8-demicubic honeycomb

D8 lattice

The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai https://books.google.com/books?id=upYwZ6cQumoC&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19 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. Both {\tilde{E}}_8 and {\tilde{D}}_8 can be seen as affine extensions of D_8 from different nodes: [[File:Affine D8 E8 relations.png]]

The D lattice (also called D) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n8). It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 27=128 from lower dimension contact progression (2n-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: 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). and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .

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.

Notes

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
  • pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
• 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)

References

1. "The Lattice D8".
2. Johnson (2015) p.177
3. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
4. Conway (1998), p. 119
5. "The Lattice D8".
6. Conway (1998), p. 120
7. Conway (1998), p. 466