Steric 5-cubes
5-cube |
Steric 5-cube |
Stericantic 5-cube |
Half 5-cube |
Steriruncic 5-cube |
Steriruncicantic 5-cube |
Orthogonal projections in B5 Coxeter plane |
---|
In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.
Steric 5-cube
Steric 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol | t0,3{3,32,1} h4{4,3,3,3} |
Coxeter-Dynkin diagram | |
4-faces | 82 |
Cells | 480 |
Faces | 720 |
Edges | 400 |
Vertices | 80 |
Vertex figure | {3,3}-t1{3,3} antiprism |
Coxeter groups | D5, [32,1,1] |
Properties | convex |
Alternate names
- Steric penteract, runcinated demipenteract
- Small prismated hemipenteract (siphin) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of
- (±1,±1,±1,±1,±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
Dimensional family of steric n-cubes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
n | 5 | 6 | 7 | 8 | |||||||
[1+,4,3n-2] = [3,3n-3,1] |
[1+,4,33] = [3,32,1] |
[1+,4,34] = [3,33,1] |
[1+,4,35] = [3,34,1] |
[1+,4,36] = [3,35,1] | |||||||
Steric figure |
|||||||||||
Coxeter | = |
= |
= |
= | |||||||
Schläfli | h4{4,33} | h4{4,34} | h4{4,35} | h4{4,36} |
Stericantic 5-cube
Stericantic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol | t0,1,3{3,32,1} h2,4{4,3,3,3} |
Coxeter-Dynkin diagram | |
4-faces | 82 |
Cells | 720 |
Faces | 1840 |
Edges | 1680 |
Vertices | 480 |
Vertex figure | |
Coxeter groups | D5, [32,1,1] |
Properties | convex |
Alternate names
- Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[2]
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:
- (±1,±1,±3,±3,±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] |
Steriruncic 5-cube
Steriruncic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol | t0,2,3{3,32,1} h3,4{4,3,3,3} |
Coxeter-Dynkin diagram | |
4-faces | 82 |
Cells | 560 |
Faces | 1280 |
Edges | 1120 |
Vertices | 320 |
Vertex figure | |
Coxeter groups | D5, [32,1,1] |
Properties | convex |
Alternate names
- Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[3]
Cartesian coordinates
The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:
- (±1,±1,±1,±3,±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] |
Steriruncicantic 5-cube
Steriruncicantic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol | t0,1,2,3{3,32,1} h2,3,4{4,3,3,3} |
Coxeter-Dynkin diagram | |
4-faces | 82 |
Cells | 720 |
Faces | 2080 |
Edges | 2400 |
Vertices | 960 |
Vertex figure | |
Coxeter groups | D5, [32,1,1] |
Properties | convex |
Alternate names
- Great prismated hemipenteract (giphin) (Jonathan Bowers)[4]
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:
- (±1,±1,±3,±5,±7)
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
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 polytera (uniform 5-polytope) 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
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
- (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)". x3o3o *b3o3x - siphin, x3x3o *b3o3x - pithin, x3o3o *b3x3x - pirhin, x3x3o *b3x3x - giphin
External links
- Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / 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 |