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Honeycombs!

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Honeycombs!

Here is a complete set of images of all 15 regular honeycombs in hyperbolic 3-space. Thanks to Tom Ruen for encouraging me to make these for wikipedia.

A honeycomb is when you take a bunch of polyhedra and pack them together with no gaps. The polyhedra in a honeycomb are called "cells".

The background colors group the honeycombs as follows:

Teal:  Cells are finite
Blue:  Cells have "ideal" vertices (vertices that live at infinity)
Green:  Cells have an infinite number of facets
Cyan:  Cells have ideal vertices and an infinite number of facets

All the images show the honeycombs in the Poincare Ball model, with the camera placed either at the origin or on the boundary of the ball. They were generated with custom C# code and rendered with POV-Ray.

Links for further study

"Regular Honeycombs in Hyperbolic Space", Coxeter:
http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf

YouTube video by Henry Segerman explaining a few of these honeycomb…

New things: Hyperbolic honeycombs.

Originally shared by Henry Segerman

New things: Hyperbolic honeycombs. These sculptures are joint work with Roice Nelson and show various tilings of 3-dimensional hyperbolic space.
http://www.youtube.com/watch?v=YzzJGeiucNg