Steric 5-cubes
Orthogonal projections in B5 Coxeter plane |
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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
[edit]Steric 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol |
|
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
[edit]- Steric penteract, runcinated demipenteract
- Small prismated hemipenteract (siphin) (Jonathan Bowers)[1]: (x3o3o *b3o3x - siphin)
Cartesian coordinates
[edit]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
[edit]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
[edit]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
[edit]Stericantic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol |
|
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
[edit]- Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[1]: (x3x3o *b3o3x - pithin)
Cartesian coordinates
[edit]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
[edit]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
[edit]Steriruncic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol |
|
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
[edit]- Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[1]: (x3o3o *b3x3x - pirhin)
Cartesian coordinates
[edit]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
[edit]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
[edit]Steriruncicantic 5-cube | |
---|---|
Type | uniform polyteron |
Schläfli symbol |
|
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
[edit]- Great prismated hemipenteract (giphin) (Jonathan Bowers)[1]: (x3x3o *b3x3x - giphin)
Cartesian coordinates
[edit]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
[edit]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
[edit]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} |
References
[edit]- ^ a b c d Klitzing, Richard. "5D uniform polytopes (polytera)".
Further reading
[edit]- Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York City: Dover. Retrieved 2022-05-19.
- Coxeter, H. S. M. (1995-05-17). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. ISBN 978-0-471-01003-6. LCCN 94047368. OCLC 632987525. OL 7598569M. Retrieved 2022-05-19.
- Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I". Mathematische Zeitschrift. 46. Springer Nature: 380–407. doi:10.1007/BF01181449. ISSN 1432-1823. S2CID 186237114. Retrieved 2022-05-19.
- Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II". Mathematische Zeitschrift. 188 (4). Springer Nature: 559–591. doi:10.1007/BF01161657. ISSN 1432-1823. S2CID 120429557. Retrieved 2022-05-19.
- Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III". Mathematische Zeitschrift. 200 (1). Springer Nature: 3–45. doi:10.1007/BF01161745. ISSN 1432-1823. S2CID 186237142. Retrieved 2022-05-19.
- Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis).
- Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD thesis). University of Toronto. Retrieved 2022-05-19.
External links
[edit]- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary