# Saul Schleimer and Henry Segerman

Saul Schleimer is a geometric topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and additionally in visualization of ideas from these fields.

Henry Segerman is an assistant professor in the Department of Mathematics at Oklahoma State University. His mathematical research is in 3-dimensional geometry and topology, and concepts from those areas often appear in his work. Other artistic interests involve procedural generation, self reference, ambigrams and puzzles.

The Klein quartic K, given by x³y + y³z + z³x = 0, naturally lives in two-dimensional complex projective space. As proved by Klein, the intrinsic conformal structure on K is covered by the (2,3,7) triangle tiling of the hyperbolic plane. The 336 triangles that tile K show that it is the maximally symmetric genus three surface.

This sculpture is a projection of the Klein quartic to three-dimensional space. The projection retains a tetrahedral symmetry from the full group of order 336. Our construction is based on a parametrization of K due to Ramanujan. We used a hill-climbing algorithm to search a space of bihomogeneous polynomials to find a projection that balances the surface being embedded against its quasiconformal distortion.

In the plane, the three angles of a triangle must add up to π: 180 degrees. However on the sphere there is a triangle with angles (π/2, π/3, π/5). As shown in the sculpture, the sphere is tiled by 120 copies of this triangle. The LED is positioned at the north pole of the sphere. The resulting shadows are the stereographic projection of the triangles to the plane. Note how the angles of the tiling are faithfully reproduced.