Pentic 6-cubes


6-demicube
(half 6-cube)
=

Pentic 6-cube
=

Penticantic 6-cube
=

Pentiruncic 6-cube
=

Pentiruncicantic 6-cube
=

Pentisteric 6-cube
=

Pentistericantic 6-cube
=

Pentisteriruncic 6-cube
=

Pentisteriruncicantic 6-cube
=
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

There are 8 pentic forms of the 6-cube.

Pentic 6-cube

Pentic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,4{3,34,1}
h5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges1440
Vertices192
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentic 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]

Penticantic 6-cube

Penticantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,4{3,34,1}
h2,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges9600
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a stericantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±3,±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]

Pentiruncic 6-cube

Pentiruncic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,2,4{3,34,1}
h3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges10560
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentiruncic 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]

Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,2,4{3,32,1}
h2,3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges20160
Vertices5760
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),

Alternate names

Cartesian coordinates

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

(±1,±1,±3,±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]

Pentisteric 6-cube

Pentisteric 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,3,4{3,34,1}
h4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges5280
Vertices960
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteric 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]

Pentistericantic 6-cube

Pentistericantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,3,4{3,34,1}
h2,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges23040
Vertices5760
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .

Alternate names

Cartesian coordinates

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

(±1,±1,±3,±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]

Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,2,3,4{3,34,1}
h3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges15360
Vertices3840
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteriruncic 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]

Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Typeuniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,32,1}
h2,3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges34560
Vertices11520
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .

Alternate names

Cartesian coordinates

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

(±1,±1,±3,±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 BC6 symmetry, and 16 are unique:

Notes

  1. Klitzing, (x3o3o *b3o3x3o3o - sochax)
  2. Klitzing, (x3x3o *b3o3x3o3o - cathix)
  3. Klitzing, (x3o3o *b3x3x3o3o - crohax)
  4. Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
  5. Klitzing, (x3o3o *b3o3x3x3x - cophix)
  6. Klitzing, (x3x3o *b3o3x3x3x - capthix)
  7. Klitzing, (x3o3o *b3x3x3x3x - caprohax)
  8. Klitzing, (x3x3o *b3x3x3x3o - gochax)

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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