Quarter 6-cubic honeycomb
# Quarter 6-cubic honeycomb

quarter 6-cubic honeycomb :

| (No image) ||
| - | - |
| Type | Uniform 6-honeycomb |
| Family | Quarter hypercubic honeycomb |
| Schläfli symbol | q{4,3,3,3,3,4} |
| Coxeter-Dynkin diagram | = |
| 5-face type | $h{4,3^4},$ $h_4{4,3^4},$ {3,3}×{3,3} duoprism |
| Vertex figure | |
| Coxeter group | ${\tilde {D}}_{6}$×2 = $[[3^{1,1},3,3,3^{1,1}]]$ |
| Dual | |
| Properties | vertex-transitive |

In six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb. Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.

## Related honeycombs

This honeycomb is one of 41 uniform honeycombs constructed by the ${\tilde {D}}_{6}$ Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$ constructions:

D6 honeycombs :

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