Truncated 7-cubes

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File:7-cube t0.svg
7-cube
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-cube t01.svg
Truncated 7-cube
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-cube t12.svg
Bitruncated 7-cube
File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-cube t23.svg
Tritruncated 7-cube
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:7-cube t6.svg
7-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png 		File:7-cube t56.svg
Truncated 7-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png 		File:7-cube t45.svg
Bitruncated 7-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 		File:7-cube t34.svg
Tritruncated 7-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Orthogonal projections in B7 Coxeter plane

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

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

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Truncated 7-cube
Type uniform 7-polytope
Schläfli symbol t{4,35}
Coxeter-Dynkin diagrams File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 3136
Vertices 896
Vertex figure Elongated 5-simplex pyramid
Coxeter groups B7, [35,4]
Properties convex

Alternate names

[edit | edit source]
  • Truncated hepteract (Jonathan Bowers)[1]

Coordinates

[edit | edit source]

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

[edit | edit source]
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph File:7-cube t01.svg File:7-cube t01 B6.svg File:7-cube t01 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph File:7-cube t01 B4.svg File:7-cube t01 B3.svg File:7-cube t01 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph File:7-cube t01 A5.svg File:7-cube t01 A3.svg
Dihedral symmetry [6] [4]
[edit | edit source]

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

Truncated hypercubes
Image File:Regular polygon 8 annotated.svg File:3-cube t01.svgFile:Truncated hexahedron.png File:4-cube t01.svgFile:Schlegel half-solid truncated tesseract.png File:5-cube t01.svgFile:5-cube t01 A3.svg File:6-cube t01.svgFile:6-cube t01 A5.svg File:7-cube t01.svgFile:7-cube t01 A5.svg File:8-cube t01.svgFile:8-cube t01 A7.svg ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure ( )v( ) File:Truncated cube vertfig.svg
( )v{ } 		File:Truncated 8-cell verf.png
( )v{3} 		File:Truncated 5-cube verf.png
( )v{3,3} 		( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Bitruncated 7-cube

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Bitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol 2t{4,35}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 9408
Vertices 2688
Vertex figure { }v{3,3,3}
Coxeter groups B7, [35,4]
D7, [34,1,1]
Properties convex

Alternate names

[edit | edit source]
  • Bitruncated hepteract (Jonathan Bowers)[2]

Coordinates

[edit | edit source]

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

[edit | edit source]
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph File:7-cube t12.svg File:7-cube t12 B6.svg File:7-cube t12 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph File:7-cube t12 B4.svg File:7-cube t12 B3.svg File:7-cube t12 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph File:7-cube t12 A5.svg File:7-cube t12 A3.svg
Dihedral symmetry [6] [4]
[edit | edit source]

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

Bitruncated hypercubes
Image File:3-cube t12.svgFile:Truncated octahedron.png File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png File:5-cube t12.svgFile:5-cube t12 A3.svg File:6-cube t12.svgFile:6-cube t12 A5.svg File:7-cube t12.svgFile:7-cube t12 A5.svg File:8-cube t12.svgFile:8-cube t12 A7.svg ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure File:Truncated octahedron vertfig.svg
( )v{ } 		File:Bitruncated 8-cell verf.png
{ }v{ } 		File:Bitruncated penteract verf.png
{ }v{3} 		File:Bitruncated 6-cube verf.png
{ }v{3,3} 		{ }v{3,3,3} { }v{3,3,3,3}

Tritruncated 7-cube

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Tritruncated 7-cube
Type uniform 7-polytope
Schläfli symbol 3t{4,35}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 13440
Vertices 3360
Vertex figure {4}v{3,3}
Coxeter groups B7, [35,4]
D7, [34,1,1]
Properties convex

Alternate names

[edit | edit source]
  • Tritruncated hepteract (Jonathan Bowers)[3]

Coordinates

[edit | edit source]

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

[edit | edit source]
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph File:7-cube t23.svg File:7-cube t23 B6.svg File:7-cube t23 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph File:7-cube t23 B4.svg File:7-cube t23 B3.svg File:7-cube t23 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph File:7-cube t23 A5.svg File:7-cube t23 A3.svg
Dihedral symmetry [6] [4]

Notes

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

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, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). [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.
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
[edit | edit source]
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 • Polytope operations
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