Truncated cubic honeycomb

Truncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t0,1{4,3,4}
Coxeter-Dynkin diagrams
Cell type	3.8.8, {3,4}
Face type	{3}, {4}, {8}
Cells/edge	(3.8.8)4 {3,4}.(3.8.8)2
Faces/edge	{8}4 {3}2.{8}
Cells/vertex	3.8.8 (4) {3,4} (1)
Faces/vertex	{8}4+{3}4
Edges/vertex	5
Euler characteristic	0
Vertex figure	square pyramid
Space group	Pm3m
Coxeter group	${\tilde{C}}_3$, [4,3,4]
Dual	Hexakis cubic honeycomb
Properties	vertex-transitive

The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1.

Edge framework

Symmetry

There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.

Construction	Truncated cubic honeycomb	Bicantellated alternate cubic
Coxeter group	[4,3,4], ${\tilde{C}}_3$	[4,31,1], ${\tilde{B}}_3$
Space group	Pm3m	Fm3m
Coloring
Coxeter-Dynkin diagram
Vertex figure

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, x4x3o4o - tich - O14
• Uniform Honeycombs in 3-Space: 03-Tich