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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{4,3}, h5{4,3}, {3}×{3,3} duoprism
Vertex figure
Coxeter group	${\displaystyle {\tilde {D}}_{7}}$×2 = ]
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. Its facets are 7-demicubes, pentellated 7-demicubes, and {3}×{3,3} duoprisms.

Related honeycombs

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
		×1	, , , , , ,
]		×2	, , ,
<> ↔	↔	×2	...
<<>> ↔	↔	×4	...
>>] ↔ ]	↔	×8	...

See also

Regular and uniform honeycombs in 7-space:

• 7-cube honeycomb
• 7-demicube honeycomb
• 7-simplex honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-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, See p318
• Klitzing, Richard. "7D Euclidean tesselations#7D".

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E	Uniform tiling	0	δ3	hδ3	qδ3	Hexagonal
E	Uniform convex honeycomb	0	δ4	hδ4	qδ4
E	Uniform 4-honeycomb	0	δ5	hδ5	qδ5	24-cell honeycomb
E	Uniform 5-honeycomb	0	δ6	hδ6	qδ6
E	Uniform 6-honeycomb	0	δ7	hδ7	qδ7	222
E	Uniform 7-honeycomb	0	δ8	hδ8	qδ8	133 • 331
E	Uniform 8-honeycomb	0	δ9	hδ9	qδ9	152 • 251 • 521
E	Uniform 9-honeycomb	0	δ10	hδ10	qδ10
E	Uniform 10-honeycomb	0	δ11	hδ11	qδ11
E	Uniform (n-1)-honeycomb	0	δn	hδn	qδn	1k2 • 2k1 • k21

• Honeycombs (geometry)
• 8-polytopes