Bitruncated cubic honeycomb

Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) was the optimal soap bubble foam.

The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4-space, specifically permutations of (1,2,3,4).

The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling.

The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb.

(1), 8, 9 The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: This honeycomb can be alternated, creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps.

The dual honeycomb is made of cells called ten-of-diamonds decahedra.