Quarter 5-cubic honeycomb
# Quarter 5-cubic honeycomb

quarter 5-cubic honeycomb :

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

In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb. Its facets are 5-demicubes and runcinated 5-demicubes.

## Related honeycombs

This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde {D}}_{5}$ Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

D5 honeycombs :

| Extended symmetry | Extended diagram | Extended group | Honeycombs |
| - | - | - | - |
| $[3^{1,1},3,3^{1,1}]$ | | ${\tilde {D}}_{5}$ | |
| $<[3^{1,1},3,3^{1,1}]>$ ↔ $[3^{1,1},3,3,4]$ | ↔ | ${\tilde {D}}_{5}$$×2_1$ = ${\tilde {B}}_{5}$ | , , , , , , |
| $[[3^{1,1},3,3^{1,1}]]$ | | ${\tilde {D}}_{5}$$×2_2$ | , |
| $<2[3^{1,1},3,3^{1,1}]>$ ↔ [4,3,3,3,4] | ↔ | ${\tilde {D}}_{5}$$×4_1$ = ${\tilde {C}}_{5}$ | , , , , , |
| $[<2[3^{1,1},3,3^{1,1}]>]$ ↔ [[4,3,3,3,4]] | ↔ | ${\tilde {D}}_{5}$×8 = ${\tilde {C}}_{5}$×2 | , , |
