Truncated 7-cubes
Uniform 7- polytope
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
Alternate names
- Truncated hepteract (Jonathan Bowers)
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 projectionsCoxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
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Graph | | | |
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Dihedral symmetry | [14] | [12] | [10] |
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Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
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Graph | | | |
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Dihedral symmetry | [8] | [6] | [4] |
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Coxeter plane | A5 | A3 |
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Graph | | |
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Dihedral symmetry | [6] | [4] |
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Related polytopes
The truncated 7-cube, is sixth in a sequence of truncated hypercubes:
Truncated hypercubes
Image | | | | | | | | ... |
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Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube |
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Coxeter diagram | | | | | | | |
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Vertex figure | ( )v( ) | ( )v{ } | ( )v{3} | ( )v{3,3} | ( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |
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Bitruncated 7-cube
Alternate names
- Bitruncated hepteract (Jonathan Bowers)
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 projectionsCoxeter 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 |
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Graph | | |
---|
Dihedral symmetry | [6] | [4] |
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Related polytopes
The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:
Bitruncated hypercubes
Tritruncated 7-cube
Alternate names
- Tritruncated hepteract (Jonathan Bowers)
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 projectionsCoxeter 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] |
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Notes
- 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. "7D uniform polytopes (polyexa)". o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
External links
References
Klitizing (x3x3o3o3o3o4o - taz)
Klitizing (o3x3x3o3o3o4o - botaz)
Klitizing (o3o3x3x3o3o4o - totaz)