Rectified 9-simplexes


9-simplex

Rectified 9-simplex

Birectified 9-simplex

Trirectified 9-simplex

Quadrirectified 9-simplex
Orthogonal projections in A9 Coxeter plane

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.

There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.

Rectified 9-simplex

Rectified 9-simplex
Typeuniform 9-polytope
Schläfli symbol t1{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces20
7-faces135
6-faces480
5-faces1050
4-faces1512
Cells1470
Faces960
Edges360
Vertices45
Vertex figure8-simplex prism
Petrie polygondecagon
Coxeter groupsA9, [3,3,3,3,3,3,3,3]
Propertiesconvex

The rectified 9-simplex is the vertex figure of the 10-demicube.

Alternate names

Coordinates

The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.

Images

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph
Dihedral symmetry [6] [5] [4] [3]

Birectified 9-simplex

Birectified 9-simplex
Typeuniform 9-polytope
Schläfli symbol t2{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges1260
Vertices120
Vertex figure{3}x{3,3,3,3,3}
Coxeter groupsA9, [3,3,3,3,3,3,3,3]
Propertiesconvex

This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 10-dimensional sphere packing.

Alternate names

Coordinates

The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.

Images

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph
Dihedral symmetry [6] [5] [4] [3]

Trirectified 9-simplex

Trirectified 9-simplex
Typeuniform 9-polytope
Schläfli symbol t3{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{3,3}x{3,3,3,3}
Coxeter groupsA9, [3,3,3,3,3,3,3,3]
Propertiesconvex

Alternate names

Coordinates

The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.

Images

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph
Dihedral symmetry [6] [5] [4] [3]

Quadrirectified 9-simplex

Quadrirectified 9-simplex
Typeuniform 9-polytope
Schläfli symbol t4{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{3,3,3}x{3,3,3}
Coxeter groupsA9, [3,3,3,3,3,3,3,3]
Propertiesconvex

Alternate names

Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.

Images

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph
Dihedral symmetry [6] [5] [4] [3]

Notes

  1. Klitzing, (o3x3o3o3o3o3o3o3o - reday)
  2. Klitzing, (o3o3x3o3o3o3o3o3o - breday)
  3. Klitzing, (o3o3o3x3o3o3o3o3o - treday)
  4. Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

References

External links

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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