Bitruncated cubic honeycomb
- 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.
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 [4,31,1] 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.
Gallery
A larger partial honeycomb with random colors.
ee also
*cubic honeycomb
External links
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