Quarter 6-cubic honeycomb

quarter 6-cubic honeycomb
(No image)
Type	Uniform 6-honeycomb
Family	Quarter hypercubic honeycomb
Schläfli symbol	q{4,3,3,3,3,4}
Coxeter-Dynkin diagram	=
5-face type	h{4,34}, h4{4,34}, {3,3}×{3,3} duoprism
Vertex figure
Coxeter group	${\tilde {D}}_{6}$×2 = [[31,1,3,3,31,1]]
Dual
Properties	vertex-transitive

In six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb. Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.

Related honeycombs

This honeycomb is one of 41 uniform honeycombs constructed by the ${\tilde {D}}_{6}$ Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$ constructions:

D6 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,3,3,31,1]		×1	,
[[31,1,3,3,31,1]]		×2	, , ,
<[31,1,3,3,31,1]> ↔ [31,1,3,3,3,4]	↔	×2	, , , , , , , , , , , , , , ,
<2[31,1,3,3,31,1]> ↔ [4,3,3,3,3,4]	↔	×4	,, ,, , , , , , , ,
[<2[31,1,3,3,31,1]>] ↔ [[4,3,3,3,3,4]]	↔	×8	, , , , , ,

See also

Regular and uniform honeycombs in 5-space:

• 6-cube honeycomb
• 6-demicube honeycomb
• 6-simplex honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-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]
• Klitzing, Richard. "6D Euclidean tesselations#6D".

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2	Uniform tiling	0[3]	δでるた3	hδでるた3	qδでるた3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δでるた4	hδでるた4	qδでるた4
E4	Uniform 4-honeycomb	0[5]	δでるた5	hδでるた5	qδでるた5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δでるた6	hδでるた6	qδでるた6
E6	Uniform 6-honeycomb	0[7]	δでるた7	hδでるた7	qδでるた7	222
E7	Uniform 7-honeycomb	0[8]	δでるた8	hδでるた8	qδでるた8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δでるた9	hδでるた9	qδでるた9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δでるた10	hδでるた10	qδでるた10
E10	Uniform 10-honeycomb	0[11]	δでるた11	hδでるた11	qδでるた11
En-1	Uniform (n-1)-honeycomb	0[n]	δでるたn	hδでるたn	qδでるたn	1k2 • 2k1 • k21