Visualizing Hyperbolic Honeycombs

Abstract

We explore visual representations of tilings corresponding to Schläfli symbols. In three dimensions, we call these tilings "honeycombs". Schläfli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces in all dimensions. In three dimensions, there are only a finite number of spherical and euclidean honeycombs, but infinitely many hyperbolic honeycombs. Moreover, there are only four hyperbolic honeycombs with material vertices and material cells (the cells are entirely inside of hyperbolic space), eleven with ideal vertices or cells (the cells touch the boundary of hyperbolic space in some way), and all others have either hyperideal vertices or hyperideal cells (the cells go outside of the boundary of hyperbolic space in some way). We develop strategies for visualizing honeycombs in all of these categories, either via rendered images or 3D prints. High resolution images are available at hyperbolichoneycombs.org.