Steric 6-cubes


6-demicube
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Steric 6-cube
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Stericantic 6-cube
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Steriruncic 6-cube
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Stericruncicantic 6-cube
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Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

Steric 6-cube

Steric 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,3{3,33,1}
h4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges3360
Vertices480
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Stericantic 6-cube

Stericantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,3{3,33,1}
h2,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges12960
Vertices2880
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Steriruncic 6-cube

Steriruncic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,2,3{3,33,1}
h3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges7680
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Steriruncicantic 6-cube

Steriruncicantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,2,3{3,32,1}
h2,3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges17280
Vertices5760
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

Notes

  1. Klitzing, (x3o3o *b3o3x3o - sophax)
  2. Klitzing, (x3x3o *b3o3x3o - pithax)
  3. Klitzing, (x3o3o *b3x3x3o - prohax)
  4. Klitzing, (x3x3o *b3x3x3o - gophax)

References

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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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