# Bitruncated cubic honeycomb

Bitruncated cubic honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymBatch
Coxeter diagramo4x3x4o ()
Elements
CellsN truncated octahedra
Faces3N squares, 4N hexagons
Edges24N
Vertices6N
Vertex figureTetragonal disphenoid, edge lengths 2 (bases) and 3 (sides)
Measures (edge length 1)
Vertex density${\displaystyle {\frac {3{\sqrt {2}}}{8}}\approx 0.53033}$
Dual cell volume${\displaystyle {\frac {4{\sqrt {2}}}{3}}\approx 1.88562}$
Related polytopes
ArmyBatch
RegimentBatch
DualBicubic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryR4×2
ConvexYes
NatureTame

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

The vertices of a bitruncated cubic honeycomb of edge length 1 are given by all permutations of:

• ${\displaystyle \left(2{\sqrt {2}}i,\,\pm {\frac {\sqrt {2}}{2}}+2{\sqrt {2}}j,\,{\sqrt {2}}+2{\sqrt {2}}k\right)}$,

where i , j , and k  range over the integers.

## Representations

A bitruncated cubic honeycomb has the following Coxeter diagrams:

• o4x3x4o () (full symmetry)
• o4x3x2x3*b () (S4 symmetry)
• x3x3x3x3*a () (P4 symmetry, as omnitruncated tetrahedral honeycomb)
• s4x3x4o () (as alternated faceting)
• s4x3x2x3*b ()