Visualizing Hyperbolic Honeycombs

TL;DR

This paper develops visualization methods for hyperbolic honeycombs, a class of regular tilings in hyperbolic space, including rendering images and 3D printing strategies for various types of vertices and cells.

Contribution

It introduces visualization techniques for hyperbolic honeycombs, covering all categories of vertices and cells, and provides high-resolution images and 3D printing methods.

Findings

Finite spherical and euclidean honeycombs; infinite hyperbolic honeycombs.

Classification of hyperbolic honeycombs by vertex and cell types.

Visualization strategies for all categories of hyperbolic honeycombs.

Abstract

We explore visual representations of tilings corresponding to Schl\"afli symbols. In three dimensions, we call these tilings "honeycombs". Schl\"afli 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.