Cubic honeycomb

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Cubic honeycomb

Type	Regular honeycomb
Family	Hypercube honeycomb
Indexing^[1]	J_11,15, A₁ W₁, G₂₂
Schläfli symbol	{4,3,4}
Coxeter-Dynkin diagram
Cell type	{4,3}
Face type	{4}
Vertex figure	(octahedron)
Space group Fibrifold notation	Pm3m (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

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
Space group Rotation subgroup	Pm (6) P1 (1)	Pmmm (47) P222 (16)	P4/mmm (123) P422 (89)	R3m (160) R3 (146)	Pm3m (221) P432 (207)
Coxeter notation	-	[∞]_a×[∞]_b×[∞]_c	[4,4]_a×[∞]_c	-	[4,3,4]_a
Coxeter diagram	-			-

Uniform colorings

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

Coxeter notation Space group	Schläfli symbol	Colors by letters
[4,3,4] Pm3m (221)	{4,3,4}	1: aaaa/aaaa
[4,3^1,1] Fm3m (225)	{4,3^1,1}	2: abba/baab
[4,3,4] Pm3m (221)	t_0,3{4,3,4}	4: abbc/bccd
[[4,3,4]] Pm3m (229)	t_0,3{4,3,4}	4: abbb/bbba
[4,3,4,2,∞]	{4,4}×t{∞}	2: aaaa/bbbb
[4,3,4,2,∞]	t₁{4,4}×{∞}	2: abba/abba
[∞,2,∞,2,∞]	t{∞}×t{∞}×{∞}	4: abcd/abcd
[∞,2,∞,2,∞]	t{∞}×t{∞}×t{∞}	8: abcd/efgh

Related 3-space tesellations

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] = [4,3^1,1]	=	Half	₇, ₁₁, ₁₂, ₁₃
I43m (217)	4^o:2	[[(4,3,4,2⁺)]]		Half × 2	₍₇₎,
Fd3m (227)	2⁺:2	[[1⁺,4,3,4,1⁺]] = [[3^[4]]]	=	Quarter × 2	₁₀,
Im3m (229)	8^o: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	[4,3^1,1] = [4,3,4,1⁺]	=	×1	₁, ₂, ₃, ₄
Fm3m (225)	2⁻:2	<[1⁺,4,3^1,1]> = <[3^[4]]>	=	×2	₍₁₎, ₍₃₎
Pm3m (221)	4⁻:2	<[4,3^1,1]>		×2	₅, ₆, ₇, ₍₆₎, ₉, ₁₀, ₁₁

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.

{p,3,4}
Space	S³	E³	H³
Name	{3,3,4}	{4,3,4}	{5,3,4}	{6,3,4}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}
Vertex figure

It in a sequence of regular polychora and honeycombs with cubic cells.

{4,3,p}
Space	S³	E³	H³
Name	{4,3,3}	{4,3,4}	{4,3,5}	{4,3,6}
Image
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}

References

↑ For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).

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
- (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

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family	${{\tilde {A}}}_{{n-1}}$	${{\tilde {C}}}_{{n-1}}$	${{\tilde {B}}}_{{n-1}}$	${{\tilde {D}}}_{{n-1}}$	${{\tilde {G}}}_{2}$ / ${{\tilde {F}}}_{4}$ / ${{\tilde {E}}}_{{n-1}}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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