title: 8-demicubic honeycomb
text: # 8-demicubic honeycomb
  
  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 | ${\tilde {B}}_{8}$ $[4,3,3,3,3,3,3^{1,1}]$ ${\tilde {D}}_{8}$ $[3^{1,1},3,3,3,3,3^{1,1}]$ |
  
  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 .
  
  ## D8 lattice
  
  The vertex arrangement of the 8-demicubic honeycomb is the $D_8$ lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice. The best known is 240, from the $E_8$ lattice and the $5_{21}$ honeycomb.
  
  ${\tilde {E}}_{8}$ contains ${\tilde {D}}_{8}$ as a subgroup of index 270. Both ${\tilde {E}}_{8}$ and ${\tilde {D}}_{8}$ can be seen as affine extensions of $D_{8}$ from different nodes:
  
  The $D^+$ $_8$ lattice (also called $D^2$ $_8)$ can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. $(2^{n-1}$ for n<8, 240 for n=8, and 2n(n-1) for n>8). 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: 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). and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .
  
  ## 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 |
  | - | - | - | - | - |
  | ${\tilde {B}}_{8}$ = $[3^{1,1},3,3,3,3,3,4]$ = $[1^+,4,3,3,3,3,3,3,4]$ | h{4,3,3,3,3,3,3,4} | = | [3,3,3,3,3,3,4] | 256: 8-demicube 16: 8-orthoplex |
  | ${\tilde {D}}_{8}$ = $[3^{1,1},3,3,3,3^{1,1}]$ = $[1^+,4,3,3,3,3,3^{1,1}]$ | $h{4,3,3,3,3,3,3^{1,1}}$ | = | $[3^{6,1,1}]$ | 128+128: 8-demicube 16: 8-orthoplex |
  | 2×½${\tilde {C}}_{8}$ = $[[(4,3,3,3,3,3,4,2^+)]]$ | $ht_{0,8}{4,3,3,3,3,3,3,4}$ | | | 128+64+64: 8-demicube 16: 8-orthoplex |
  