Runcicantic cubic honeycomb

Runcicantic cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	h2,3{4,3,4}
Coxeter-Dynkin diagrams	=
Coxeter group	, [4,31,1]
Vertex figure
Dual	half pyramidille
Symmetry group	Fm3m (225)
Properties	vertex-transitive

The runcicantic cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cuboctahedra, truncated cubes and truncated tetrahedra in a ratio of 1:1:2. It is related to the runcicantellated cubic honeycomb.

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

Images

Runcicantic cubic

Runcicantellated cubic

Related 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.

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

See also

• 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, x3x3o *b4x - gratoh - O28
• Uniform Honeycombs in 3-Space: 14-Gratoh