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8-demicubic honeycomb

## Summary

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]

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{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. (2n-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 (2n-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

## Notes

1. ^ "The Lattice D8".
2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
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. ^ "The Lattice D8".
7. ^ Conway (1998), p. 120
8. ^ Conway (1998), p. 466

## References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• 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, ISBN 978-0-471-01003-6 [2]
• (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)
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.