A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling of the plane.

However, besides honeycombs in Euclidean space, we can also have honeycombs in hyperbolic space, which is a 3-dimensional Riemannian manifold with constant negative curvature. The {6,3,3} honeycomb lives in hyperbolic space. Here Roice Nelson has drawn it in the upper half space model of hyperbolic space, which is the 3d analogue of Poincaré’s famous upper half-plane model of the hyperbolic plane. As usual, you can click the image for a better view!

The {6,3,3} honeycomb is also called the **hexagonal tiling honeycomb**. The reason is that three hexagonal tilings of the plane meet at any edge of this honeycomb. This fact is recorded in the notation {6,3,3}, which is an example of a Schläfli symbol. The Schläfli symbol is defined in a recursive way. The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Similarly, the symbol for the hexagonal tiling honeycomb is {6,3,3} because 3 hexagonal tilings meet along each edge.

The {6,3,3} honeycomb is one of 15 regular honeycombs in hyperbolic space. For a complete list, with links to pictures, see:

• Tesselations of hyperbolic 3-space, Wikipedia.

Roice Nelson has a blog with lots of articles about geometry, and he makes plastic models of interesting geometrical objects using a 3d printer:

• Roice.

*Visual Insight* is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!