Saul Schleimer and Henry Segerman

Associate Professor (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.
Triple Gear
19.8 x 11.3 x 13.0 cm
PA 2200 Plastic, Selective-Laser-Sintered, with motorised base.
A relatively common sight in graphic design is a planar arrangement of three gears in contact. However, since neighbouring gears must rotate in opposite directions, none of the gears can move. This sculpture gives a non-planar, and non-frozen, arrangement of three linked gears. We were inspired by existing arrangements with two linked gears, due independently to Helaman Ferguson and Oskar van Deventer.

The gears are powered by a motor in the base of the sculpture, which rotates a central axle. We thank Adrian Goldwaser and Stuart Young for prototyping and designing the motorised base.
Seifert surfaces for (3,3) and (4,4) torus links
13.1 x 12.1 x 10.5 cm and 14.3 x 14.3 x 14.6 cm
PA 2200 Plastic, Selective-Laser-Sintered
A torus link is a link that can be drawn on a torus. A Seifert surface spans its link, somewhat like a soap-film clinging to its supporting wire-frame. The surface acts as a bridge between the 1-dimensional link and the 3-dimensional space it lives in.

The torus links and their Seifert surfaces live most naturally in the 3-sphere, a higher dimensional version of the more familiar sphere. We transfer our sculptures to Euclidean 3-space using stereographic projection. The Seifert surface is cut out of the 3-sphere by the Milnor fibers of the corresponding algebraic singularity. We parametrize the Milnor fiber, following the work of Tsanov, via fractional automorphic forms. These give a map from SL(2,R), the canonical geometry of the torus link complement, to the 3-sphere. The patterns on each Seifert surface arise from two applications of the Schwarz-Christoffel theory of complex analysis, turning a Euclidean triangle into a hyperbolic one.