title: Omnitruncated 6-simplex honeycomb
text: # Omnitruncated 6-simplex honeycomb
  
  Omnitruncated 6-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Omnitruncated simplectic honeycomb |
  | Schläfli symbol | ${3^{[8]}}$ |
  | Coxeter–Dynkin diagrams | |
  | Facets | $t_{0,1,2,3,4,5}{3,3,3,3,3}$ |
  | Vertex figure | Irr. 6-simplex |
  | Symmetry | ${\tilde {A}}_{7}$×14, $[7[3^{[7]}]]$ |
  | Properties | vertex-transitive |
  
  In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-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).
  
  ## $A^*$ $_6$ lattice
  
  The $A^*$ $_6$ lattice (also called $A^7$ $_6)$ is the union of seven $A_6$ lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
  
  ∪ ∪ ∪ ∪ ∪ ∪ = dual of
  
  ## Related polytopes and honeycombs
  
  This honeycomb is one of 17 unique uniform honeycombs constructed by the ${\tilde {A}}_{6}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
  
  A6 honeycombs :
  
  | Heptagon symmetry | Extended symmetry | Extended group | Honeycombs |
  | - | - | - | - |
  | a1 | $[3^{[7]}]$ | ${\tilde {A}}_{6}$ | |
  | i2 | $[[3^{[7]}]]$ | ${\tilde {A}}_{6}$×2 | $_1$ $_2$ |
  | r14 | $[7[3^{[7]}]]$ | ${\tilde {A}}_{6}$×14 | $_3$ |
  