, 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
in any number of dimensions.
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 polytope
can be projected to its circumsphere to form a uniform honeycomb in spherical space.
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.
The simplest honeycombs to build are formed from stacked layers or slabs
based on some tessellation 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. Another interesting family is the Hill tetrahedra
and their generalizations, which can also tile the space.
A uniform honeycomb
is a honeycomb in Euclidean 3-space composed of uniform polyhedral cells
, and having all vertices the same (i.e., the group of is transitive on vertices
). There are 28 convex
examples, also called the Archimedean honeycombs
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