Bitruncated_cubic_honeycomb

Bitruncated cubic honeycomb



Type	Uniform honeycomb
Schläfli symbol	t_1,2{4,3,4}
Coxeter-Dynkin diagrams	>-	Cell type	(4.6.6)
Face types	square {4} hexagon {6}
Edge figure	isosceles triangle {3}
Vertex figure	4 (4.6.6) (disphenoid tetrahedron)
Cells/edge	(4.6.6)³
Cells/vertex	(4.6.6)⁴
Faces/edge	4.6.6
Faces/vertex	4².6⁴
Edges/vertex	4
Coxeter groups	B^~₃ or [4,3,4] C^~₃ or [4,3^1,1] A^~₃ or [៛]
Dual	Disphenoid tetrahedral honeycomb
Properties	cell-transitive, edge-transitive, vertex-transitive

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.

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.

Symmetry

This honeycomb has three uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions:

B^~₃ or [4,3,4] group - Two types of truncated octahedra in 1:1 ratio. Half from the original cells of a cubic honeycomb, and half are centered on the vertices of the original honeycomb.
C^~₃ or [4,3^1,1] group - Three types of truncated octahedra in 2:1:1 ratios.
A^~₃ or [៛] group - Four types of truncated octahedra in 1:1:1:1 ratios.

These uniform symmetries can be represented by coloring differently the cells in each construction.