Runcic 5-cubes
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| Orthogonal projections in B5 Coxeter plane | |
|---|---|
| 5-cube | Runcic 5-cube = |
| 5-demicube = | Runcicantic 5-cube = |
In five-dimensional geometry, a runcic 5-cube, runcic 5-demicube or 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)
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 |
| Runcic n-cubes | |||||
|---|---|---|---|---|---|
| n | 4 | 5 | 6 | 7 | 8 |
| [1+,4,3n − 2] = [3,3n − 3,1] | [1+,4,32] = [3,31,1] | [1+,4,33] = [3,32,1] | [1+,4,34] = [3,33,1] | [1+,4,35] = [3,34,1] | [1+,4,36] = [3,35,1] |
| Runcic figure | |||||
| Coxeter | = | = | = | = | = |
| Schläfli | h3{4,32} | h3{4,33} | h3{4,34} | h3{4,35} | h3{4,36} |
Runcicantic 5-cube
| Runcicantic 5-cube | |
|---|---|
| Type | uniform 5-polytope |
| Schläfli symbol | t0,1,2{3,32,1} h2,3{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)
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
These polytopes are based on the 5-demicube, a member 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.
| D5 polytopes | |||||||
|---|---|---|---|---|---|---|---|
| h{4,3,3,3} | h2{4,3,3,3} | h3{4,3,3,3} | h4{4,3,3,3} | h2,3{4,3,3,3} | h2,4{4,3,3,3} | h3,4{4,3,3,3} | h2,3,4{4,3,3,3} |
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 Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN 978-0-471-01003-6 (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. . x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin
External links
| vteFundamental convex regular and uniform polytopes in dimensions 2–10 | |||||
|---|---|---|---|---|---|
| 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 polychoron | Pentachoron | 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 • Polytope operations |