Honeycomb (geometry)

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.

Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra:

For every honeycomb there is a dual honeycomb, which may be obtained by exchanging:

These are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems.