Quarter 5-cubic honeycomb

quarter 5-cubic honeycomb
(No image)
Type	Uniform 5-honeycomb
Family	Quarter hypercubic honeycomb
Schläfli symbol	q{4,3,3,3,4}
Coxeter-Dynkin diagram	=
5-face type	h{4,33}, h4{4,33},
Vertex figure	60px Rectified 5-cell antiprism or Stretched birectified 5-simplex
Coxeter group	${\tilde{D}}_5$×2 = [[31,1,3,31,1]]
Dual
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb. Its facets are 5-demicubes and runcinated 5-demicubes.

Related honeycombs

This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde{D}}_5$ Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

D5 honeycombs
Extended symmetry	Extended diagram	Extended group	Honeycombs
[31,1,3,31,1]		${\tilde{D}}_5$
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	${\tilde{D}}_5$×21 = ${\tilde{B}}_5$	, , , , , ,
[[31,1,3,31,1]]		${\tilde{D}}_5$×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	${\tilde{D}}_5$×41 = ${\tilde{C}}_5$	, , , , ,
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]	↔	${\tilde{D}}_5$×8 = ${\tilde{C}}_5$×2	, ,

See also

Regular and uniform honeycombs in 5-space:

• 5-cube honeycomb
• 5-demicube honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb

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]
• Richard Klitzing, 5D, Euclidean tesselations#5D x3o3o x3o3o *b3*e - spaquinoh