8-demicubic honeycomb

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

8-demicubic honeycomb
(No image)
Type	Uniform 8-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,3,3,4}
Coxeter diagrams	= =
Facets	{3,3,3,3,3,3,4} h{4,3,3,3,3,3,3}
Vertex figure	Rectified 8-orthoplex
Coxeter group	${\displaystyle {\tilde {B}}_{8}}$ [4,3,3,3,3,3,31,1] ${\displaystyle {\tilde {D}}_{8}}$ [31,1,3,3,3,3,31,1]

It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets .

D8 lattice

The vertex arrangement of the 8-demicubic honeycomb is the D₈ lattice.^[1] The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.^[2] The best known is 240, from the E₈ lattice and the 5₂₁ honeycomb.

${\displaystyle {\tilde {E}}_{8}}$ contains ${\displaystyle {\tilde {D}}_{8}}$ as a subgroup of index 270.^[3] Both ${\displaystyle {\tilde {E}}_{8}}$ and ${\displaystyle {\tilde {D}}_{8}}$ can be seen as affine extensions of ${\displaystyle D_{8}}$ from different nodes:

The D⁺
₈ lattice (also called D²
₈) can be constructed by the union of two D8 lattices.^[4] This packing is only a lattice for even dimensions. The kissing number is 240. (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[5] It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 2⁷=128 from lower dimension contact progression (2^n-1), and 16*7=112 from higher dimensions (2n(n-1)).

∪ = .

The D^*
₈ lattice (also called D⁴
₈ and C²
₈) can be constructed by the union of all four D8 lattices:^[6] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

∪ ∪ ∪ = ∪ .

The kissing number of the D^*
₈ lattice is 16 (2n for n≥5).^[7] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .^[8]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256 8-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${\displaystyle {\tilde {B}}_{8}}$ = [31,1,3,3,3,3,3,4] = [1+,4,3,3,3,3,3,3,4]	h{4,3,3,3,3,3,3,4}	=	[3,3,3,3,3,3,4]	256: 8-demicube 16: 8-orthoplex
${\displaystyle {\tilde {D}}_{8}}$ = [31,1,3,3,3,31,1] = [1+,4,3,3,3,3,31,1]	h{4,3,3,3,3,3,31,1}	=	[36,1,1]	128+128: 8-demicube 16: 8-orthoplex
2×½${\displaystyle {\tilde {C}}_{8}}$ = [[(4,3,3,3,3,3,4,2+)]]	ht0,8{4,3,3,3,3,3,3,4}			128+64+64: 8-demicube 16: 8-orthoplex

See also

• 8-cubic honeycomb
• Uniform polytope