Visualizing three-dimensional geometries

Prof. Jeremy Tyson

Mathematics Department

University of Illinois

I'll describe an interesting way to do geometry in a three-dimensional space which is completely non-Euclidean yet entirely distinct from more "traditional" non-Euclidean geometries such as spherical geometry or hyperbolic geometry. This notion, which typically goes by the name Heisenberg geometry or nilgeometry, has its historical motivations from many different areas of pure and applied mathematics including complex variables, quantum mechanics, and the mechanics of nonintegrable physical systems (e.g., motion of a bicycle or car, robotic motion). It is one of the 8 flavors of 3D geometry, as conjectured by Bill Thurston in the 1980's and recently proved by G. Perelman in arguably the most spectacular mathematical advance of the 21st century. The ALICE project is a plan to develop computer visualization software for virtual reality environments in each of the 8 3D geometries. Three of the eight geometries have previously been implemented; we have five to go. Students would work with me to develop working computer models for nilgeometry.