This real-time interactive CAVE application takes you on a visit to the
post-Euclidean geometry of Gauss, Riemann, Klein, Poincare, and Thurston.
Here you can walk into a rectangular dodecahedron, a shape which is
possible only in negatively curved hyperbolic space. With a wand, you can
summon and play with the
snail-shaped 3D shadows
of soap films in positively curved elliptic space. You can see how to sew the edges of
hyperbolic octagons together into the surface of a 2-holed donut. The CAVE
becomes a spaceship you can navigate with the wand, as it glides through
the phantasmic shapes that populate the 3-sphere.
The purpose of this project is to perfect persuasive visual and sonic
environments in which to exhibit geometrical wonders and their startling
metamorphoses, which interest research geometers. Convincing
visualizations of multi-dimensional, time-varying geometrical structures
are equally useful in applied and pure mathematics.
MATHEMATICS
Post-Euclidean image (full size)