7-demicubic honeycomb explained

bgcolor=#e7dcc3 colspan=2	7-demicubic honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform 7-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h h ht0,7
Coxeter-Dynkin diagram	= =
Facets	h
Vertex figure	Rectified 7-orthoplex
Coxeter group	{\tilde{B}} _{7} [4,3,3,3,3,3<sup>1,1</sup>] {\tilde{D}} _{7} , [3<sup>1,1</sup>,3,3,3,3<sup>1,1</sup>]

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h and the alternated vertices create 7-orthoplex facets.

D7 lattice

The vertex arrangement of the 7-demicubic honeycomb is the D₇ lattice.^[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.^[2] The best known is 126, from the E₇ lattice and the 3₃₁ honeycomb.

The D packing (also called D) can be constructed by the union of two D₇ lattices. The D packings form lattices only in even dimensions. The kissing number is 2⁶=64 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[3]

∪

The D lattice (also called D and C) can be constructed by the union of all four 7-demicubic lattices:^[4] 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 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb,, containing all with tritruncated 7-orthoplex, Voronoi cells.^[5]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 128 7-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
{\tilde{B}} _{7} = [3<sup>1,1</sup>,3,3,3,3,4] = [1<sup>+</sup>,4,3,3,3,3,3,4]	h	=	[3,3,3,3,3,4]	128: 7-demicube 14: 7-orthoplex
{\tilde{D}} _{7} = [3<sup>1,1</sup>,3,3,3<sup>1,1</sup>] = [1<sup>+</sup>,4,3,3,3,3<sup>1,1</sup>]	h	=	[3<sup>5,1,1</sup>]	64+64: 7-demicube 14: 7-orthoplex
2×½{\tilde{C}} _{7} = (4,3,3,3,3,4,2^+)	ht0,7			64+32+32: 7-demicube 14: 7-orthoplex

See also

• 7-cubic honeycomb

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]
• 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 D7.
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. Conway (1998), p. 119
4. Web site: The Lattice D7.
5. Conway (1998), p. 466