Differential Geometry Carved in Stone
I work on geometric computation and graphics in Mathematica, and for Mathematica 6 I was responsible for our new surface-drawing capabilities. When I talk about my work at university mathematics departments, I am often told that I just have to see what the department has tucked away in some corner of its building: plaster casts of intriguing mathematical surfaces, created in the early part of the twentieth century to illustrate the achievements of the field of differential geometry.
It’s been very difficult even to reproduce those plaster casts, let alone to go beyond them—each one represents a sophisticated combination of symbolic mathematics, numerics and geometry. But with Mathematica, we now have just the combination of capabilities that are needed. And I always find it fun to reproduce those plaster-cast surfaces—often with single lines of Mathematica code, usually centered on the function ParametricPlot3D. With 3D printing, I’ve even been able to make my own physical versions of lots of these surfaces.
In the last twenty years, there’s been quite a rebirth of interest in differential geometry—and Mathematica has been right in the middle of it. When Mathematica was young, I’m told that one of the first things several mathematicians wanted to do was to create an image of a hot new discovery in differential geometry: the Costa minimal surface.
For more than a century it had been believed that the minimal-area surfaces made by things like soap films had to follow a small number of possible forms. But in the 1980s it was discovered that another form, known as the Costa surface, was also possible. A few years after Mathematica was released, pioneering Mathematica user Alfred Gray managed to find a way to construct the Costa surface just using ParametricPlot3D, together with the function WeierstrassP.
Here is a recently published Demonstration of The Topology of Costa’s Minimal Surface:
After Alfred Gray died in 1998, longtime Mathematica users Stan Wagon* and Helaman Ferguson decided as a memorial to Gray to create a 12-foot-high Costa surface for the Team Minnesota entry to the 1999 International Snow Sculpture Championships—doing all the necessary computations with Mathematica. The resulting sculpture, titled “Invisible Handshake,” was widely admired—but of course melted in a few days.
The Costa surface has many interesting properties, which you can explore with Mathematica. One of them is that the surface has the same topology as a torus—and one can see this by finding a way to deform a torus into the surface, as in the Mathematica movie below.
Nine years after their snow sculpture, Helaman Ferguson and Stan Wagon again teamed up to generate a Costa model—but this time the work was done in solid stone. Ferguson began with an 11-ton block of granite and sculpted it down to a 3-ton Costa shape. In February 2008 it was installed in the Science Center at Macalester College in St. Paul, Minnesota.
It’s a fascinating application of Mathematica: going from an analytical theory, to numerics, to geometry and finally to a physical model. And with the physical model, it’s possible to get direct experience of some of the mathematics—as this human demonstration of the genus-1 character of the surface illustrates:
We’re finally at the point where we can firmly surpass the remarkable work done in creating plaster casts of differential geometry a century ago. Particularly with the new geometric algorithms in Mathematica 6, it has become straightforward to create sophisticated, smooth, parametric surfaces. They make great art, but more than that, they give us new insights into the whole field of differential geometry.
* Special thanks to Stan Wagon for providing many details and images used in this post.