Cubic honeycomb

Cubic honeycomb
Type	Regular honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3,4}
Coxeter-Dynkin diagram
Cell type	{4,3}
Face type	{4}
Vertex figure	(octahedron)
Cells/edge	{4,3}4
Faces/edge	44
Cells/vertex	{4,3}8
Faces/vertex	412
Edges/vertex	6
Euler characteristic	0
Coxeter groups	${\tilde{C}}_3$, [4,3,4]
Dual	self-dual
Properties	vertex-transitive

edge framework

The cubic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron.

It is a self-dual tessellation with Schläfli symbol {4,3,4}. It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.

It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

Related polytopes and tesellations

It is related to the regular 4-polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4-space, and only has 3 cubes around each edge. It's also related to the order-5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge.

Uniform colorings

There is a large number of uniform colorings, derived from different symmetries. These include:

Coxeter-Dynkin diagram	Schläfli symbol	Partial honeycomb	Colors by letters
	{4,3,4}		1: aaaa/aaaa
	{4,4}xt{∞}		2: aaaa/bbbb
	t1{4,4}x{∞}		2: abba/abba
	{4,31,1}		2: abba/baab
	t{∞}xt{∞}x{∞}		4: abcd/abcd
	t0,3{4,3,4}		4: abbc/bccd
	t{∞}xt{∞}xt{∞}		8: abcd/efgh

See also

• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
• 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, x4o3o4o - chon - O1
• Uniform Honeycombs in 3-Space: 01-Chon