Math Video a Winner of 2007 "Science and Engineering Visualization"

October 19, 2007

A short video depicting the beauty of Möbius transformations was a winner of the 2007 Science and Engineering Visualization Challenge. The National Science Foundation, along with the journal Science, announced the winners of the fifth annual edition of this contest at the end of September.

Created by Douglas N. Arnold of the Institute for Mathematics and its Applications and Jonathan Rogness of the University of Minnesota, "Möbius Transformations Revealed" shows how moving to a higher dimension unveils the essential unity of these transformations. A low-resolution version of the video was released on YouTube in June 2007, where it has been seen by about 50,000 viewers.

Möbius transformations are among the most fundamental mappings in geometry, with applications from brain mapping to relativity theory. A Möbius transformation acts on the plane, sending each point to a corresponding point somewhere else on the plane, either by rotation, translation, inversion, or dilation. The video illustrates the notion that even the most complicated Möbius transformations can be understood as simple motions of a sphere.

As the sphere moves and rotates above the plane, suddenly all the transformations become linked, in a way that conveys visually in minutes what would otherwise take "pages of algebraic manipulations" to explain, Rogness said.

The transformation video earned an honorable mention in the "non-interactive media" category. First place went to a video titled "Nicotine: The Physiologic Mechanism of Tobacco Dependence."

This year's NSF visualization challenge attracted more than 200 entries from about two dozen countries. "Breakthroughs in science and engineering are often portrayed in movies and literature as 'ah-ha!' moments," said Jeff Nesbit, director of the NSF's Office of Legislative and Public Affairs. "What these artists and communicators have given us are similar experiences, showing us how bats fly or how nicotine becomes physically addictive."

"We look at their visualizations," he added, "and we understand."

Source: National Science Foundation, Sept. 27, 2007; Science, Sept. 28, 2007.