8-demicubic honeycomb explained

bgcolor=#e7dcc3 colspan=2	8-demicubic honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
bgcolor=#e7dcc3 width=100	Type	Uniform 8-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h
Coxeter diagrams	= =
Facets	h
Vertex figure	Rectified 8-orthoplex
Coxeter group	{\tilde{B}} _{8} [4,3,3,3,3,3,3<sup>1,1</sup>] {\tilde{D}} _{8} [3<sup>1,1</sup>,3,3,3,3,3<sup>1,1</sup>]

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.

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

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

The D lattice (also called D) 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 lattice (also called D and C) 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 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
{\tilde{B}} _{8} = [3<sup>1,1</sup>,3,3,3,3,3,4] = [1<sup>+</sup>,4,3,3,3,3,3,3,4]	h	=	[3,3,3,3,3,3,4]	256: 8-demicube 16: 8-orthoplex
{\tilde{D}} _{8} = [3<sup>1,1</sup>,3,3,3,3<sup>1,1</sup>] = [1<sup>+</sup>,4,3,3,3,3,3<sup>1,1</sup>]	h	=	[3<sup>6,1,1</sup>]	128+128: 8-demicube 16: 8-orthoplex
2×½{\tilde{C}} _{8} = (4,3,3,3,3,3,4,2^+)	ht0,8			128+64+64: 8-demicube 16: 8-orthoplex

See also

• 8-cubic honeycomb
• Uniform polytope

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
  • pp. 154–156: Partial truncation or alternation, represented by h prefix: h=; h=, h=, ...
• 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: Geometries and Transformations, (2018)
• Book: Conway JH, Sloane NJH . 1998 . Sphere Packings, Lattices and Groups . 3rd . 0-387-98585-9 . registration .

Notes and References

1. Web site: The Lattice D8.
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannaihttps://books.google.com/books?id=upYwZ6cQumoC&amp;dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&amp;pg=PR19
3. Johnson (2015) p.177
4. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
5. Conway (1998), p. 119
6. Web site: The Lattice D8.
7. Conway (1998), p. 120
8. Conway (1998), p. 466