| 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 |  [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 (orhoneycomb) 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 
.
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.
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]
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 |
|---|---|---|---|---|
| = [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 |
| = [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 = [[(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 |
|
Fundamental convex regular and uniform honeycombs in dimensions 2–9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | 
|
|
|
|
 /  / 
|
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |