Bitruncated cubic honeycomb

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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	R₄ or [4,3,4] S₄ or [4,3^1,1] P₄ or [៛]
Dual	Disphenoid tetrahedral honeycomb
Properties	cell-transitive, edge-transitive, vertex-transitive

Edge-drawn 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.

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.

1 Symmetry
2 Gallery
3 See also
4 External links

[edit] Symmetry

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

R₄ 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.
S₄ or [4,3^1,1] group - Three types of truncated octahedra in 2:1:1 ratios.
P₄ 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.