Rectified 5-cubes
![]() 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Rectified 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Birectified 5-cube Birectified 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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![]() 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Rectified 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Orthogonal projections in A5 Coxeter plane |
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In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.
There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-ocube are located in the square face centers of the 5-cube.
Contents
Rectified 5-cube
Rectified 5-cube rectified penteract (rin) |
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Type | uniform 5-polytope | |
Schläfli symbol | r{4,3,3,3} | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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4-faces | 42 | |
Cells | 200 | |
Faces | 400 | |
Edges | 320 | |
Vertices | 80 | |
Vertex figure | ![]() tetrahedral prism |
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Coxeter group | BC5, [4,33], order 3840 | |
Dual | ||
Base point | (0,1,1,1,1,1)√2 | |
Circumradius | sqrt(2) = 1.414214 | |
Properties | convex, isogonal |
Alternate names
- Rectified penteract (acronym: rin) (Jonathan Bowers)
Construction
The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 5-cube with edge length is given by all permutations of:
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
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Graph | ![]() |
150px | 150px |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | 150px | 150px | |
Dihedral symmetry | [4] | [4] |
Birectified 5-cube
Birectified 5-cube birectified penteract (nit) |
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Type | uniform 5-polytope | |
Schläfli symbol | 2r{4,3,3,3} | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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4-faces | 42 | 10 {3,4,3} 32 t1{3,3,3} |
Cells | 280 | |
Faces | 640 | |
Edges | 480 | |
Vertices | 80 | |
Vertex figure | ![]() {3}×{4} |
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Coxeter group | BC5, [4,33], order 3840 D5, [32,1,1], order 1920 |
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Dual | ||
Base point | (0,0,1,1,1,1)√2 | |
Circumradius | sqrt(3/2) = 1.224745 | |
Properties | convex, isogonal |
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 as a second rectification of a 5-dimensional cross polytope.
Alternate names
- Birectified 5-cube/penteract
- Birectified pentacross/5-orthoplex/triacontiditeron
- Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
- Rectified 5-demicube/demipenteract
Construction and coordinates
The birectified 5-cube may be constructed by birectifing the vertices of the 5-cube at of the edge length.
The Cartesian coordinates of the vertices of a birectified 5-cube having edge length 2 are all permutations of:
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
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Graph | ![]() |
150px | 150px |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | 150px | 150px | |
Dihedral symmetry | [4] | [4] |
Related polytopes
Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
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Name | t{4} | r{4,3} | 2t{4,3,3} | 2r{4,3,3,3} | 3t{4,3,3,3,3} | 3r{4,3,3,3,3,3} | 4t{4,3,3,3,3,3,3} | |
Coxeter diagram |
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Images | 60px | ![]() ![]() |
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60px60px | 60px60px | ... |
Facets | {3} ![]() {4} ![]() |
t{3,3} ![]() t{3,4} ![]() |
r{3,3,3} ![]() r{3,3,4} ![]() |
2t{3,3,3,3} ![]() 2t{3,3,3,4} ![]() |
2r{3,3,3,3,3} ![]() 2r{3,3,3,3,4} ![]() |
3t{3,3,3,3,3,3} 30px 3t{3,3,3,3,3,4} ![]() |
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Vertex figure |
![]() Rectangle |
60px Disphenoid |
![]() {3}×{4} duoprism |
{3,3}×{3,4} duoprism |
Related polytopes
Thes polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
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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.
- Richard Klitzing, 5D, uniform polytopes (polytera) o3x3o3o4o - rin, o3o3x3o4o - nit
External links
- Weisstein, Eric W., "Hypercube", MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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 |