Steric 5-cubes

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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

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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

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  • Steric penteract, runcinated demipenteract
  • Small prismated hemipenteract (siphin) (Jonathan Bowers)[1]: (x3o3o *b3o3x - siphin) 

Cartesian coordinates

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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

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orthographic projections
Coxeter plane B5
Graph File:5-demicube t03 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph File:5-demicube t03 D5.svg File:5-demicube t03 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph File:5-demicube t03 D3.svg File:5-demicube t03 A3.svg
Dihedral symmetry [4] [4]
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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 		File:5-demicube t04 D5.svg File:6-demicube t04 D6.svg File:7-demicube t04 D7.svg File:8-demicube t04 D8.svg
Coxeter File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Schläfli h4{4,33} h4{4,34} h4{4,35} h4{4,36}

Stericantic 5-cube

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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

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Cartesian coordinates

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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

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orthographic projections
Coxeter plane B5
Graph File:5-demicube t013 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph File:5-demicube t013 D5.svg File:5-demicube t013 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph File:5-demicube t013 D3.svg File:5-demicube t013 A3.svg
Dihedral symmetry [4] [4]

Steriruncic 5-cube

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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

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Cartesian coordinates

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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

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orthographic projections
Coxeter plane B5
Graph File:5-demicube t023 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph File:5-demicube t023 D5.svg File:5-demicube t023 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph File:5-demicube t023 D3.svg File:5-demicube t023 A3.svg
Dihedral symmetry [4] [4]

Steriruncicantic 5-cube

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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

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Cartesian coordinates

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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

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orthographic projections
Coxeter plane B5
Graph File:5-demicube t0123 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph File:5-demicube t0123 D5.svg File:5-demicube t0123 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph File:5-demicube t0123 D3.svg File:5-demicube t0123 A3.svg
Dihedral symmetry [4] [4]
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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
File:5-demicube t0 D5.svg
h{4,3,3,3} 		File:5-demicube t01 D5.svg
h2{4,3,3,3} 		File:5-demicube t02 D5.svg
h3{4,3,3,3} 		File:5-demicube t03 D5.svg
h4{4,3,3,3} 		File:5-demicube t012 D5.svg
h2,3{4,3,3,3} 		File:5-demicube t013 D5.svg
h2,4{4,3,3,3} 		File:5-demicube t023 D5.svg
h3,4{4,3,3,3} 		File:5-demicube t0123 D5.svg
h2,3,4{4,3,3,3}

References

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  1. ^ a b c d Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

Further reading

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  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
[edit | edit source]
Fundamental 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
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