Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.

The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered.

The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane.

An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers.

Two classes can be distinguished: In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size.

The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron.

Cubic honeycomb
It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles , as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. However, not all geometers accept such hexagons.