Bitruncated cubic honeycomb

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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
Dual cell volume
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[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 () (S4 symmetry)
  • x3x3x3x3*a () (P4 symmetry, as omnitruncated tetrahedral honeycomb)
  • s4x3x4o () (as alternated faceting)
  • s4x3x2x3*b ()

Gallery[edit | edit source]

External links[edit | edit source]