title: Omnitruncated 7-simplex honeycomb
text: # Omnitruncated 7-simplex honeycomb
  
  Omnitruncated 7-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Omnitruncated simplectic honeycomb |
  | Schläfli symbol | ${3^{[8]}}$ |
  | Coxeter–Dynkin diagrams | |
  | 6-face types | $t_{0123456}{3,3,3,3,3,3}$ |
  | Vertex figure | Irr. 7-simplex |
  | Symmetry | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {{\tilde {A}}}_{8} ×16, $[8[3^{[8]}]]$ |
  | Properties | vertex-transitive |
  
  In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-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).
  
  ## $A_7^*$ lattice
  
  The $A^*$ $_7$ lattice (also called $A^8$ $_7)$ is the union of eight $A_7$ lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.
  
  ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
  
  ## Related polytopes and honeycombs
  
  This honeycomb is one of 29 unique uniform honeycombs constructed by the ${\tilde {A}}_{7}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
  
  A7 honeycombs :
  
  | Octagon symmetry | Extended symmetry | Extended group | Honeycombs |
  | - | - | - | - |
  | a1 | $[3^{[8]}]$ | ${\tilde {A}}_{7}$ | |
  | d2 | $<[3^{[8]}]>$ | ${\tilde {A}}_{7}$$×2_1$ | $_1$ |
  | p2 | $[[3^{[8]}]]$ | ${\tilde {A}}_{7}$$×2_2$ | $_2$ |
  | d4 | $<2[3^{[8]}]>$ | ${\tilde {A}}_{7}$$×4_1$ | |
  | p4 | $[2[3^{[8]}]]$ | ${\tilde {A}}_{7}$$×4_2$ | |
  | d8 | $[4[3^{[8]}]]$ | ${\tilde {A}}_{7}$×8 | |
  | r16 | $[8[3^{[8]}]]$ | ${\tilde {A}}_{7}$×16 | $_3$ |
  