Outside In
Symposium:
Session Title:
- Visualization (The Next Generation)
Presentation Title:
- Outside In
Presenter(s):
Abstract:
Abstract
Outside In is a Mathematical Visualization Project from the Geometry Center. The video illustrates an amazing mathematical discovery made in 1957: you can turn the surface of a sphere inside out without making a hole, if you think of the surface as being made of an elastic material that can pass through itself. Communicating how this process of eversion can be carried out has been a challenge to differential topologists ever since.
“Outside In” uses nontechnical language and computer animation to illustrate the process and to explain the concepts involved to a non-mathematical audience. Yet, the video retains mathematical depth: we introduce the concept of a “regular homotopy” from topology, which is traditionally not encountered until advanced undergraduate mathematics classes. The metaphor we use is that of a material that can stretch and pass through itself, but that self-destructs if punctured or even pinched sharply. Of course, there is no such material in real life! That’s where computer graphics comes in. The framework is a dialogue between a female teacher and a male student. In the first scene they work out between themselves the ground rules of what it means to turn a sphere inside out, but the student remains skeptical that the problem can be solved under these rules. If anything his skepticism increases in subsequent scenes, as the teacher persuades him that a circle cannot be turned inside out under the same rules. However, an idea that is introduced in connection with curves—namely, adding waves, or corrugations—turns out to be useful for surfaces as well. In the final scene of the twenty-minute movie, the student is shown how to turn the sphere inside out using this corrugation method. The process is shown a number of different ways to build up the student’s (and the viewer’s) intuition.
Acknowledgements:
“Outside In” is the result of collaboration between mathematicians, programmers, and designers. We developed a great deal of custom software in addition to using RenderMan, Softimage, and Mathematica. I want to thank Silvio Levy and Tamara Munzner (the other two directors), Nathaniel Thurston, Stuart Levy, David BenZvi, Daeron Meyer, and all of the other contributors. “The Geometry Center” is the informal name for the National Science and Technology Center for the Computation and Visualization of Geometric Structures, based at the University of Minnesota.
