Sphere with Chaotic Geodesic Flow

My area of research is chaotic properties of dynamical systems. Systems that I am particularly interested include geodesic flow on surfaces and billiards. My colleague Keith Burns and I created the first examples of (unusual shaped) spheres and other surfaces that can actually exist in regular three dimensional space and whose geodesic motion is chaotic (see the article Embedded surfaces with ergodic geodesic flow, Inter. J. of Bifurcation and Chaos Vol. 7, No. 7 (1997), 1509-1527.)

Below are more such surfaces.

By attaching "focusing caps" to each of the ends, we create a sphere. This is the first known example of a sphere in three dimensional space whose geodesic flow is chaotic (ergodic).

Images created by Bryn Mawr math major Louisa Winer using Mathematica, Surface Evolver and Geomview. Thanks to Rob Kusner, Ken Brakke, Bogdan Butoi, Michelle Francl and Lisa Chirlian for technical assistance.