title: Truncated 6-orthoplexes
text: # Truncated 6-orthoplexes
  
  | 6-orthoplex | Truncated 6-orthoplex | Bitruncated 6-orthoplex | Tritruncated 6-cube |
  | - | - | - | - |
  | 6-cube | Truncated 6-cube | Bitruncated 6-cube | Tritruncated 6-cube |
  | Orthogonal projections in $B_6$ Coxeter plane ||||
  
  In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.
  
  There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.
  
  ## Truncated 6-orthoplex
  
  Truncated 6-orthoplex :
  
  | Type | uniform 6-polytope |
  | - | - |
  | Schläfli symbol | t{3,3,3,3,4} |
  | Coxeter-Dynkin diagrams | |
  | 5-faces | 76 |
  | 4-faces | 576 |
  | Cells | 1200 |
  | Faces | 1120 |
  | Edges | 540 |
  | Vertices | 120 |
  | Vertex figure | ( )v{3,4} |
  | Coxeter groups | $B_6,$ [3,3,3,3,4] $D_6,$ $[3^{3,1,1}]$ |
  | Properties | convex |
  
  ### Alternate names
  
  * Truncated hexacross
  * Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)
  
  ### Construction
  
  There are two Coxeter groups associated with the truncated hexacross, one with the $C_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the $D_6$ or $[3^{3,1,1}]$ Coxeter group.
  
  ### Coordinates
  
  Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of
  
      (±2,±1,0,0,0,0)
  
  ### Images
  
  orthographic projections :
  
  | Coxeter plane | $B_6$ | $B_5$ | $B_4$ |
  | - | - | - | - |
  | Graph | | | |
  | Dihedral symmetry | [12] | [10] | [8] |
  | Coxeter plane | $B_3$ | $B_2$ |
  | Graph | | |
  | Dihedral symmetry | [6] | [4] |
  | Coxeter plane | $A_5$ | $A_3$ |
  | Graph | | |
  | Dihedral symmetry | [6] | [4] |
  
  ## Bitruncated 6-orthoplex
  
  Bitruncated 6-orthoplex :
  
  | Type | uniform 6-polytope |
  | - | - |
  | Schläfli symbol | 2t{3,3,3,3,4} |
  | Coxeter-Dynkin diagrams | |
  | 5-faces | |
  | 4-faces | |
  | Cells | |
  | Faces | |
  | Edges | |
  | Vertices | |
  | Vertex figure | { }v{3,4} |
  | Coxeter groups | $B_6,$ [3,3,3,3,4] $D_6,$ $[3^{3,1,1}]$ |
  | Properties | convex |
  
  ### Alternate names
  
  * Bitruncated hexacross
  * Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)
  
  ### Images
  
  orthographic projections :
  
  | Coxeter plane | $B_6$ | $B_5$ | $B_4$ |
  | - | - | - | - |
  | Graph | | | |
  | Dihedral symmetry | [12] | [10] | [8] |
  | Coxeter plane | $B_3$ | $B_2$ |
  | Graph | | |
  | Dihedral symmetry | [6] | [4] |
  | Coxeter plane | $A_5$ | $A_3$ |
  | Graph | | |
  | Dihedral symmetry | [6] | [4] |
  
  ## Related polytopes
  
  These polytopes are a part of a set of 63 uniform 6-polytopes generated from the $B_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
  
  B6 polytopes :
  
  | $β_6$ | $t_1β_6$ | $t_2β_6$ | $t_2γ_6$ | $t_1γ_6$ | $γ_6$ | $t_{0,1}β_6$ | $t_{0,2}β_6$ |
  | - | - | - | - | - | - | - | - |
  | $t_{1,2}β_6$ | $t_{0,3}β_6$ | $t_{1,3}β_6$ | $t_{2,3}γ_6$ | $t_{0,4}β_6$ | $t_{1,4}γ_6$ | $t_{1,3}γ_6$ | $t_{1,2}γ_6$ |
  | $t_{0,5}γ_6$ | $t_{0,4}γ_6$ | $t_{0,3}γ_6$ | $t_{0,2}γ_6$ | $t_{0,1}γ_6$ | $t_{0,1,2}β_6$ | $t_{0,1,3}β_6$ | $t_{0,2,3}β_6$ |
  | $t_{1,2,3}β_6$ | $t_{0,1,4}β_6$ | $t_{0,2,4}β_6$ | $t_{1,2,4}β_6$ | $t_{0,3,4}β_6$ | $t_{1,2,4}γ_6$ | $t_{1,2,3}γ_6$ | $t_{0,1,5}β_6$ |
  | $t_{0,2,5}β_6$ | $t_{0,3,4}γ_6$ | $t_{0,2,5}γ_6$ | $t_{0,2,4}γ_6$ | $t_{0,2,3}γ_6$ | $t_{0,1,5}γ_6$ | $t_{0,1,4}γ_6$ | $t_{0,1,3}γ_6$ |
  | $t_{0,1,2}γ_6$ | $t_{0,1,2,3}β_6$ | $t_{0,1,2,4}β_6$ | $t_{0,1,3,4}β_6$ | $t_{0,2,3,4}β_6$ | $t_{1,2,3,4}γ_6$ | $t_{0,1,2,5}β_6$ | $t_{0,1,3,5}β_6$ |
  | $t_{0,2,3,5}γ_6$ | $t_{0,2,3,4}γ_6$ | $t_{0,1,4,5}γ_6$ | $t_{0,1,3,5}γ_6$ | $t_{0,1,3,4}γ_6$ | $t_{0,1,2,5}γ_6$ | $t_{0,1,2,4}γ_6$ | $t_{0,1,2,3}γ_6$ |
  | $t_{0,1,2,3,4}β_6$ | $t_{0,1,2,3,5}β_6$ | $t_{0,1,2,4,5}β_6$ | $t_{0,1,2,4,5}γ_6$ | $t_{0,1,2,3,5}γ_6$ | $t_{0,1,2,3,4}γ_6$ | $t_{0,1,2,3,4,5}γ_6$ |
  