abbot and his wonderful novel, Flatland. Banchoff owns an impressive collection of editions of Flatland, and of Abbott memorabilia. This paper is a partially updated version of a 1990 article that appeared originally in Interdisciplinary Science Reviews.

