title: Cyclotruncated 5-simplex honeycomb
text: # Cyclotruncated 5-simplex honeycomb
  
  Cyclotruncated 5-simplex honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Family | Cyclotruncated simplectic honeycomb |
  | Schläfli symbol | $t_{0,1}{3^{[6]}}$ |
  | Coxeter diagram | or |
  | 5-face types | {3,3,3,3} t{3,3,3,3} 2t{3,3,3,3} |
  | 4-face types | {3,3,3} t{3,3,3} |
  | Cell types | {3,3} t{3,3} |
  | Face types | {3} t{3} |
  | Vertex figure | Elongated 5-cell antiprism |
  | Coxeter groups | ${\tilde {A}}_{5}$$×2_2,$ $[[3^{[6]}]]$ |
  | Properties | vertex-transitive |
  
  In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.
  
  ## Structure
  
  Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.
  
  It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.
  
  ## 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 | $_3$ |
  