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},
{3,3}×{3,3} duoprism
Vertex figure
Coxeter group×2 = [[3<sup>1,1</sup>,3,3,3<sup>1,1</sup>]]

This honeycomb is one of 41 uniform honeycombs constructed by the 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 and constructions:

See also

Regular and uniform honeycombs in 5-space:


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


  • 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 / /
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
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