Bitruncated cubic honeycomb

It can be realized as the Voronoi tessellation of the body-centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) was the optimal soap bubble foam. However, a number of less symmetrical structures have later been found to be more efficient foams of soap bubbles, among which the Weaire–Phelan structure appears to be the best.

The honeycomb represents the permutohedron tessellation for 3-space. The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4-space, specifically permutations of (1,2,3,4). The tessellation is formed by translated copies within the hyperplane.

The tessellation is the highest tessellation of parallelohedrons in 3-space.

Projections

The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling. A square symmetry projection forms two overlapping truncated square tiling, which combine together as a chamfered square tiling.

Orthogonal projections

Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the ${\displaystyle {\tilde {A}}_{3}}$ Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.

Five uniform colorings by cell

Space group	Im3m (229)	Pm3m (221)	Fm3m (225)	F43m (216)	Fd3m (227)
Fibrifold	8o:2	4−:2	2−:2	1o:2	2+:2
Coxeter group	${\displaystyle {\tilde {C}}_{3}}$×2 [[4,3,4]] =[4[3[4]]] =	${\displaystyle {\tilde {C}}_{3}}$ [4,3,4] =[2[3[4]]] =	${\displaystyle {\tilde {B}}_{3}}$ [4,31,1] =<[3[4]]> =	${\displaystyle {\tilde {A}}_{3}}$ [3[4]]	${\displaystyle {\tilde {A}}_{3}}$×2 [[3[4]]] =[[3[4]]]
Coxeter diagram
truncated octahedra	1	1:1 :	2:1:1 ::	1:1:1:1 :::	1:1 :
Vertex figure
Vertex figure symmetry	[2+,4] (order 8)	[2] (order 4)	[ ] (order 2)	[ ]+ (order 1)	[2]+ (order 2)
Image Colored by cell

The regular skew apeirohedron {6,4|4} contains the hexagons of this honeycomb.

The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.

C3 honeycombs
Space group	Fibrifold	Extended symmetry	Extended diagram	Order	Honeycombs
Pm3m (221)	4−:2	[4,3,4]		×1	1, 2, 3, 4, 5, 6
Fm3m (225)	2−:2	[1+,4,3,4] ↔ [4,31,1]	↔	Half	7, 11, 12, 13
I43m (217)	4o:2	[[(4,3,4,2+)]]		Half × 2	(7),
Fd3m (227)	2+:2	[[1+,4,3,4,1+]] ↔ [[3[4]]]	↔	Quarter × 2	10,
Im3m (229)	8o:2	[[4,3,4]]		×2	(1), 8, 9

The [4,3^1,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

B3 honeycombs
Space group	Fibrifold	Extended symmetry	Extended diagram	Order	Honeycombs
Fm3m (225)	2−:2	[4,31,1] ↔ [4,3,4,1+]	↔	×1	1, 2, 3, 4
Fm3m (225)	2−:2	<[1+,4,31,1]> ↔ <[3[4]]>	↔	×2	(1), (3)
Pm3m (221)	4−:2	<[4,31,1]>		×2	5, 6, 7, (6), 9, 10, 11

This honeycomb is one of five distinct uniform honeycombs[1] constructed by the ${\displaystyle {\tilde {A}}_{3}}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A3 honeycombs
Space group	Fibrifold	Square symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
F43m (216)	1o:2	a1	[3[4]]		${\displaystyle {\tilde {A}}_{3}}$	(None)
Fm3m (225)	2−:2	d2	<[3[4]]> ↔ [4,31,1]	↔	${\displaystyle {\tilde {A}}_{3}}$×21 ↔ ${\displaystyle {\tilde {B}}_{3}}$	1, 2
Fd3m (227)	2+:2	g2	[[3[4]]] or [2+[3[4]]]	↔	${\displaystyle {\tilde {A}}_{3}}$×22	3
Pm3m (221)	4−:2	d4	<2[3[4]]> ↔ [4,3,4]	↔	${\displaystyle {\tilde {A}}_{3}}$×41 ↔ ${\displaystyle {\tilde {C}}_{3}}$	4
I3 (204)	8−o	r8	[4[3[4]]]+ ↔ [[4,3+,4]]	↔	½${\displaystyle {\tilde {A}}_{3}}$×8 ↔ ½${\displaystyle {\tilde {C}}_{3}}$×2	(*)
Im3m (229)	8o:2		[4[3[4]]] ↔ [[4,3,4]]		${\displaystyle {\tilde {A}}_{3}}$×8 ↔ ${\displaystyle {\tilde {C}}_{3}}$×2	5

Alternated form

Alternated bitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	2s{4,3,4} 2s{4,31,1} sr{3[4]}
Coxeter diagrams	= = =
Cells	tetrahedron icosahedron
Vertex figure
Coxeter group	[[4,3+,4]], ${\displaystyle {\tilde {C}}_{3}}$
Dual	Ten-of-diamonds honeycomb Cell:
Properties	vertex-transitive

This honeycomb can be alternated, creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps. There are three constructions from three related Coxeter-Dynkin diagrams: , , and . These have symmetry [4,3⁺,4], [4,(3^1,1)⁺] and [3^[4]]⁺ respectively. The first and last symmetry can be doubled as [[4,3⁺,4]] and [[3^[4]]]⁺.

The dual honeycomb is made of cells called ten-of-diamonds decahedra.

Five uniform colorings

Space group	I3 (204)	Pm3 (200)	Fm3 (202)	Fd3 (203)	F23 (196)
Fibrifold	8−o	4−	2−	2o+	1o
Coxeter group	[[4,3+,4]]	[4,3+,4]	[4,(31,1)+]	[[3[4]]]+	[3[4]]+
Coxeter diagram
Order	double	full	half	quarter double	quarter
Image colored by cells

This honeycomb is represented in the boron atoms of the α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[2]

Wikimedia Commons has media related to Bitruncated cubic honeycomb.

• Architectonic and catoptric tessellation
• Cubic honeycomb
• Brillouin zone

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
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• Uniform Honeycombs in 3-Space: 05-Batch
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• Weisstein, Eric W. "Space-filling polyhedron". MathWorld.