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.

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<sup>+</sup>)]]	ht0,6{4,3,3,3,3,4}			32+16+16: 6-demicube 12: 6-orthoplex

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	, , , , , ,

• 6-cubic honeycomb

• 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 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
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21