title: Cyclotruncated 7-simplex honeycomb
text: # Cyclotruncated 7-simplex honeycomb
  
  Cyclotruncated 7-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Cyclotruncated simplectic honeycomb |
  | Schläfli symbol | $t_{0,1}{3^{[8]}}$ |
  | Coxeter diagram | |
  | 7-face types | ${3^6}$ $t_{0,1}{3^6}$ $t_{1,2}{3^6}$ $t_{2,3}{3^6}$ |
  | Vertex figure | Elongated 6-simplex antiprism |
  | Symmetry | ${\tilde {A}}_{7}$$×2_2,$ $[[3^{[8]}]]$ |
  | Properties | vertex-transitive |
  
  In seven-dimensional Euclidean geometry, the cyclotruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.
  
  ## Structure
  
  It can be constructed by eight sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 6-simplex honeycomb divisions on each hyperplane.
  
  ## 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 | $_3$ |
  