title: Truncated 8-orthoplexes
text: # Truncated 8-orthoplexes
  
  | 8-orthoplex | Truncated 8-orthoplex | Bitruncated 8-orthoplex |
  | - | - | - |
  | Tritruncated 8-orthoplex | Quadritruncated 8-cube | Tritruncated 8-cube |
  | Bitruncated 8-cube | Truncated 8-cube | 8-cube |
  | Orthogonal projections in $B_8$ Coxeter plane |||
  
  In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
  
  There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.
  
  ## Truncated 8-orthoplex
  
  Truncated 8-orthoplex :
  
  | Type | uniform 8-polytope |
  | - | - |
  | Schläfli symbol | $t_{0,1}{3,3,3,3,3,3,4}$ |
  | Coxeter-Dynkin diagrams | |
  | 6-faces | |
  | 5-faces | |
  | 4-faces | |
  | Cells | |
  | Faces | |
  | Edges | 1456 |
  | Vertices | 224 |
  | Vertex figure | ( )v{3,3,3,4} |
  | Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
  | Properties | convex |
  
  ### Alternate names
  
  * Truncated octacross (acronym tek) (Jonthan Bowers)
  
  ### Construction
  
  There are two Coxeter groups associated with the truncated 8-orthoplex, one with the $C_8$ or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the $D_8$ or $[3^{5,1,1}]$ Coxeter group.
  
  ### Coordinates
  
  Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of
  
      (±2,±1,0,0,0,0,0,0)
  
  ### Images
  
  orthographic projections :
  
  | $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
  | - | - | - | - | - | - |
  | [16] | [16] | [16] | [14] | [14] | [14] |
  | $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
  | [12] | [12] | [12] | [10] | [10] | [10] |
  | $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  | $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  
  ## Bitruncated 8-orthoplex
  
  Bitruncated 8-orthoplex :
  
  | Type | uniform 8-polytope |
  | - | - |
  | Schläfli symbol | $t_{1,2}{3,3,3,3,3,3,4}$ |
  | Coxeter-Dynkin diagrams | |
  | 6-faces | |
  | 5-faces | |
  | 4-faces | |
  | Cells | |
  | Faces | |
  | Edges | |
  | Vertices | |
  | Vertex figure | { }v{3,3,3,4} |
  | Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
  | Properties | convex |
  
  ### Alternate names
  
  * Bitruncated octacross (acronym batek) (Jonthan Bowers)
  
  ### Coordinates
  
  Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of
  
      (±2,±2,±1,0,0,0,0,0)
  
  ### Images
  
  orthographic projections :
  
  | $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
  | - | - | - | - | - | - |
  | [16] | [16] | [16] | [14] | [14] | [14] |
  | $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
  | [12] | [12] | [12] | [10] | [10] | [10] |
  | $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  | $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  
  ## Tritruncated 8-orthoplex
  
  Tritruncated 8-orthoplex :
  
  | Type | uniform 8-polytope |
  | - | - |
  | Schläfli symbol | $t_{2,3}{3,3,3,3,3,3,4}$ |
  | Coxeter-Dynkin diagrams | |
  | 6-faces | |
  | 5-faces | |
  | 4-faces | |
  | Cells | |
  | Faces | |
  | Edges | |
  | Vertices | |
  | Vertex figure | {3}v{3,3,4} |
  | Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
  | Properties | convex |
  
  ### Alternate names
  
  * Tritruncated octacross (acronym tatek) (Jonthan Bowers)
  
  ### Coordinates
  
  Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of
  
      (±2,±2,±2,±1,0,0,0,0)
  
  ### Images
  
  orthographic projections :
  
  | $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
  | - | - | - | - | - | - |
  | [16] | [16] | [16] | [14] | [14] | [14] |
  | $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
  | [12] | [12] | [12] | [10] | [10] | [10] |
  | $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  | $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
  | [8] | [8] | [6] | [6] | [4] | [4] |
  