5-demicubic honeycomb explained

bgcolor=#e7dcc3 colspan=2	Demipenteractic honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform 5-honeycomb
Family	Alternated hypercubic honeycomb
Schläfli symbols	h h ht0,5 hh hh ht0,4h hhh hhh
Coxeter diagrams	= =
Facets
Vertex figure
Coxeter group	{\tilde{B}} _{5} [4,3,3,3<sup>1,1</sup>] {\tilde{D}} _{5} [3<sup>1,1</sup>,3,3<sup>1,1</sup>]

The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h and the alternated vertices create 5-orthoplex facets.

D5 lattice

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 packing (also called D) 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 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[3]

∪

The D^[4] lattice (also called D and C) 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 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]

Symmetry constructions

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.

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

See also

• 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

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 D5.
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 D5.
5. Conway (1998), p. 120
6. Conway (1998), p. 466