6-demicubic honeycomb

6-demicubic honeycomb
(No image)
Type	Uniform 6-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,4} h{4,3,3,3,31,1} ht0,6{4,3,3,3,3,4}
Coxeter diagram	Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD
Facets	{3,3,3,3,4} h{4,3,3,3,3}
Vertex figure	r{3,3,3,3,4}
Coxeter group	${\tilde{B}}_6$ [4,3,3,3,31,1] ${\tilde{D}}_6$ [31,1,3,3,31,1]

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.

D6 lattice

The vertex arrangement of the 6-demicubic honeycomb is the D₆ lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice. The best known is 72, from the E₆ lattice and the 2₂₂ honeycomb.

The DTemplate:Sup sub lattice (also called DTemplate:Sup sub) can be constructed by the union of two D₆ lattices. This packing is only a lattice for even dimensions. The kissing number is 2⁵=32 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).

Template:CDD ∪ Template:CDD

The DTemplate:Sup sub lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.

Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD = Template:CDD ∪ Template:CDD.

The kissing number of the D₆^* lattice is 12 (2n for n≥5). and its Voronoi tessellation is a trirectified 6-cubic honeycomb, Template:CDD, containing all birectified 6-orthoplex Voronoi cell, Template:CDD.

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 64 6-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${\tilde{B}}_6$ = [31,1,3,3,3,4] = [1+,4,3,3,3,3,4]	h{4,3,3,3,3,4}	Template:CDD = Template:CDD	Template:CDD [3,3,3,4]	64: 6-demicube 12: 6-orthoplex
${\tilde{D}}_6$ = [31,1,3,31,1] = [1+,4,3,3,31,1]	h{4,3,3,3,31,1}	Template:CDD = Template:CDD	Template:CDD [33,1,1]	32+32: 6-demicube 12: 6-orthoplex
½${\tilde{C}}_6$ = [[(4,3,3,3,4,2+)]]	ht0,6{4,3,3,3,3,4}	Template:CDD		32+16+16: 6-demicube 12: 6-orthoplex

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]	Template:CDD	×1	Template:CDD, Template:CDD
[[31,1,3,3,31,1]]	Template:CDD	×2	Template:CDD, Template:CDD, Template:CDD, Template:CDD
<[31,1,3,3,31,1]> ↔ [31,1,3,3,3,4]	Template:CDD ↔ Template:CDD	×2	Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD
<2[31,1,3,3,31,1]> ↔ [4,3,3,3,3,4]	Template:CDD ↔ Template:CDD	×4	Template:CDD,Template:CDD, Template:CDD,Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD
[<2[31,1,3,3,31,1]>] ↔ [[4,3,3,3,3,4]]	Template:CDD ↔ Template:CDD	×8	Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD

See also

• 6-cubic honeycomb

External links

• 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, Template:ISBN [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Template:Cite book

Template:Navbar-collapsible
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]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21