Honeycomb (geometry) - Duality of Honeycombs

Duality of Honeycombs

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

cells for vertices.

walls for edges.

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

The more regular honeycombs dualise neatly:

• The cubic honeycomb is self-dual.
• That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.
• The slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are.
• The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald.