Cubic honeycomb
From Wikipedia, the free encyclopedia
| Cubic honeycomb | |
|---|---|
 |
|
| Type | Regular honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | {4,3,4} t0,3{4,3,4} {4,4} x {∞} {∞} x {∞} x {∞} t0{4,31,1} |
| 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 |
| Coxeter groups | [4,3,4] [4,31,1] |
| Dual | self-dual |
| Properties | vertex-transitive |
 |
Please help improve this article or section by expanding it. Further information might be found on the talk page or at requests for expansion. (April 2007) |
The cubic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.
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.
[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] See also
[edit] References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
