Runcic 5-Cubes
From Handwiki  5-cube    
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 Runcic 5-cube   = 
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 5-demicube =
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 Runcicantic 5-cube =
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| Orthogonal projections in B5 Coxeter plane | |||
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In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.
Runcic 5-cube
| Runcic 5-cube | |
|---|---|
| Type | uniform 5-polytope |
| Schläfli symbol | h3{4,3,3,3} |
| Coxeter-Dynkin diagram |
|
| 4-faces | 42 |
| Cells | 360 |
| Faces | 880 |
| Edges | 720 |
| Vertices | 160 |
| Vertex figure | |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
Alternate names
- Cantellated 5-demicube/demipenteract
- Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:
- (±1,±1,±1,±3,±3)
with an odd number of plus signs.
Images
| Coxeter plane | B5 | |
|---|---|---|
| Graph | 
| |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph | 
|
|
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph | 
|
|
| Dihedral symmetry | [4] | [4] |
Related polytopes
It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:
 Runcic 5-cube |
 Runcinated 5-cube |
Runcicantic 5-cube
| Runcicantic 5-cube | |
|---|---|
| Type | uniform 5-polytope |
| Schläfli symbol | t0,1,2{3,32,1} h3{4,33} |
| Coxeter-Dynkin diagram |
|
| 4-faces | 42 |
| Cells | 360 |
| Faces | 1040 |
| Edges | 1200 |
| Vertices | 480 |
| Vertex figure | |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
Alternate names
- Cantitruncated 5-demicube/demipenteract
- Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:
- (±1,±1,±3,±5,±5)
with an odd number of plus signs.
Images
| Coxeter plane | B5 | |
|---|---|---|
| Graph | 
| |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph | 
|
|
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph | 
|
|
| Dihedral symmetry | [4] | [4] |
Related polytopes
It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:
 Runcicantic 5-cube |
 Runcicantellated 5-cube |
Related polytopes
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform 5-polytopes that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
Notes
- ↑ Klitzing, (x3o3o *b3x3o - sirhin)
- ↑ Klitzing, (x3x3o *b3x3o - girhin)
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. x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin
External links
- Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html.
- 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 | ||||||||||||
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Categories: [5-polytopes]
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