Judging what a sculpture will look like after it is finished can present many difficulties for an artist just beginning a piece. Will it look "right" from every angle? Would a little tweaking of its shape or structure create a more pleasing or artistically interesting form?
In the past, sculptors have painstakingly constructed models out of wire and wax and other materials to try out their designs before fabricating the full-scale object in wood, stone, or metal. That sort of prototyping can take many weeks and involve considerable labor.
Computer scientist Carlo Séquin of the University of California, Berkeley, specializes in the computer-assisted design of objects, such as complicated machine parts, and the production of plastic or ceramic prototypes using stereolithography and other computer-driven fabrication techniques. He also creates three-dimensional art as a hobby.
Working with sculptor Brent Collins of Gower, Mo., Séquin and several of his students have developed software for manipulating certain geometric forms—specifically, shapes known as Scherk minimal surfaces (see http://mathworld.wolfram.com/ScherksMinimalSurfaces.html or http://www.math.uni-bonn.de/people/weber/minimal/gallery/scherk.html). They are named for the German mathematician Heinrich F. Scherk, who discovered them in the early part of the 19th century.
A minimal surface fills in a boundary with the smallest possible area. In three dimensions, these surfaces have a saddle shape, as seen in soap films spanning wire frameworks. A Scherk minimal surface consists of an array of holes and interlocking saddle forms set at 90 degrees to each other.
Séquin’s sculpture generator starts with a section of a Scherk minimal surface, having a specific number of holes and saddles.
That unit can then be stretched, twisted, bent into an arc, or joined end-to-end in a ring. The artist can also adjust color, surface texture, and edge thickness and even add a background scene to see how the final sculpture would look in a particular setting.
I recently had a chance to try my hand at creating a mathematical "art" object using Séquin’s sculpture generator. Here’s the result of about an hour of manipulation.
Collins and Séquin have collaborated on a number of art projects. The sculpture generator software has allowed them to try out shapes of much greater complexity than those possible using traditional prototyping techniques.
Séquin can also output these sculpture representations in a format that drives a machine capable of creating small, three-dimensional models. Séquin’s department recently acquired a fused deposition modeling machine, which allows Séquin to obtain plastic pieces of the desired geometry in a matter of hours or days (see http://www.cs.berkeley.edu/~sequin/CAFFE/cyberbuild.html).
Constructing a solid model from a virtual prototype has some advantages. It can demonstrate, for example, that a complex structure is physically realizable when even an inspection on the computer screen leaves some doubt, Séquin remarks.
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Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He and his wife, Nancy Henderson, have just completed Math Trek: Adventures in the MathZone, a book for children of middle-school age to be published in the fall by Wiley.MATHEMUSEMENTS: Look for math-related articles by Ivars Peterson every month in the children's general-interest magazine Muse (http://www.musemag.com) from the publishers of Cricket and Smithsonian magazine.