Truncated 7-cubes

In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.


7-cube
Truncated 7-cube
Bitruncated 7-cube
Tritruncated 7-cube

7-orthoplex
Truncated 7-orthoplex
Bitruncated 7-orthoplex
Tritruncated 7-orthoplex
Orthogonal projections in B7 Coxeter plane

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.

Truncated 7-cube
Truncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges3136
Vertices896
Vertex figureElongated 5-simplex pyramid
Coxeter groupsB7, [35,4]
Propertiesconvex

Alternate names
  • Truncated hepteract (Jonathan Bowers)[1]

Coordinates

Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

Images
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

The truncated 7-cube, is sixth in a sequence of truncated hypercubes:

Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}		 ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Bitruncated 7-cube
Bitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol2t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges9408
Vertices2688
Vertex figure{ }v{3,3,3}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex

Alternate names
  • Bitruncated hepteract (Jonathan Bowers)[2]

Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±2,±1,0)

Images
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}		 { }v{3,3,3} { }v{3,3,3,3}

Tritruncated 7-cube
Tritruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol3t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges13440
Vertices3360
Vertex figure{4}v{3,3}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex

Alternate names
  • Tritruncated hepteract (Jonathan Bowers)[3]

Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0)

Images
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Notes
  1. Klitizing (x3x3o3o3o3o4o - taz)
  2. Klitizing (o3x3x3o3o3o4o - botaz)
  3. Klitizing (o3o3x3x3o3o4o - totaz)

References
  • 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
      • (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. "7D uniform polytopes (polyexa)". o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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