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
Type	uniform 9-polytope
Schläfli symbol	t1{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces	20
7-faces	135
6-faces	480
5-faces	1050
4-faces	1512
Cells	1470
Faces	960
Edges	360
Vertices	45
Vertex figure	8-simplex prism
Petrie polygon	decagon
Coxeter groups	A9, [3,3,3,3,3,3,3,3]
Properties	convex

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

Alternate names

• Rectified decayotton (reday) (Jonathan Bowers)^[1]

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
Type	uniform 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
Edges	1260
Vertices	120
Vertex figure	{3}x{3,3,3,3,3}
Coxeter groups	A9, [3,3,3,3,3,3,3,3]
Properties	convex

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

• Birectified decayotton (breday) (Jonathan Bowers)^[2]

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
Type	uniform 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 groups	A9, [3,3,3,3,3,3,3,3]
Properties	convex

Alternate names

• Trirectified decayotton (treday) (Jonathan Bowers)^[3]

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
Type	uniform 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 groups	A9, [3,3,3,3,3,3,3,3]
Properties	convex

Alternate names

• Quadrirectified decayotton
• Icosayotton (icoy) (Jonathan Bowers)^[4]

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

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Richard Klitzing, 9D, uniform polytopes (polyyotta) o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy

External links

• Olshevsky, George, Simplex at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary