title: Cyclotruncated 6-simplex honeycomb
text: # Cyclotruncated 6-simplex honeycomb
  
  Cyclotruncated 6-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Cyclotruncated simplectic honeycomb |
  | Schläfli symbol | $t_{0,1}{3^{[7]}}$ |
  | Coxeter diagram | |
  | 6-face types | ${3^5}$ $t{3^5}$ $2t{3^5}$ $3t{3^5}$ |
  | Vertex figure | Elongated 5-simplex antiprism |
  | Symmetry | ${\tilde {A}}_{6}$×2, $[[3^{[7]}]]$ |
  | Properties | vertex-transitive |
  
  In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
  
  ## Structure
  
  It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.
  
  ## 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 | $_3$ |
  