Mathematical beauties
The American Mathematical Society (AMS) made us a great gift last week. It announced that a collaboration between a French mathematician and a Belgian artist has resulted in math animations that herald new era in visualization. And I must say that these animations about research in dynamical systems theory are absolutely astonishing. The AMS is even more enthusiast and says that these graphic animations "point to a new era for the use of computer graphics in communicating and carrying out mathematical research." But judge by yourself...
Let's start with a short excerpt of this AMS news release.
The two collaborators are Etienne Ghys, a mathematician at the Ecole Normale Superieure in Lyon, France, and Jos Leys, a Belgian graphic artist and engineer with strong mathematical interests. They have written an online animated exposition of an important new theorem Ghys has proved in dynamical systems theory.
I don't want to discuss here the discovery Ghys made by proving "a deep connection between two seemingly unrelated phenomena in dynamical systems theory: Lorenz attractors and modular flows." Instead, I would like to insist on the personal connection between a mathematician and an artist.
While Ghys was preparing an -- enthusiastically received -- lecture about his new theorem for the International Congress of Mathematicians in Madrid, Spain, in August 2006, he discovered intriguing mathematical images on Leys's website. And soon they decided to work together on animations.
All of the work was done on their personal computers, and their interaction has been almost entirely through email. "It was an intense collaboration," said Leys. "We exchanged about 1,800 email messages in 4 months, a lot of them sent late at night when we were close to a breakthrough." While Leys wrote code in a program called UltraFractal that is good for pictures but not ideal for mathematics, Ghys would work in parallel using specialized mathematical software packages to provide hints about how the computations should be done.
Now, it's time to look at some of the great images created by Ghys and Leys. Lets's start with this one: "For a given matrix A, one draws at the same time a Seifert surface and the knot kA. As the construction of kA evolves, it intersects the surface from time to time, positively or negatively. The linking number we are looking for is the sum of these signs." This image is obtained with two plus signs. (Credit: Etienne Ghys/Jos Leys/AMS)
For an explanation about how this second image was generated, please read the paragraph 3.2 of the full AMS column. Basically, you can see a stable manifold in green and an unstable in red. "This structure of stable and unstable manifolds creating intricate nets in the phase space has become a central tool of study in dynamical systems: the so-called hyperbolic theory." (Credit: Etienne Ghys/Jos Leys/AMS)
This third image is an example of a modular flow called the horocyclic flow. You should really watch the animation associated with the wonderful dynamics of this flow. For a detailed explanation about this image has been created, please read the paragraph 4.2 of the full AMS column. Here is a single shot. (Credit: Etienne Ghys/Jos Leys/AMS)
These images have been extracted from the AMS November 2006 Feature Column. Here is a link to this very long paper. It contains links to absolutely stunning animations in QuickTime format. And please note that if you read this post in a few months that the links would no longer be valid when AMS transfers this column to its archive section.
And don't miss Jos Leys's image galleries. Here is a link to the Knots and dynamics gallery which contains many images releated to the subject of this post.
Sources: American Mathematical Society news release, via EurekAlert!, November 1, 2006; and various websites
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