Truncated 6-orthoplexes

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6-cube t5.svg
6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t45.svg
Truncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t34.svg
Bitruncated 6-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t23.svg
Tritruncated 6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t0.svg
6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
6-cube t01.svg
Truncated 6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
6-cube t12.svg
Bitruncated 6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Orthogonal projections in B6 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[edit]

Truncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

5-faces 76
4-faces 576
Cells 1200
Faces 1120
Edges 540
Vertices 120
Vertex figure Elongated 16-cell pyramid
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names[edit]

  • Truncated hexacross
  • Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)[1]

Construction[edit]

There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t45.svg 6-cube t45 B5.svg 6-cube t45 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t45 B3.svg 6-cube t45 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t45 A5.svg 6-cube t45 A3.svg
Dihedral symmetry [6] [4]

Bitruncated 6-orthoplex[edit]

Bitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol 2t{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names[edit]

  • Bitruncated hexacross
  • Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t34.svg 6-cube t34 B5.svg 6-cube t34 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t34 B3.svg 6-cube t34 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t34 A5.svg 6-cube t34 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

Thes polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

Notes[edit]

  1. ^ Klitzing, (x3x3o3o3o4o - tag)
  2. ^ Klitzing, (o3x3x3o3o4o - botag)

References[edit]

  • 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.
  • Klitzing, Richard. "6D uniform polytopes (polypeta)".  x3x3o3o3o4o - tag, o3x3x3o3o4o - botag

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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