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.

KEYWORDS:

• Hyperbolic geometry
• honeycombs
• polytopes
• tilings
• tessellations
• visualization
• 3D printing
• Coxeter groups
• reflection groups
• Schläfli symbols
• kleinian groups
• apollonian gasket
• fractals

Acknowledgements

We would like to thank Don Hatch and Nan Ma for early discussion about visualizing hyperideal honeycombs. The exciting thread that seeded our approach of boundary images is available online [7]. We thank Tom Ruen for useful discussions, and for suggesting the possibility of rendering {∞, ∞, ∞}. We thank Saul Schleimer and Christopher Tuffley for discussions on calculating distance in the cell adjacency graph of a honeycomb. We also thank the reviewers for their valuable feedback.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author was supported in part by National Science Foundation [grant number DMS-1308767].