5-demicubic honeycomb

The vertex arrangement of the 5-demicubic honeycomb is the D₅ lattice which is the densest known sphere packing in 5 dimensions.^[1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.^[2]

The D⁺
₅ packing (also called D²
₅) can be constructed by the union of two D₅ lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2⁴=16 (2ⁿ⁻¹ for n<8, 240 for n=8, and 2n(n−1) for n>8).^[3]

∪

The D^*
₅^[4] lattice (also called D⁴
₅ and C²
₅) can be constructed by the union of all four 5-demicubic lattices:^[5] It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.

∪ ∪ ∪ = ∪ .

The kissing number of the D^*
₅ lattice is 10 (2n for n≥5) and its Voronoi tessellation is a tritruncated 5-cubic honeycomb, , containing all bitruncated 5-orthoplex, Voronoi cells.^[6]

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 32 5-demicube facets around each vertex.

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Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${\displaystyle {\tilde {B}}_{5}}$ = [31,1,3,3,4] = [1+,4,3,3,4]	h{4,3,3,3,4}	=	[3,3,3,4]	32: 5-demicube 10: 5-orthoplex
${\displaystyle {\tilde {D}}_{5}}$ = [31,1,3,31,1] = [1+,4,3,31,1]	h{4,3,3,31,1}	=	[32,1,1]	16+16: 5-demicube 10: 5-orthoplex
2×½${\displaystyle {\tilde {C}}_{5}}$ = [[(4,3,3,3,4,2+)]]	ht0,5{4,3,3,3,4}			16+8+8: 5-demicube 10: 5-orthoplex

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This honeycomb is one of 20 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{5}}$ Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

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D5 honeycombs
Extended symmetry	Extended diagram	Extended group	Honeycombs
[31,1,3,31,1]		${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×21 = ${\displaystyle {\tilde {B}}_{5}}$	, , , , , ,
[[31,1,3,31,1]]		${\displaystyle {\tilde {D}}_{5}}$×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×41 = ${\displaystyle {\tilde {C}}_{5}}$	, , , , ,
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]	↔	${\displaystyle {\tilde {D}}_{5}}$×8 = ${\displaystyle {\tilde {C}}_{5}}$×2	, ,

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• Uniform polytope

Regular and uniform honeycombs in 5-space:

• 5-cube honeycomb
• 5-demicube honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb