8-demicubic honeycomb
| 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 |     or      |
| 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 D+
8 lattice (also called D2
8) 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 D*
8 lattice (also called D4
8 and C2
8) 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 D*
8 lattice is 16 (2n for n≥5).[6] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, 
, containing all trirectified 8-orthoplex Voronoi cell, .[7]
See also
Notes
- ↑ 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
- ↑ Johnson (2015) 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
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
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- N.W. Johnson: Geometries and Transformations, (2015)
- Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
External links
- Olshevsky, George, Half measure polytope at Glossary for Hyperspace.
| Fundamental convex regular and uniform honeycombs in dimensions 2–10 | |||||
|---|---|---|---|---|---|
| Family |  |
 |
 |
 |
 /  /  |
| Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| Uniform 5-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| Uniform 6-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| Uniform 7-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| Uniform 8-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| Uniform 9-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| Uniform n-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |