# Roice Nelson and Henry Segerman

Roice Nelson is a recreational mathematician who draws inspiration from higher dimensional and non-euclidean geometry, and applies related ideas to digital art, permutation puzzles, and 3D printing. He appreciates software's ability to realize geometrical structure not physically possible in our 3D universe.

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.

For any 3 integers greater than 1, one can construct a regular {p,q,r} honeycomb with {p,q} cells and a {q,r} vertex figure. We render the honeycomb in the limit where p, q, and r all go to infinity, showing its intersection with the boundary of the upper half space model of hyperbolic 3-space. The cells have infinite volume: the vertices are "ultra-ideal", living beyond the boundary of hyperbolic space. The intersection of each cell with the boundary is an infinite collection of apeirogons (polygon with an infinite number of sides), together with a disk: all red apeirogons are intersections with the same cell, and its disk is the exterior of the image. The colour of the other cells change according to their distance from the red cell.

This picture visualizes the regular, self-dual {3,7,3} honeycomb in the upper half space model of hyperbolic 3-space. The cells are {3,7} tilings and the vertex figure is a {7,3} tiling. The cells have infinite volume: the vertices are "ultra-ideal", living beyond the boundary of hyperbolic space. The intersection of each cell with the boundary is an infinite collection of heptagons, together with a disk. The white ceiling and each red “creature” are isometric cells; for all other cells we only show the intersection with the boundary of hyperbolic space, on the floor of the catacombs. Every disk on the floor containing a {7,3} tiling is associated with an ultra-ideal vertex of the honeycomb.