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From Notices of the AMS

The Hexagonal Tiling Honeycomb

by John C. Baez Communicated by Han-Bom Moon

This picture by Roice Nelson shows the "hexagonal tiling honeycomb." But what is it? Roughly speaking, a honeycomb is a way of filling three-dimensional space with polyhedra. The most familiar one is the usual way of filling Euclidean space with cubes. This cubic honeycomb is denoted by the symbol $\{4,3,4\}$, because a square has 4 edges, 3 squares meet at each corner of a cube, and 4 cubes meet along each edge of this honeycomb. We can also define honeycombs in hyperbolic space, which is a three-dimensional Riemannian manifold with constant negative curvature. For example, there is a hyperbolic honeycomb denoted $\{4,3,5\}$, which is a way of filling hyperbolic space with cubes where 5 meet along each edge.

Coxeter has classified the most symmetrical hyperbolic honeycombs 3, called the "regular" ones, and the hexagonal tiling honeycomb is one of these. But it does not contain polyhedra of the usual sort, with finitely many faces.

Read more»

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Stay up-to-date with mathematics in current events and get ideas for classroom activities! Also read Tony Phillips' deeper dives into recent mathematical developments.