Bitruncated cubic honeycomb
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Bitruncated cubic honeycomb | |
---|---|
Rank | 4 |
Type | Uniform |
Space | Euclidean |
Notation | |
Bowers style acronym | Batch |
Coxeter diagram | o4x3x4o () |
Elements | |
Cells | N truncated octahedra |
Faces | 3N squares, 4N hexagons |
Edges | 24N |
Vertices | 6N |
Vertex figure | Tetragonal disphenoid, edge lengths √2 (bases) and √3 (sides) |
Measures (edge length 1) | |
Vertex density | |
Dual cell volume | |
Related polytopes | |
Army | Batch |
Regiment | Batch |
Dual | Bicubic honeycomb |
Conjugate | None |
Topological properties | |
Orientable | Yes |
Properties | |
Symmetry | R_{4}×2 |
Convex | Yes |
The bitruncated cubic honeycomb, or batch, is a convex noble uniform honeycomb. 4 truncated octahedra join at each vertex of this honeycomb. As the name suggests, it is the bitruncation of the cubic honeycomb, the medial stage in the series of truncations between a cubic honeycomb and its dual.
This honeycomb can be alternated into a bisnub cubic honeycomb, although it cannot be made uniform.
Before the discovery of the Weaire-Phelan structure, it was the most efficient known tiling of 3D Euclidean space.
Vertex coordinates[edit | edit source]
The vertices of a bitruncated cubic honeycomb of edge length 1 are given by all permutations of:
where i, j, and k range over the integers.
Representations[edit | edit source]
A bitruncated cubic honeycomb has the following Coxeter diagrams:
- (full symmetry)
- (S4 symmetry)
- (P4 symmetry, as omnitruncated tetrahedral honeycomb)
- (as alternated faceting)
Gallery[edit | edit source]
Related polytopes[edit | edit source]
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Cubic honeycomb | chon | {4,3,4} | ||
Truncated cubic honeycomb | tich | t{4,3,4} | ||
Rectified cubic honeycomb | rich | r{4,3,4} | ||
Bitruncated cubic honeycomb | batch | 2t{4,3,4} | ||
Rectified cubic honeycomb | rich | r{4,3,4} | ||
Truncated cubic honeycomb | tich | t{4,3,4} | ||
Cubic honeycomb | chon | {4,3,4} | ||
Small rhombated cubic honeycomb | srich | rr{4,3,4} | ||
Great rhombated cubic honeycomb | grich | tr{4,3,4} | ||
Small rhombated cubic honeycomb | srich | rr{4,3,4} | ||
Great rhombated cubic honeycomb | grich | tr{4,3,4} | ||
Small prismated cubic honeycomb = Cubic honeycomb | chon | t_{0,3}{4,3,4} | ||
Prismatorhombated cubic honeycomb | prich | t_{0,1,3}{4,3,4} | ||
Prismatorhombated cubic honeycomb | prich | t_{0,1,3}{4,3,4} | ||
Great prismated cubic honeycomb | gippich | t_{0,1,2,3}{4,3,4} |
External links[edit | edit source]
- Klitzing, Richard. "batch".
- Wikipedia Contributors. "Bitruncated cubic honeycomb".
- Binnendyk, Eric. "Category 2: Truncates" (#21).