Pentellated 6-cube


6-cube

6-orthoplex

Pentellated 6-cube

Pentitruncated 6-cube

Penticantellated 6-cube

Penticantitruncated 6-cube

Pentiruncitruncated 6-cube

Pentiruncicantellated 6-cube

Pentiruncicantitruncated 6-cube

Pentisteritruncated 6-cube

Pentistericantitruncated 6-cube

Omnitruncated 6-cube
Orthogonal projections in BC6 Coxeter plane

In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.

There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.

Contents

Pentellated 6-cube

Pentellated 6-cube
Type Uniform polypeton
Schläfli symbol t0,5{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 1920
Vertices 384
Vertex figure 5-cell antiprism
Coxeter group BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentitruncated 6-cube

Pentitruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 8640
Vertices 1920
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Penticantellated 6-cube

Penticantellated 6-cube
Type uniform polypeton
Schläfli symbol t0,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 21120
Vertices 3840
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Penticantitruncated 6-cube

Penticantitruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 30720
Vertices 7680
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncitruncated 6-cube

Pentiruncitruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 151840
Vertices 11520
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncicantellated 6-cube

Pentiruncicantellated 6-cube
Type uniform polypeton
Schläfli symbol t0,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 46080
Vertices 11520
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncicantitruncated 6-cube

Pentiruncicantitruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 80640
Vertices 23040
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentisteritruncated 6-cube

Pentisteritruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 30720
Vertices 7680
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentistericantitruncated 6-cube

Pentistericantitruncated 6-cube
Type uniform polypeton
Schläfli symbol t0,1,2,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 80640
Vertices 23040
Vertex figure
Coxeter groups BC6, [4,3,3,3,3]
Properties convex

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Omnitruncated 6-cube

Omnitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 138240
Vertices 46080
Vertex figure irregular 5-simplex
Coxeter group BC6, [4,3,3,3,3]
Properties convex, isogonal

The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.

Alternate names

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Related polytopes

These polytopes are from a set of 63 uniform polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.


β6

t1β6

t2β6

t2γ6

t1γ6

γ6

t0,1β6

t0,2β6

t1,2β6

t0,3β6

t1,3β6

t2,3γ6

t0,4β6

t1,4γ6

t1,3γ6

t1,2γ6

t0,5γ6

t0,4γ6

t0,3γ6

t0,2γ6

t0,1γ6

t0,1,2β6

t0,1,3β6

t0,2,3β6

t1,2,3β6

t0,1,4β6

t0,2,4β6

t1,2,4β6

t0,3,4β6

t1,2,4γ6

t1,2,3γ6

t0,1,5β6

t0,2,5β6

t0,3,4γ6

t0,2,5γ6

t0,2,4γ6

t0,2,3γ6

t0,1,5γ6

t0,1,4γ6

t0,1,3γ6

t0,1,2γ6

t0,1,2,3β6

t0,1,2,4β6

t0,1,3,4β6

t0,2,3,4β6

t1,2,3,4γ6

t0,1,2,5β6

t0,1,3,5β6

t0,2,3,5γ6

t0,2,3,4γ6

t0,1,4,5γ6

t0,1,3,5γ6

t0,1,3,4γ6

t0,1,2,5γ6

t0,1,2,4γ6

t0,1,2,3γ6

t0,1,2,3,4β6

t0,1,2,3,5β6

t0,1,2,4,5β6

t0,1,2,4,5γ6

t0,1,2,3,5γ6

t0,1,2,3,4γ6

t0,1,2,3,4,5γ6

Notes

  1. ^ Klitzing, (x4o3o3o3o3x - stoxog)
  2. ^ Klitzing, (x4x3o3o3o3x - tacog)
  3. ^ Klitzing, (x4o3x3o3o3x - topag)
  4. ^ Klitzing, (x4x3x3o3o3x - togrix)
  5. ^ Klitzing, (x4x3o3x3o3x - tocrag)
  6. ^ Klitzing, (x4o3x3x3o3x - tiprixog)
  7. ^ Klitzing, (x4x3x3o3x3x - tagpox)
  8. ^ Klitzing, (x4x3o3o3x3x - tactaxog)
  9. ^ Klitzing, (x4x3x3o3x3x - tocagrax)
  10. ^ Klitzing, (x4x3x3x3x3x - gotaxog)

References

External links