In geometry, 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.
Each cell in a section of honeycomb has a hexagonal (‘hexagon-like’) opening and is in the shape of a type of irregular polyhedron called a decahedra. This means that the honeycomb cell has 10 flat sides – the prefix ‘deca-’ means ‘10’ and the suffix ‘-hedra’ means ‘geometric figure’. Also, all the flat sides of honeycomb cell except for the hexagonal opening are quadrilaterals.
There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytopecan be projected to its circumsphere to form a uniform honeycomb in spherical space.
The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space.