Pentellated 6-orthoplexes

Orthogonal projections in B6 Coxeter plane

6-orthoplex

Pentellated 6-orthoplex
Pentellated 6-cube

6-cube

Pentitruncated 6-orthoplex

Penticantellated 6-orthoplex

Penticantitruncated 6-orthoplex

Pentiruncitruncated 6-orthoplex

Pentiruncicantellated 6-cube

Pentiruncicantitruncated 6-orthoplex

Pentisteritruncated 6-cube

Pentistericantitruncated 6-orthoplex

Pentisteriruncicantitruncated 6-orthoplex
(Omnitruncated 6-cube)

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

There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown, with the other 6 more easily constructed as a pentellated 6-cube. The simple pentellated 6-orthoplex (Same as pentellated 5-cube) is also called an expanded 6-orthoplex, constructed by an expansion operation applied to the regular 6-orthoplex. The highest form, the pentisteriruncicantitruncated 6-orthoplex, is called an omnitruncated 6-orthoplex with all of the nodes ringed.

Pentitruncated 6-orthoplex

Pentitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,1,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges8640
Vertices1920
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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-orthoplex

Penticantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,2,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges21120
Vertices3840
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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-orthoplex

Penticantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,1,2,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges30720
Vertices7680
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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-orthoplex

Pentiruncitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,1,3,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges51840
Vertices11520
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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-orthoplex

Pentiruncicantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,1,2,3,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges80640
Vertices23040
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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-orthoplex

Pentistericantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol t0,1,2,4,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges80640
Vertices23040
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

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]


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

Notes

  1. Klitzing, (x4o3o3o3x3x - tacox)
  2. Klitzing, (x4o3o3x3o3x - tapox)
  3. Klitzing, (x4o3o3x3x3x - togrig)
  4. Klitzing, (x4o3x3o3x3x - tocrax)
  5. Klitzing, (x4x3o3x3x3x - tagpog)
  6. Klitzing, (x4x3o3x3x3x - tecagorg)

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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.