6-demicubic honeycomb

6-demicubic honeycomb
(No image)
Type	Uniform honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,4}
Coxeter diagram	or
Facets	{3,3,3,3,4} h{4,3,3,3,3}
Vertex figure	t1{3,3,3,3,4}
Coxeter group	${{\tilde {B}}}_{6}$ [4,3,3,3,3^1,1] ${\tilde {D}}_{6}$ [3^1,1,3,3,3^1,1]

The 6-demicubic honeycomb or demihexeractic honeycube 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 differened colors on the 64 6-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${{\tilde {B}}}_{6}$ = [3^1,1,3,3,3,4] = [1^+,4,3,3,3,3,4]	= h{4,3,3,3,3,4}	=	[3,3,3,4]	64: 6-demicube 12: 6-orthoplex
${\tilde {D}}_{6}$ = [3^1,1,3,3^1,1] = [1^+,4,3,3,3^1,1]	= h{4,3,3,3,3^1,1}	=	[3^3,1,1]	32+32: 6-demicube 12: 6-orthoplex
${{\tilde {C}}}_{6}$ = [[(4,3,3,3,4,2<sup>+</sup>)]]	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 honycombs 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
[3^1,1,3,3,3^1,1]		×1	,
[[3^1,1,3,3,3^1,1]]		×2	, , ,
<[3^1,1,3,3,3^1,1]> ↔ [3^1,1,3,3,3,4]	↔	×2	, , , , , , , , , , , , , , ,
<2[3^1,1,3,3,3^1,1]> ↔ [4,3,3,3,3,4]	↔	×4	,, ,, , , , , , , ,
[<2[3^1,1,3,3,3^1,1]>] ↔ [[4,3,3,3,3,4]]	↔	×8	, , , , , ,

See also

• 6-cubic honeycomb

Notes

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.html
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds6.html
5. Conway (1998), p. 120
6. Conway (1998), p. 466

External links

• Olshevsky, George. "Half measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• 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.

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21