Truncated alternated cubic honeycomb

Truncated alternated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t0,1{31,1,4}
Coxeter-Dynkin diagram
Vertex figure
Coxeter groups	[4,31,1], ${\tilde{B}}_3$
Dual	-
Properties	vertex-transitive

The truncated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid.

Edge framework

Symmetry

It has two different uniform constructions. The ${\tilde{A}}_3$ construction can be seen with alternately colored truncated tetrahedra.

Symmetry	[4,31,1], ${\tilde{B}}_3$	[3[4]], ${\tilde{A}}_3$
Name	Truncated alternate cubic	Cantitruncated quarter cubic
Coloring
Coxeter
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.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. 
• Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1. 
• 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.
• D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
• Richard Klitzing, 3D Euclidean Honeycombs, x3x3o *b4o - tatoh - O25
• Uniform Honeycombs in 3-Space: 13-Tatoh