Arrhenius plot
| 8-demicubic honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 8-space honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | h{4,3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagram | Template:CDD or Template:CDD Template:CDD Template:CDD |
| Facets | {3,3,3,3,3,3,4} h{4,3,3,3,3,3,3} |
| Vertex figure | Rectified octacross |
| Coxeter group | [4,3,3,3,3,3,31,1] [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.
contains as a subgroup of index 270.[3] Both and can be seen as affine extensions of from different nodes: 
The DTemplate:Sup sub lattice (also called DTemplate:Sup sub) 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 n<8, 240 for n=8, and 2n(n-1) for n>8).[4] 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 DTemplate:Sup sub lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four D8 lattices:[5] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.
The kissing number of the DTemplate:Sup sub lattice is 16 (2n for n≥5).[6] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, Template:CDD, containing all trirectified 8-orthoplex Voronoi cell, Template:CDD.[7]
See also
Notes
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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 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- N.W. Johnson: Geometries and Transformations, Manuscript, (2011)
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External links
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- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D8.html
- ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [2]
- ↑ Johnson (2011) p.177
- ↑ Conway (1998), p. 119
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466