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}, {31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group	${\displaystyle {\tilde {D}}_{7}}$×2 = [[31,1,3,3,3,31,1]]
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.

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

D7 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,3,3,3,31,1]		×1	, , , , , ,
[[31,1,3,3,3,31,1]]		×2	, , ,
<[31,1,3,3,3,31,1]> ↔ [31,1,3,3,3,3,4]	↔	×2	...
<<[31,1,3,3,3,31,1]>> ↔ [4,3,3,3,3,3,4]	↔	×4	...
[<<[31,1,3,3,3,31,1]>>] ↔ [[4,3,3,3,3,3,4]]	↔	×8	...

Regular and uniform honeycombs in 7-space:

• 7-cube honeycomb
• 7-demicube honeycomb
• 7-simplex honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb

• 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	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2	Uniform tiling	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
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21