Bitruncated cubic honeycomb

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	$\frac{3\sqrt2}8 \approx 0.53033008589$
Dual cell volume	$\frac{4\sqrt2}3 \approx 1.88561808316$
Related polytopes
Army	Batch
Regiment	Batch
Dual	Bicubic honeycomb
Conjugate	None
Topological properties
Orientable	Yes
Properties
Symmetry	R4×2
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

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

• $\left(2\sqrt2i,\,±\frac{\sqrt2}{2}+2\sqrt2j,\,\sqrt2+2\sqrt2k\right),$

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

Representations

A bitruncated cubic honeycomb has the following Coxeter diagrams:

• (full symmetry)
• (S4 symmetry)
• (P4 symmetry, as omnitruncated tetrahedral honeycomb)
• (as alternated faceting)
• 

Gallery

• 

  Wireframe

• 
• 

Related polytopes

o4o3o4o truncations

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	t0,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t0,1,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t0,1,3{4,3,4}
Great prismated cubic honeycomb	gippich	t0,1,2,3{4,3,4}

External links

• Klitzing, Richard. "batch".

• Wikipedia Contributors. "Bitruncated cubic honeycomb".
• Binnendyk, Eric. "Category 2: Truncates" (#21).