Honeycomb Tessellations
A honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as \(n\)-honeycomb for a honeycomb of \(n\)-dimensional space.
Convex Uniform Honeycomb
A convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs are known:
- the familiar cubic honeycomb and 7 truncations thereof;
- the alternated cubic honeycomb and 4 truncations thereof;
- 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
- 5 modifications of some of the above by elongation and/or gyration.
They can be considered the three-dimensional analogue to the uniform tilings of the plane.
The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra.