# Quarter 6-cubic honeycomb

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.[1] Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.

quarter 6-cubic honeycomb
(No image)
TypeUniform 6-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,3,4}
Coxeter-Dynkin diagram =
5-face typeh{4,34},
h4{4,34},
{3,3}×{3,3} duoprism
Vertex figure
Coxeter group${\displaystyle {\tilde {D}}_{6}}$×2 = [[3<sup>1,1</sup>,3,3,3<sup>1,1</sup>]]
Dual
Propertiesvertex-transitive

This honeycomb is one of 41 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{6}}$ 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 ${\displaystyle {\tilde {B}}_{6}}$ and ${\displaystyle {\tilde {C}}_{6}}$ constructions:

Regular and uniform honeycombs in 5-space:

## Notes

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

## References

• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "6D Euclidean tesselations#6D".
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 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21