7-cubic honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.
| 7-cubic honeycomb | |
|---|---|
| (no image) | |
| Type | Regular 7-honeycomb Uniform 7-honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | {4,35,4} {4,34,31,1} {∞}(7) |
| Coxeter-Dynkin diagrams | |
| 7-face type | {4,3,3,3,3,3} |
| 6-face type | {4,3,3,3,3} |
| 5-face type | {4,3,3,3} |
| 4-face type | {4,3,3} |
| Cell type | {4,3} |
| Face type | {4} |
| Face figure | {4,3} (octahedron) |
| Edge figure | 8 {4,3,3} (16-cell) |
| Vertex figure | 128 {4,35} (7-orthoplex) |
| Coxeter group | [4,35,4] |
| Dual | self-dual |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(7).
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