Cubic honeycomb

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Cubic honeycomb

Type	Regular honeycomb
Schläfli symbol	{4,3,4} t_0,3{4,3,4} {4,4} x {∞} {∞} x {∞} x {∞}
Coxeter-Dynkin diagram
Cell type	{4,3}
Face type	{4}
Vertex figure	8 {4,3} (octahedron)
Cells/edge	{4,3}⁴
Faces/edge	4⁴
Cells/vertex	{4,3}⁸
Faces/vertex	4¹²
Edges/vertex	6
Euler characteristic	0
Symmetry group	group [4,3,4]
Dual	self-dual
Properties	vertex-transitive

Vertex figure: octahedron

edge framework

The cubic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is an analog of the square tiling of the plane.

It is one of 28 uniform honeycombs using regular and semiregular polyhedral cells.

Four cubes exist on each edge, and 8 cubes around each vertex. It is a self-dual tessellation.

It is related to the regular tesseract which exists in 4-space with 3 cubes on each edge.

1 Uniform colorings
2 Related tessellations
- 2.1 Hypercube tessellations
- 2.2 Alternated hypercube tessellations
3 See also
4 References

[edit] Uniform colorings

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

Coxeter-Dynkin diagram	Partial honeycomb	Colors by letters
		1: aaaa/aaaa
		2: aaaa/bbbb
		2: abba/abba
		2: abba/baab
		4: abcd/abcd
		4: abbcbccd
		8: abcd/efgh

[edit] Related tessellations

[edit] Hypercube tessellations

The cubic honeycomb is part of a dimensional family of regular honeycombs with the Schläfli symbols {4,3...3,4}, constructed from 4 n-hypercubes per ridge. The vertex figure for every honeycomb is a cross-polytope {3...3,4}.

These are also named as - δ_n for an (n-1)-dimensional honeycomb.

Apeirogon {∞} - δ₂ -
Square tiling {4,4} - δ₃ - four squares/vertex. =
Cubic honeycomb {4,3,4} - δ₄ - four cubes/edge. =
Tesseractic tetracomb {4,3,3,4} - δ₅ - four tesseracts/face. =
Penteractic pentacomb {4,3,3,3,4} - δ₆ - four penteracts/cell. =
...

[edit] Alternated hypercube tessellations

A second infinite family is based on an alternation of the regular family, given a Schläfli symbols h{4,3...3,4} representing the regular form with half the vertices removed. The hypercube facets become demihypercubes, and the deleted vertices create new cross-polytope facets. The vertex figure for honeycombs of this family are rectified hypercubes.

These are also named as - hδ_n for an (n-1)-dimensional honeycomb.

Alternated square tiling - hδ₃ - h{4,4} - square faces alternate into edges, deleted vertices become new squares. (Same as regular square tiling {4,4})
- ( = )
Alternated cubic honeycomb - hδ₄ - h{4,3,4} - Cube cells alternate into tetrahedra, deleted vertices become octahedra.
- =
Alternated tesseractic tetracomb or demitesseractic tetracomb- hδ₅ - h{4,3,3,4} - Tesseracts alternate into 16-cells and deleted vertices also become 16-cells. (Same as regular {3,3,4,3})
- = ( = as a regular honeycomb)
Demipenteractic pentacomb - hδ₆ - h{4,3,3,3,4} - penteract alternate into demipenteracts and deleted vertices form pentacrosses.
- =
...