8-demicubic honeycomb | |
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(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 | [4,3,3,3,3,3,3^{1,1}] [3^{1,1},3,3,3,3,3^{1,1}] |
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 .
The vertex arrangement of the 8-demicubic honeycomb is the D_{8} 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 E_{8} lattice and the 5_{21} 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 D^{2}
_{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. (2^{n-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 2^{7}=128 from lower dimension contact progression (2^{n-1}), and 16*7=112 from higher dimensions (2n(n-1)).
The D^{*}
_{8} lattice (also called D^{4}
_{8} and C^{2}
_{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 |
---|---|---|---|---|
= [3^{1,1},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 |
= [3^{1,1},3,3,3,3^{1,1}] = [1^{+},4,3,3,3,3,3^{1,1}] |
h{4,3,3,3,3,3,3^{1,1}} | = | [3^{6,1,1}] |
128+128: 8-demicube 16: 8-orthoplex |
2×½ = [[(4,3,3,3,3,3,4,2^{+})]] | ht_{0,8}{4,3,3,3,3,3,3,4} | 128+64+64: 8-demicube 16: 8-orthoplex |
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |