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 
Abstract & topological properties  
Orientable  Yes 
Properties  
Symmetry  R_{4}×2 
Convex  Yes 
Nature  Tame 
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 WeairePhelan 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:
 o4x3x4o () (full symmetry)
 o4x3x2x3*b () (S_{4} symmetry)
 x3x3x3x3*a () (P_{4} symmetry, as omnitruncated tetrahedral honeycomb)
 s4x3x4o () (as alternated faceting)
 s4x3x2x3*b ()
Gallery[edit  edit source]

Wireframe
External links[edit  edit source]
 Klitzing, Richard. "batch".
 Wikipedia contributors. "Bitruncated cubic honeycomb".
 Binnendyk, Eric. "Category 2: Truncates" (#21).