# Saul Schleimer and Henry Segerman

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.

Tetrahedral racks
10 x 10 x 10 cm
3D printed nylon plastic
2016

This sculpture consists of four identical pieces. Each has three identical rods; each rod is a triangular prism, with racks on each of its three sides. The racks each mesh with one or two racks on the other pieces, for a total of 24 meshings. The pieces (and rods) are arranged using orientation preserving symmetries of the tetrahedron. At its middle position it realizes its maximal symmetry, the orientation preserving symmetries of the cube.

Challenge: Does this pattern of racks extend to tile three-space? If so, how many degrees of freedom does it have?

Borromean racks
10 x 10 x 10 cm
3D printed nylon plastic
2013

This sculpture consists of three identical pieces. Each has two identical rods; each rod is a rectangular prism with racks on three of its four sides. These racks mesh with one or two racks on the other pieces, for a total of 12 meshings. The pieces interlink in the fashion of the Borromean rings; their long axes form a standard orthogonal frame. When we take the directions of the racks into account, the sculpture has a handedness. At its middle position it realizes its maximal symmetry, a dihedral group of order six. Curiously, the Borromean Racks give an example of a triple of gears that mesh pairwise, but are not frozen.

Challenge: Does this pattern of racks extend to tile three-space? If so, how many degrees of freedom does it have?