6-demicube
=
Steric 6-cube
=
Stericantic 6-cube
=

Steriruncic 6-cube
=
Stericruncicantic 6-cube
=
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
Type uniform 6-polytope
Schläfli symbol t0,3{3,33,1}
h4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 480
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

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]

Related polytopes

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 6-cube

Stericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,33,1}
h2,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2880
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,33,1}
h3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 7680
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

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
Type uniform 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
Edges 17280
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

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]

Related polytopes

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

D6 polytopes

h{4,34}
h2{4,34}
h3{4,34}
h4{4,34}
h5{4,34}
h2,3{4,34}
h2,4{4,34}
h2,5{4,34}

h3,4{4,34}
h3,5{4,34}
h4,5{4,34}
h2,3,4{4,34}
h2,3,5{4,34}
h2,4,5{4,34}
h3,4,5{4,34}
h2,3,4,5{4,34}

Notes

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

References

External links

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
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