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	= =
Facets	{3,3,3,3,4} h{4,3,3,3,3}
Vertex figure	r{3,3,3,3,4}
Coxeter group	${\displaystyle {\tilde {B}}_{6}}$ [4,3,3,3,31,1] ${\displaystyle {\tilde {D}}_{6}}$ [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. ^[1] The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice. ^[2] The best known is 72, from the E₆ lattice and the 2₂₂ honeycomb.

The D⁺
₆ lattice (also called D²
₆) 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). ^[3]

∪

The D^*
₆ lattice (also called D⁴
₆ and C²
₆) can be constructed by the union of all four 6-demicubic lattices: ^[4] It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.

∪ ∪ ∪ = ∪ .

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

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
${\displaystyle {\tilde {B}}_{6}}$ = [31,1,3,3,3,4] = [1+,4,3,3,3,3,4]	h{4,3,3,3,3,4}	=	[3,3,3,4]	64: 6-demicube 12: 6-orthoplex
${\displaystyle {\tilde {D}}_{6}}$ = [31,1,3,31,1] = [1+,4,3,3,31,1]	h{4,3,3,3,31,1}	=	[33,1,1]	32+32: 6-demicube 12: 6-orthoplex
½${\displaystyle {\tilde {C}}_{6}}$ = [[(4,3,3,3,4,2+)]]	ht0,6{4,3,3,3,3,4}			32+16+16: 6-demicube 12: 6-orthoplex

Related honeycombs

This honeycomb is one of 41 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{6}}$ 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 ${\displaystyle {\tilde {B}}_{6}}$ and ${\displaystyle {\tilde {C}}_{6}}$ 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

• 6-cubic honeycomb

Notes

1. "The Lattice D6".
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. "The Lattice D6".
5. Conway (1998), p. 120
6. Conway (1998), p. 466

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, ISBN   978-0-471-01003-6
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN   0-387-98585-9.

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21