Quarter_6-cubic_honeycomb Knowpia

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{4,3⁴}, h₄{4,3⁴}, {3,3}×{3,3} duoprism
Vertex figure
Coxeter group	${\tilde {D}}_{6}$ ×2 = [[3^1,1,3,3,3^1,1]]
Dual
Properties	vertex-transitive

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.

Related honeycombs

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

Notes

^ 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 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [2]
Klitzing, Richard. "6D Euclidean tesselations#6D".

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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

[1]

Summary

Related honeycombs

See also

Notes

References