6-demicubic honeycomb

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]

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.

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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

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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:

More information D6 honeycombs, Extendedsymmetry ...

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	, , , , , ,

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• 6-cubic honeycomb