Steric 6-cubes

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Steric 6-cubes

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

More information Orthogonal projections in D5 Coxeter plane ...
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6-demicube
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Steric 6-cube
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Thumb
Stericantic 6-cube
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Thumb
Steriruncic 6-cube
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Thumb
Stericruncicantic 6-cube
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Orthogonal projections in D5 Coxeter plane
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Steric 6-cube

Steric 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,3{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

  • Runcinated demihexeract
  • Runcinated 6-demicube
  • Small prismated hemihexeract (Acronym: sophax) (Jonathan Bowers)[1]

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

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph Thumb
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph Thumb Thumb
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
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More information Dimensional family of steric n-cubes, n ...
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
Thumb Thumb Thumb Thumb
Coxeter
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=

=

=
Schläfli h4{4,33} h4{4,34} h4{4,35} h4{4,36}
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Stericantic 6-cube

More information Stericantic 6-cube ...
Stericantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,3{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
Close

Alternate names

  • Runcitruncated demihexeract
  • Runcitruncated 6-demicube
  • Prismatotruncated hemihexeract (Acronym: pithax) (Jonathan Bowers)[2]

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

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph Thumb
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph Thumb Thumb
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
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Steriruncic 6-cube

More information Steriruncic 6-cube ...
Steriruncic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,2,3{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
Close

Alternate names

  • Runcicantellated demihexeract
  • Runcicantellated 6-demicube
  • Prismatorhombated hemihexeract (Acronym: prohax) (Jonathan Bowers)[3]

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

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph Thumb
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph Thumb Thumb
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
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Steriruncicantic 6-cube

More information Steriruncicantic 6-cube ...
Steriruncicantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3{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
Close

Alternate names

  • Runcicantitruncated demihexeract
  • Runcicantitruncated 6-demicube
  • Great prismated hemihexeract (Acronym: gophax) (Jonathan Bowers)[4]

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

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph Thumb
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph Thumb Thumb
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph Thumb Thumb
Dihedral symmetry [6] [4]
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There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

Notes

References

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