title: Omnitruncated 5-simplex honeycomb
text: # Omnitruncated 5-simplex honeycomb
  
  Omnitruncated 5-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Omnitruncated simplectic honeycomb |
  | Schläfli symbol | $t_{012345}{3^{[6]}}$ |
  | Coxeter–Dynkin diagram | |
  | 5-face types | $t_{01234}{3,3,3,3}$ |
  | 4-face types | $t_{0123}{3,3,3}×t_{012}{3,3}$ {6}×{6} |
  | Cell types | $t_{012}{3,3}$ {4,3}x{6} |
  | Face types | {4} {6} |
  | Vertex figure | Irr. 5-simplex |
  | Symmetry | ${\tilde {A}}_{5}$×12, $[6[3^{[6]}]]$ |
  | Properties | vertex-transitive |
  
  In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-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).
  
  ## $A_5^*$ lattice
  
  The $A^*$ $_5$ lattice (also called $A^6$ $_5)$ is the union of six $A_5$ lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
  
  ∪ ∪ ∪ ∪ ∪ = dual of
  
  ## Related polytopes and honeycombs
  
  This honeycomb is one of 12 unique uniform honeycombs constructed by the ${\tilde {A}}_{5}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde {A}}_{5}$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:
  
  A5 honeycombs :
  
  | Hexagon symmetry | Extended symmetry | Extended group | Honeycomb diagrams |
  | - | - | - | - |
  | a1 | $[3^{[6]}]$ | ${\tilde {A}}_{5}$ | |
  | d2 | $<[3^{[6]}]>$ | ${\tilde {A}}_{5}$$×2_1$ | $_1,$ , , , |
  | p2 | $[[3^{[6]}]]$ | ${\tilde {A}}_{5}$$×2_2$ | $_2,$ |
  | i4 | $[<[3^{[6]}]>]$ | ${\tilde {A}}_{5}$$×2_1×2_2$ | , |
  | d6 | $<3[3^{[6]}]>$ | ${\tilde {A}}_{5}$$×6_1$ | |
  | r12 | $[6[3^{[6]}]]$ | ${\tilde {A}}_{5}$×12 | $_3$ |
  
  ### Projection by folding
  
  The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:
  
  * ${\tilde {A}}_{5}$
  * ${\tilde {C}}_{3}$
  