Bitruncated Cubic Honeycomb

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra.

It is one of 28 uniform honeycombs. It has 4 truncated octahedra around each vertex.

It can be realized as the Voronoi tessellation of the body-centred cubic lattice.

Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive.

Although a regular tetrahedron can not tessellate space alone, the dual of this honeycomb has identical tetrahedral cells with isosceles triangle faces (called a disphenoid tetrahedron) and these do tessellate space. The dual of this honeycomb is the disphenoid tetrahedral honeycomb.

Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.