Stericated 6-simplexes


6-simplex

Stericated 6-simplex

Steritruncated 6-simplex

Stericantellated 6-simplex

Stericantitruncated 6-simplex

Steriruncinated 6-simplex

Steriruncitruncated 6-simplex

Steriruncicantellated 6-simplex

Steriruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 6-simplex

Stericated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces700
Cells1470
Faces1400
Edges630
Vertices105
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Steritruncated 6-simplex

Steritruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces945
Cells2940
Faces3780
Edges2100
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Stericantellated 6-simplex

Stericantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1050
Cells3465
Faces 5040
Edges3150
Vertices630
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Stericantitruncated 6-simplex

stericantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1155
Cells4410
Faces7140
Edges5040
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Steriruncinated 6-simplex

steriruncinated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces700
Cells1995
Faces2660
Edges1680
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Steriruncitruncated 6-simplex

steriruncitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces945
Cells3360
Faces5670
Edges4410
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Steriruncicantellated 6-simplex

steriruncicantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1050
Cells3675
Faces5880
Edges4410
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Steriruncicantitruncated 6-simplex

Steriuncicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1155
Cells4620
Faces8610
Edges7560
Vertices 2520
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

Coordinates

The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

Notes

  1. Klitzing, (x3o3o3o3x3o - scal)
  2. Klitzing, (x3x3o3o3x3o - catal)
  3. Klitzing, (x3o3x3o3x3o - cral)
  4. Klitzing, (x3x3x3o3x3o - cagral)
  5. Klitzing, (x3o3o3x3x3o - copal)
  6. Klitzing, (x3x3o3x3x3o - captal)
  7. Klitzing, ( x3o3x3x3x3o - copril)
  8. Klitzing, (x3x3x3x3x3o - gacal)

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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