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)
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}, h4{4,3^4}, {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
Properties	vertex-transitive

Related honeycombs

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:

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	, , , , , ,

See also

Regular and uniform honeycombs in 5-space:

• 6-cube honeycomb
• 6-demicube honeycomb
• 6-simplex honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb

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	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21