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	Rectified 5-cell antiprism or Stretched birectified 5-simplex
Coxeter group	×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.^[1] Its facets are 5-demicubes and runcinated 5-demicubes.

Related honeycombs

This honeycomb is one of 20 uniform honycombs constructed by the 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:

Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,3,31,1]		×1
[[31,1,3,31,1]]		×2	,
<[31,1,3,31,1]> = [31,1,3,3,4]	=	×2	, , , , , ,
<<[31,1,3,31,1]>> = [4,3,3,3,4]	=	×4	, , , , ,
[<<[31,1,3,1,1]>>] = [[4,3,3,3,4]]	=	×8	, ,

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

Notes

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, [Math. Zeit. 200 (1988) 3-45] See p318
• Richard Klitzing, 5D, Euclidean tesselations#5D x3o3o x3o3o *b3*e - spaquinoh

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family					/ /
Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform n-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21