Seminar

We survey what we know about the dynamics of billiards in rational prisms, that is, right prisms over rational polygons. We will discuss how to use beautiful ideas of Furstenberg and Veech to make connections between mixing properties of billiards in polygons to ergodic properties of billiards in prisms. There will be lots of examples, and lots of pictures, and no prior knowledge of ergodic theory is needed. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

The exponential map is one of the most common functions in many areas of mathematics. However, when it comes to illustration, it is often a source of struggle. Even when we draw its graph, we have only a limited window, which makes it difficult to grasp all the features of this function. When we represent fractals, we very quickly create a mass of details that we are unable to render properly. In computations, it can be a source of numerical errors due to the limited precision of floating-point numbers, etc. In this talk, I will present some attempts (failures?) of mathematical illustration where, to my own misfortune, I came across the exponential function. 

How do you visualize something which doesn't fit on a page, like a high dimensional space or an entire field of math? You lie and cheat, use shortcuts and shorthand.  In this talk, I'll share my favorite tips and tricks for bringing math to life in digital painting.

The Illustrating Mathematics Seminar Online celebrates the life and work of Roger Antonsen.

The theoretical discovery of hyperbolic geometry first got its actual tactile example in 1868 when Eugenio Beltrami created a negatively curved surface from paper annuli and named it a pseudosphere. Later the name pseudosphere got attached to a surface created by a tractrix rotating around its axis. However, mathematicians found more useful for theoretical purposes using different, non-tactile models such as Klein or Poincare disc models or half-plane model. Those are traditionally used in college textbooks. However, to experience deeper understanding of hyperbolic geometry, these models were not enough for Bill Thurston when he was a college student. Since in 1901 Hilbert proved that hyperbolic plane cannot be described analytically in 3-space, Thurston together with his peers at informal seminar decided to make a tactile model of hyperbolic plane and created it by gluing together paper annuli without knowing about Beltrami’s paper model created hundred years earlier. I learned about Thurston’s model in 1997 and decided to make it more durable by crocheting it. Crocheted hyperbolic planes have turned out to be a useful tool in tactile explorations of hyperbolic geometry giving to theoretical knowledge a different perspective. 
 

I'll tell a few mathematical stories from my personal experience with mathematical illustration as a research tool in number theory, sharing some of my experience with the process, not just results.  Topics include Apollonian circle packings, Mobius transformations, continued fractions, and algebraic integers.

I'll tell a few mathematical stories from my personal experience with mathematical illustration as a research tool in number theory, sharing some of my experience with the process, not just results.  Topics include Apollonian circle packings, Mobius transformations, continued fractions, and algebraic integers.

When most people picture mathematical art, they envision regular shapes formed by plotting precise equations. In this talk, I will discuss my approach to creating art through a very different process, where mathematics is used to guide a virtual growth process.  The results frequently mirror shapes found in nature, both in form and aesthetic appeal.

Nelson Max will show segments of several computer animated films from the 1970s, on turning a circle or sphere inside out by a regular homotopy, and on space filling curves, and demonstrate a web-based system for uniform tilings of the sphere, plane, and hyperbolic plane.

I’ve been illustrating mathematics since I’ve known any, and will be showing a variety of work in a range of media.