title: Cyclotruncated 8-simplex honeycomb
text: # Cyclotruncated 8-simplex honeycomb
  
  Cyclotruncated 8-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Cyclotruncated simplectic honeycomb |
  | Schläfli symbol | $t_{0,1}{3^{[9]}}$ |
  | Coxeter diagram | |
  | 8-face types | ${3^7}$ , $t_{0,1}{3^7}$ $t_{1,2}{3^7}$ , $t_{2,3}{3^7}$ $t_{3,4}{3^7}$ |
  | Vertex figure | Elongated 7-simplex antiprism |
  | Symmetry | ${\tilde {A}}_{8}$×2, $[[3^{[9]}]]$ |
  | Properties | vertex-transitive |
  
  In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.
  
  ## Structure
  
  It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.
  
  ## Related polytopes and honeycombs
  
  This honeycomb is one of 45 unique uniform honeycombs constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
  
  A8 honeycombs :
  
  | Enneagon symmetry | Symmetry | Extended group | Honeycombs |
  | - | - | - | - |
  | a1 | $[3^{[9]}]$ | ${\tilde {A}}_{8}$ | |
  | i2 | $[[3^{[9]}]]$ | ${\tilde {A}}_{8}$×2 | $_1$ $_2$ |
  | i6 | $[3[3^{[9]}]]$ | ${\tilde {A}}_{8}$×6 | |
  | r18 | $[9[3^{[9]}]]$ | ${\tilde {A}}_{8}$×18 | $_3$ |
  