Quarter 7-cubic honeycomb

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.

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

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
[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	...

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