Quarter 7-cubic honeycomb
# Quarter 7-cubic honeycomb

quarter 7-cubic honeycomb :

| (No image) ||
| - | - |
| Type | Uniform 7-honeycomb |
| Family | Quarter hypercubic honeycomb |
| Schläfli symbol | q{4,3,3,3,3,3,4} |
| Coxeter diagram | = |
| 6-face type | $h{4,3^5},$ $h_5{4,3^5},$ ${3^{1,1,1}}×{3,3}$ duoprism |
| Vertex figure | |
| Coxeter group | ${\tilde {D}}_{7}$×2 = $[[3^{1,1},3,3,3,3^{1,1}]]$ |
| Dual | |
| Properties | vertex-transitive |

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and ${3^{1,1,1}}×{3,3}$ duoprisms.

## Related honeycombs

This honeycomb is one of 77 uniform honeycombs constructed by the ${\tilde {D}}_{7}$ Coxeter group, all but 10 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 77 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{7}$ and ${\tilde {C}}_{7}$ constructions:

D7 honeycombs :

| Extended symmetry | Extended diagram | Order | Honeycombs |
| - | - | - | - |
| $[3^{1,1},3,3,3,3^{1,1}]$ | | ×1 | , , , , , , |
| $[[3^{1,1},3,3,3,3^{1,1}]]$ | | ×2 | , , , |
| $<[3^{1,1},3,3,3,3^{1,1}]>$ ↔ $[3^{1,1},3,3,3,3,4]$ | ↔ | ×2 | ... |
| $<<[3^{1,1},3,3,3,3^{1,1}]>>$ ↔ [4,3,3,3,3,3,4] | ↔ | ×4 | ... |
| $[<<[3^{1,1},3,3,3,3^{1,1}]>>]$ ↔ [[4,3,3,3,3,3,4]] | ↔ | ×8 | ... |
