Steric 5-cubes

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
  • Stericantic 5-cube
  • Steriruncic 5-cube
  • Steriruncicantic 5-cube
Orthogonal projections in B5 Coxeter plane

Steric 5-cube

Steric 5-cube
Typeuniform polyteron
Schläfli symbol
  • t0,3{3,32,1}
  • h4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells480
Faces720
Edges400
Vertices80
Vertex figure{3,3}-t1{3,3} antiprism
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

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

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

orthographic projections
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]
Dimensional family of steric n-cubes
n5678
[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
Typeuniform polyteron
Schläfli symbol
  • t0,1,3{3,32,1}
  • h2,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells720
Faces1840
Edges1680
Vertices480
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

  • Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[1]:(x3x3o *b3o3x - pithin)

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

orthographic projections
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
Typeuniform polyteron
Schläfli symbol
  • t0,2,3{3,32,1}
  • h3,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells560
Faces1280
Edges1120
Vertices320
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

  • Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[1]:(x3o3o *b3x3x - pirhin)

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

orthographic projections
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
Typeuniform polyteron
Schläfli symbol
  • t0,1,2,3{3,32,1}
  • h2,3,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells720
Faces2080
Edges2400
Vertices960
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

  • Great prismated hemipenteract (giphin) (Jonathan Bowers)[1]:(x3x3o *b3x3x - giphin)

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

orthographic projections
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]

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

Further reading

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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