title: Omnitruncated 8-simplex honeycomb
text: # Omnitruncated 8-simplex honeycomb
  
  Omnitruncated 8-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Omnitruncated simplectic honeycomb |
  | Schläfli symbol | ${3^{[9]}}$ |
  | Coxeter–Dynkin diagrams | |
  | 7-face types | $t_{01234567}{3,3,3,3,3,3,3}$ |
  | Vertex figure | Irr. 8-simplex |
  | Symmetry | ${\tilde {A}}_{9}$×18, $[9[3^{[9]}]]$ |
  | Properties | vertex-transitive |
  
  In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets.
  
  The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
  
  ## $A^*$ $_8$ lattice
  
  The $A^*$ $_8$ lattice (also called $A^9$ $_8)$ is the union of nine $A_8$ lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex
  
  ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
  
  ## Related polytopes and honeycombs
  
  This honeycomb is one of 45 unique uniform honeycombs constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
  
  A8 honeycombs :
  
  | Enneagon symmetry | Symmetry | Extended group | Honeycombs |
  | - | - | - | - |
  | a1 | $[3^{[9]}]$ | ${\tilde {A}}_{8}$ | |
  | i2 | $[[3^{[9]}]]$ | ${\tilde {A}}_{8}$×2 | $_1$ $_2$ |
  | i6 | $[3[3^{[9]}]]$ | ${\tilde {A}}_{8}$×6 | |
  | r18 | $[9[3^{[9]}]]$ | ${\tilde {A}}_{8}$×18 | $_3$ |
  