# 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{4,35},
h5{4,35},
{31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group ${\displaystyle {\tilde {D}}_{7}}$×2 = [[31,1,3,3,3,31,1]]
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.[1] Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

## Related honeycombs

This honeycomb is one of 77 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{7}}$ 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 ${\displaystyle {\tilde {B}}_{7}}$ and ${\displaystyle {\tilde {C}}_{7}}$ constructions:

Regular and uniform honeycombs in 7-space:

## Notes

1. ^ Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318

## References

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21