Rectified 9-simplexes

Orthogonal projections in A₉ Coxeter plane
9-simplex	Rectified 9-simplex
Birectified 9-simplex	Trirectified 9-simplex	Quadrirectified 9-simplex

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
A_k Coxeter plane	A₉	A₈	A₇	A₆
Graph
Dihedral symmetry	[10]	[9]	[8]	[7]
A_k 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}×{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 9-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
A_k Coxeter plane	A₉	A₈	A₇	A₆
Graph
Dihedral symmetry	[10]	[9]	[8]	[7]
A_k 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}×{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
A_k Coxeter plane	A₉	A₈	A₇	A₆
Graph
Dihedral symmetry	[10]	[9]	[8]	[7]
A_k 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	or
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3,3,3}×{3,3,3}
Coxeter groups	A₉×2, [[3⁸]]
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
A_k Coxeter plane	A₉	A₈	A₇	A₆
Graph
Dihedral symmetry	[10]	[9]	[8]	[7]
A_k Coxeter plane	A₅	A₄	A₃	A₂
Graph
Dihedral symmetry	[6]	[5]	[4]	[3]

Notes

^ Klitzing, (o3x3o3o3o3o3o3o3o - reday)
^ Klitzing, (o3o3x3o3o3o3o3o3o - breday)
^ Klitzing, (o3o3o3x3o3o3o3o3o - treday)
^ 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 [1]
  - (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)
Klitzing, Richard. "9D uniform polytopes (polyyotta)". o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Klitzing, (o3x3o3o3o3o3o3o3o - reday)

[2] Klitzing, (o3o3x3o3o3o3o3o3o - breday)

[3] Klitzing, (o3o3o3x3o3o3o3o3o - treday)

[4] Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

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