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}, {3^1,1,1}×{3,3} duoprism
Vertex figure
Coxeter group	${\tilde {D}}_{7}$ ×2 = [[3^1,1,3,3,3,3^1,1]]
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 {3^1,1,1}×{3,3} duoprisms.

Related honeycombs

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

D7 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[3^1,1,3,3,3,3^1,1]		×1	, , , , , ,
[[3^1,1,3,3,3,3^1,1]]		×2	, , ,
<[3^1,1,3,3,3,3^1,1]> ↔ [3^1,1,3,3,3,3,4]	↔	×2	...
<<[3^1,1,3,3,3,3^1,1]>> ↔ [4,3,3,3,3,3,4]	↔	×4	...
[<<[3^1,1,3,3,3,3^1,1]>>] ↔ [[4,3,3,3,3,3,4]]	↔	×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. "7D Euclidean tesselations#7D".

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

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