Omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t0,1,2,3{4,3,4}
Coxeter-Dynkin diagram
Vertex figure	Phyllic disphenoid
Space group Fibrifold notation Coxeter notation	Im3m (229) 8o:2 [[4,3,4]]
Coxeter group	[4,3,4],
Dual	eighth pyramidille
Properties	vertex-transitive

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

John Horton Conway calls this honeycomb a b-tCO-trille, and its dual eighth pyramidille.

Symmetry

Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octahedral prisms. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octahedral prism cells.

Two uniform colorings

Symmetry	, [4,3,4]	×2, [[4,3,4]]
Space group	Pm3m (221)	Im3m (229)
Fibrifold	4−:2	8o:2
Coloring
Coxeter diagram
Vertex figure

Related honeycombs

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.

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

Alternation

A snub cubic honeycomb can be constructed by alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given Coxeter-Dynkin diagram: and has symmetry [4,3,4]⁺. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms and with new tetrahedral cells created in the gaps.

See also

Wikimedia Commons has media related to Omnitruncated cubic honeycomb.

• Architectonic and catoptric tessellation

References

• 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)
• 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
  • (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, x4x3x4x - otch - O20
• Uniform Honeycombs in 3-Space: 08-Otch