Truncated 5-cubes
![]() 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Truncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bitruncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Truncated 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bitruncated 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Orthogonal projections in B5 Coxeter plane |
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In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.
There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.
Truncated 5-cube
Truncated 5-cube | ||
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Type | uniform 5-polytope | |
Schläfli symbol | t{4,3,3,3} | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 42 | 10 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 32 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | 200 | 40 ![]() ![]() ![]() ![]() ![]() ![]() 160 ![]() ![]() ![]() ![]() ![]() ![]() |
Faces | 400 | 80 ![]() ![]() ![]() ![]() 320 ![]() ![]() ![]() ![]() |
Edges | 400 | 80 ![]() 320 ![]() |
Vertices | 160 | |
Vertex figure | ![]() ( )v{3,3} | |
Coxeter group | B5, [3,3,3,4], order 3840 | |
Properties | convex |
Alternate names
- Truncated penteract (Acronym: tan) (Jonathan Bowers)
Construction and coordinates
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at [math]\displaystyle{ 1/(\sqrt{2}+2) }[/math] of the edge length. A regular 5-cell is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
- [math]\displaystyle{ \left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right) }[/math]
Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
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Graph | ![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [4] |
Related polytopes
The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
Bitruncated 5-cube
Bitruncated 5-cube | ||
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Type | uniform 5-polytope | |
Schläfli symbol | 2t{4,3,3,3} | |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 42 | 10 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 32 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | 280 | 40 ![]() ![]() ![]() ![]() ![]() ![]() 160 ![]() ![]() ![]() ![]() ![]() 80 ![]() ![]() ![]() ![]() ![]() ![]() |
Faces | 720 | 80 ![]() ![]() ![]() ![]() 320 ![]() ![]() ![]() 320 ![]() ![]() ![]() ![]() |
Edges | 800 | 320 ![]() 480 ![]() |
Vertices | 320 | |
Vertex figure | ![]() { }v{3} | |
Coxeter groups | B5, [3,3,3,4], order 3840 | |
Properties | convex |
Alternate names
- Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)
Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at [math]\displaystyle{ \sqrt{2} }[/math] of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
- [math]\displaystyle{ \left(0,\ \pm1,\ \pm2,\ \pm2,\ \pm2\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Related polytopes
The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
Related polytopes
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.
Notes
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 [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. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm. o3o3o3x4x - tan, o3o3x3x4o - bittin
External links
Fundamental convex regular and uniform polytopes in dimensions 2–10
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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 | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 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 |
![]() | Original source: https://en.wikipedia.org/wiki/Truncated 5-cubes.
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