# Saul Schleimer and Henry Segerman

Reader (SS), Assistant Professor (HS)

Mathematics Institute, University of Warwick (SS), Department of Mathematics, Oklahoma State University (HS)

Coventry, United Kingdom (SS), Stillwater, Oklahoma (HS)

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.

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.

Klein quartic

17 x 17 x 17 cm

3D printed nylon plastic

2015

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.

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.