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 B8 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

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Truncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t0,1{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 B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names

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  • Truncated octacross (acronym tek) (Jonthan Bowers)[1]

Construction

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There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.

Coordinates

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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

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orthographic projections
B8 B7
   
[16] [14]
B6 B5
   
[12] [10]
B4 B3 B2
     
[8] [6] [4]
A7 A5 A3
     
[8] [6] [4]

Bitruncated 8-orthoplex

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Bitruncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t1,2{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 B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names

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  • Bitruncated octacross (acronym batek) (Jonthan Bowers)[2]

Coordinates

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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

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orthographic projections
B8 B7
   
[16] [14]
B6 B5
   
[12] [10]
B4 B3 B2
     
[8] [6] [4]
A7 A5 A3
     
[8] [6] [4]

Tritruncated 8-orthoplex

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Tritruncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t2,3{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 B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names

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  • Tritruncated octacross (acronym tatek) (Jonthan Bowers)[3]

Coordinates

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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

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orthographic projections
B8 B7
   
[16] [14]
B6 B5
   
[12] [10]
B4 B3 B2
     
[8] [6] [4]
A7 A5 A3
     
[8] [6] [4]

Notes

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  1. ^ Klitizing, (x3x3o3o3o3o3o4o - tek)
  2. ^ Klitizing, (o3x3x3o3o3o3o4o - batek)
  3. ^ Klitizing, (o3o3x3x3o3o3o4o - tatek)

References

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  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)". x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek
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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds