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Visualizing Mathematical Surfaces
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“Still Life,” created by Luc Bénard using visualizations from Richard Palais’ 3D-XplorMath program, was awarded first place in the illustration category of the Science/NSF 2006 Visualization Challenge. The surfaces in the image, clockwise from lower left, are: Klein Bottle, Symmetric 4-Noid, Breather, Bryant–Kusner Boy’s Surface, and the Sievert–Enneper Surface. Readers are encouraged to visit vmm.math.uci.edu/collection.jpg, where a high-resolution, color version of the image can be found. Courtesy of Richard Palais, University of California, Irvine, and Luc Bénard. |
by Michelle Sipics
Richard Palais, a mathematician at the University of California, Irvine, dates his work in mathematical visualization to the early 1990s, when his wife, Chuulian Terng, was researching the relationship between soliton mathematics and submanifold geometry. Terng wanted to visualize certain surfaces in order to better understand the relationship; Palais was able to convert mathematical ideas into numerical algorithms and, eventually, to bring visual representations of the surfaces into being. That experience provided the inspiration for 3D-XplorMath, a program for the Macintosh platform that Palais developed and has been updating ever since.
Palais has made 3D-XplorMath freely available since the late 1990s. One day early this year, he recalls, a message arrived "out of the blue" from one user, a graphic artist named Luc Bénard. Attached to Bénard's message was a digital image whose contents looked extremely familiar to Palais. Bénard had taken visualizations created by 3D-XplorMath, imported them into the 3D rendering software Bryce, and transformed what Palais recognized as "his children" into "glass sculpture"---a transformation that, he says, "seemed like pure magic."
The judges of the Science/NSF Visualization Challenge, seeming to agree, awarded Palais and Bénard first place in the illustration category for their image "Still Life."
The image, which contains visualizations of five mathematical surfaces, appeared on the cover of the September 22 issue of Science. (See black and white version above.) Shortly afterward, Palais had a curious experience: The network security group at UC Irvine alerted him to what appeared to be a "massive attack" on his Web site. But the mass of visitors to the site---far exceeding the 1000 to 1500 visitors of a typical month---had no malicious intent: They were downloading 3D-XplorMath and viewing the images in the Web site's galleries, to the tune of 26 gigabytes worth of downloads in a single day.
Palais, of course, was thrilled.
"Luc and I were delighted," he says of the Visualization Challenge award. "When we submitted it, I thought it might have a shot at an honorable mention---I didn't think that something that was of a purely mathematical nature had a serious chance at better than that. So the first prize came as a big and welcome surprise."
As for having the image appear on the cover of Science, Palais adds, "I think the feeling for both of us was a little like learning that you had won a big payoff in the lottery."
Visitors to the 3D-XplorMath Web site (http://vmm.math.uci.edu/3D-XplorMath/) can download a copy of the program, view galleries of mathematical visualizations, and find out more about the 3D-XplorMath Consortium, the group devoted to the development and evolution of the program.