Runcinated 5-cubes
5-cube |
Runcinated 5-cube |
Runcinated 5-orthoplex |
Runcitruncated 5-cube |
Runcicantellated 5-cube |
Runcicantitruncated 5-cube |
Runcitruncated 5-orthoplex |
Runcicantellated 5-orthoplex |
Runcicantitruncated 5-orthoplex |
Orthogonal projections in B5 Coxeter plane |
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In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.
There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.
Runcinated 5-cube
[edit]Runcinated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,3{4,3,3,3} | |
Coxeter diagram | ||
4-faces | 202 | 10 80 80 32 |
Cells | 1240 | 40 240 320 160 320 160 |
Faces | 2160 | 240 960 640 320 |
Edges | 1440 | 480+960 |
Vertices | 320 | |
Vertex figure | ||
Coxeter group | B5 [4,3,3,3] | |
Properties | convex |
Alternate names
[edit]- Small prismated penteract (Acronym: span) (Jonathan Bowers)
Coordinates
[edit]The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:
Images
[edit]Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | |||
Dihedral symmetry | [4] | [4] |
Runcitruncated 5-cube
[edit]Runcitruncated 5-cube | ||
---|---|---|
Type | Uniform 5-polytope | |
Schläfli symbol | t0,1,3{4,3,3,3} | |
Coxeter-Dynkin diagrams | ||
4-faces | 202 | 10 80 80 32 |
Cells | 1560 | 40 240 320 320 160 320 160 |
Faces | 3760 | 240 960 320 960 640 640 |
Edges | 3360 | 480+960+1920 |
Vertices | 960 | |
Vertex figure | ||
Coxeter group | B5, [3,3,3,4] | |
Properties | convex |
Alternate names
[edit]- Runcitruncated penteract
- Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)
Construction and coordinates
[edit]The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:
Images
[edit]Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | |||
Dihedral symmetry | [4] | [4] |
Runcicantellated 5-cube
[edit]Runcicantellated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,2,3{4,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 202 | 10 80 80 32 |
Cells | 1240 | 40 240 320 320 160 160 |
Faces | 2960 | 240 480 960 320 640 320 |
Edges | 2880 | 960+960+960 |
Vertices | 960 | |
Vertex figure | ||
Coxeter group | B5 [4,3,3,3] | |
Properties | convex |
Alternate names
[edit]- Runcicantellated penteract
- Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)
Coordinates
[edit]The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:
Images
[edit]Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | |||
Dihedral symmetry | [4] | [4] |
Runcicantitruncated 5-cube
[edit]Runcicantitruncated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,1,2,3{4,3,3,3} | |
Coxeter-Dynkin diagram |
||
4-faces | 202 | |
Cells | 1560 | |
Faces | 4240 | |
Edges | 4800 | |
Vertices | 1920 | |
Vertex figure | Irregular 5-cell | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex, isogonal |
Alternate names
[edit]- Runcicantitruncated penteract
- Biruncicantitruncated pentacross
- great prismated penteract (gippin) (Jonathan Bowers)
Coordinates
[edit]The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
[edit]Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | |||
Dihedral symmetry | [4] | [4] |
Related polytopes
[edit]These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
References
[edit]- 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)". o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
External links
[edit]- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), Jonathan Bowers
- Multi-dimensional Glossary