Bitruncated cubic honeycomb

Bitruncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t1,2{4,3,4}
Coxeter-Dynkin diagram
Cell type	(4.6.6)
Face types	square {4} hexagon {6}
Edge figure	isosceles triangle {3}
Vertex figure	(disphenoid tetrahedron)
Cells/edge	(4.6.6)3
Cells/vertex	(4.6.6)4
Faces/edge	4.6.6
Faces/vertex	42.64
Edges/vertex	4
Space group	Im3m
Symmetry	[[4,3,4]]
Coxeter groups	${\tilde{C}}_3$ or [4,3,4]
Dual	Disphenoid tetrahedral honeycomb
Properties	isogonal, isotoxal, isochoric

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra.

It is one of 28 uniform honeycombs. It has 4 truncated octahedra around each vertex.

It can be realized as the Voronoi tessellation of the body-centred cubic lattice.

Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive.

Although a regular tetrahedron can not tessellate space alone, the dual of this honeycomb has identical tetrahedral cells with isosceles triangle faces (called a disphenoid tetrahedron) and these do tessellate space. The dual of this honeycomb is the disphenoid tetrahedral honeycomb.

Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.

Symmetry

This honeycomb has three 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.

Four uniform colorings by cell

Coxeter group	[[4,3,4]]	${\tilde{C}}_3$, [4,3,4]	${\tilde{B}}_3$, [4,31,1]	${\tilde{A}}_3$, [3[4]]
Space group	Im3m	Pm3m	Fm3m	?
Coxeter-Dynkin diagrams
truncated octahedra	1	1:1	2:1:1	1:1:1:1
Vertex figure
Image

Related honeycombs

Edge framework of the bitruncated cubic honeycomb seen in perspective

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

This honeycomb can be alternated, creating regular icosahedron from the truncated octahedra with irregular 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 symmetry can be doubled as [[4,3⁺,4]]. This honeycomb is represented in the boron atoms of the α-rhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.

Projection by folding

The bitruncated cubic honeycomb can be orthogonally projected into the planar truncated square tiling by a geometric folding operation that maps two pairs of mirrors into each other. The projection of the bitruncated cubic honeycomb creating two offset copies of the truncated square tiling vertex arrangement of the plane:

Coxeter group	${\tilde{A}}_3$	${\tilde{C}}_2$
Coxeter diagram
Graph	Bitruncated cubic honeycomb	Truncated square tiling

See also

• cubic honeycomb
• Brillouin zone

References

• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
• Richard Klitzing, 3D Euclidean Honeycombs, o4x3x4o - batch - O16
• Uniform Honeycombs in 3-Space: 05-Batch
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. 

External links

• Weisstein, Eric W., "Space-filling polyhedron" from MathWorld.