How did it come about that an invited 45-minute address on computer graphics in mathematical research occurred at the International Congress of Mathematicians in Helsinki in 1978? What did we know at that time and what did we expect? What has actually happened over the past 24 years with respect to the topics featured in that first computer-illustrated invited talk at an ICM? This report at the ICMS will address the changes that have taken place in the author's collaborations with colleagues and students, particularly concerning advances in software for research and teaching in the areas of geometry and topology.
In the early 1970s, the only medium for presenting computer animation to an audience was film. In 1967, when I arrived at Brown University, I was fortunate to meet an applied mathematician, Charles Strauss, just finishing his Ph.D. in three-dimensional computer graphics, the first doctoral thesis supervised by the world expert on the subject,
Andries van Dam. We began using the techniques from his thesis to analyze surfaces in four-dimensional space, projected orthographically or stereographically into three-space, and within one year, we had produced our first film, "The Hypertorus" (described above). It was a laborious process, waiting the better part of a minute for each image of a wire-frame model with 400 vertices, then taking two pictures and instructing the computer to produce the next image. It took all night in a darkened room before a storage tube to generate a few seconds worth of film. What we saw convinced us that not only was it worth the effort. We would never be satisfied again with still images.
It was nearly five years before we succeeded in producing our next films, preliminary versions of "The Hypercube: Projections and Slicing" and "Complex Function Graphs". I showed these in a two-part series at Berkeley during my first sabbatical, 1973-4. The first talk was titled "Neue Polyhedralische Methoden in der Differentialgeometrie" and the second was "Disquisitiones Generales super Superficies Polyhedrales". My Ph.D. advisor, Professor
S.-S. Chern, showed interest in the project.
Two years later I gave a presentation at a symposium in honor of Prof. Chern. Afterwards he said to me that he thought it was time to present this to the world. He would put in a word, he said, with the selection committee for the ICM coming up in Helsinki. In late summer, 1977, I received a call from Prof.
Armand Borelat the Institute for Advanced Study in Princeton. Could I tell him something about these films that had been mentioned? I said I would be coming to New Jersey in two weeks for a talk and I could show some films to him then. He said he would have to know about them before that, and perhaps I could describe them on the phone. "I'll be there tomorrow," I said. Prof. Borel put up an announcement, and over forty people arrived for a presentation in the IAS theater. I narrated films on the hypercube, the Gauss mapping, and an early version of the Veronese surface, and presented two films on complex functions with their own soundtracks. The audience was appreciative.
Afterwards in his office, Prof. Borel said he thought that it would be good to have a showing of these films at the ICM. He indicated that there could be a side event some evening, or there could be an invited talk in
Section XIX, on History and Pedagogy. "You would probably prefer the latter?" he asked. I agreed, and that was the invitation.
Charles Strauss and I worked hard that next year, producing the version of "The Hypercube: Projections and Slicing" that is still used to this day. Since there were no color monitors available at that time, we achieved color by using color filters and a quadruple exposure for each frame, a semi-automated process that was still quite labor-intensive. The other films were all black-and-white, using vector graphics images with no filled surfaces. We worked especially on the Veronese surface, a series of images of the real projective plane which some exceptional singular projections.
At the ICM in Helsinki in 1978, the invited presentation was early in the meeting and over four hundred mathematicians attended. A few days later,
Prof. Michael Atiyahasked if I would give another showing since many people had heard about the films only after the address. Another full audience watched a second showing, in which I included the original "Hypertorus" film. The article for the Proceedings used screen photographs of stills from the films as illustrations. Most of the article is reproduced above, along with annotations and new renditions of many of the images.
The purpose of the talk in Helsinki was to show how this new medium could offer new insights and new possibilities for teaching and research. In this article, we make particular mention of advances that have taken place using computer graphics in various aspects of mathematics, especially when visualization contributes new insights into geometric phenomena, new conjectures, and new methods of proof.