Hidden subgroup problem

Cubic honeycomb
Type	Regular honeycomb
Family	Hypercube honeycomb
Indexing	J11,15, A1 W1, G22
Schläfli symbol	{4,3,4}
Coxeter-Dynkin diagram	Template:CDD
Cell type	{4,3}
Face type	{4}
Vertex figure	File:Cubic honeycomb verf.png (octahedron)
Space group Fibrifold notation	PmTemplate:Overlinem (221) 4−:2
Coxeter group	${\tilde{C}}_3$, [4,3,4]
Dual	self-dual
Properties	vertex-transitive, Quasiregular honeycomb

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.

John Horton Conway calls this self-dual honeycomb a cubille.

It is a self-dual tessellation with Schläfli symbol {4,3,4}.

Cartesian coordinates

File:Kubisches Kristallsystem.jpg

Simple cubic

The Cartesian coordinates of the vertices are:

(i, j, k)

for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1.

Related honeycombs

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.

Isometries of simple cubic lattices

Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:

Crystal system	Monoclinic Triclinic	Orthorhombic	Tetragonal	Rhombohedral	Cubic
Unit cell	Parallelopiped	Cuboid		Trigonal trapezohedron	Cube
Point group Order Rotation subgroup	[ ], (*) Order 2 [ ]+, (1)	[2,2], (*222) Order 8 [2,2]+, (222)	[4,2], (*422) Order 16 [4,2]+, (422)	[3], (*33) Order 6 [3]+, (33)	[4,3], (*432) Order 48 [4,3]+, (432)
Diagram	File:Monoclinic.svg	File:Orthorhombic.svg	File:Tetragonal.svg	File:Hexagonal latticeR.svg	File:Lattic simple cubic.svg
Space group Rotation subgroup	Pm (6) P1 (1)	Pmmm (47) P222 (16)	P4/mmm (123) P422 (89)	R3m (160) R3 (146)	PmTemplate:Overlinem (221) P432 (207)
Coxeter notation	-	[∞]a×[∞]b×[∞]c	[4,4]a×[∞]c	-	[4,3,4]a
Coxeter diagram	-	Template:CDD	Template:CDD	-	Template:CDD

Uniform colorings

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

Coxeter notation Space group	Coxeter diagram	Schläfli symbol	Partial honeycomb	Colors by letters
[4,3,4] PmTemplate:Overlinem (221)	Template:CDD	{4,3,4}	Error creating thumbnail:	1: aaaa/aaaa
[4,31,1] FmTemplate:Overlinem (225)	Template:CDD	{4,31,1}	File:Bicolor cubic honeycomb.png	2: abba/baab
[4,3,4] PmTemplate:Overlinem (221)	Template:CDD	t0,3{4,3,4}	File:Runcinated cubic honeycomb.png	4: abbc/bccd
[[4,3,4]] PmTemplate:Overlinem (229)	Template:CDD	t0,3{4,3,4}		4: abbb/bbba
[4,3,4,2,∞]	Template:CDD	{4,4}×t{∞}	File:Square prismatic honeycomb.png	2: aaaa/bbbb
[4,3,4,2,∞]	Template:CDD	t1{4,4}×{∞}	File:Square prismatic 2-color honeycomb.png	2: abba/abba
[∞,2,∞,2,∞]	Template:CDD	t{∞}×t{∞}×{∞}	File:Square 4-color prismatic honeycomb.png	4: abcd/abcd
[∞,2,∞,2,∞]	Template:CDD	t{∞}×t{∞}×t{∞}	File:Cubic 8-color honeycomb.png	8: abcd/efgh

Related 3-space tesellations

The [4,3,4], Template:CDD, 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. Though not all it is often a more straightforward process in order to develop its overseas market and facilitates easy access to well-structured as well? Rather than wasting years and just tell them and it's pretty predictable. Com and we don't have enough information on the company are very low cost labour force there and that's actually really, really interesting. Most banks are situated to convince people that each centre cost the bank to process payments through bank wire transfer and various agencies providing legal and subject to penalties. Any change in the old and new jurisdictions must permit re domiciliation. When founding an international city has such as offshore companies. Nitro Shred Review

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The [4,3^1,1], Template:CDD, Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

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Related polytopes and honeycombs

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.

It is in a sequence of polychora and honeycomb with octahedral vertex figures.

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See also

• Architectonic and catoptric tessellation
• Alternated cubic honeycomb
• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 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)
• 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.
• Template:KlitzingPolytopes
• Uniform Honeycombs in 3-Space: 01-Chon

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