Rectified 10-simplexes

Orthogonal projections in A₉ Coxeter plane
10-simplex	Rectified 10-simplex	Birectified 10-simplex
Trirectified 10-simplex	Quadrirectified 10-simplex

In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.

These polytopes are part of a family of 527 uniform 10-polytopes with A₁₀ symmetry.

There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.

Rectified 10-simplex

Rectified 10-simplex
Type	uniform polyxennon
Schläfli symbol	t₁{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
9-faces	22
8-faces	165
7-faces	660
6-faces	1650
5-faces	2772
4-faces	3234
Cells	2640
Faces	1485
Edges	495
Vertices	55
Vertex figure	9-simplex prism
Petrie polygon	decagon
Coxeter groups	A₁₀, [3,3,3,3,3,3,3,3,3]
Properties	convex

The rectified 10-simplex is the vertex figure of the 11-demicube.

Alternate names

Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)^[1]

Coordinates

The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₁₀	A₉	A₈
Graph
Dihedral symmetry	[11]	[10]	[9]
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Birectified 10-simplex

Birectified 10-simplex
Type	uniform 9-polytope
Schläfli symbol	t₂{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1980
Vertices	165
Vertex figure	{3}x{3,3,3,3,3,3}
Coxeter groups	A₁₀, [3,3,3,3,3,3,3,3,3]
Properties	convex

Alternate names

Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)^[2]

Coordinates

The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₁₀	A₉	A₈
Graph
Dihedral symmetry	[11]	[10]	[9]
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Trirectified 10-simplex

Trirectified 10-simplex
Type	uniform polyxennon
Schläfli symbol	t₃{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	4620
Vertices	330
Vertex figure	{3,3}x{3,3,3,3,3}
Coxeter groups	A₁₀, [3,3,3,3,3,3,3,3,3]
Properties	convex

Alternate names

Trirectified hendecaxennon (Jonathan Bowers)^[3]

Coordinates

The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₁₀	A₉	A₈
Graph
Dihedral symmetry	[11]	[10]	[9]
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Quadrirectified 10-simplex

Quadrirectified 10-simplex
Type	uniform polyxennon
Schläfli symbol	t₄{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	6930
Vertices	462
Vertex figure	{3,3,3}x{3,3,3,3}
Coxeter groups	A₁₀, [3,3,3,3,3,3,3,3,3]
Properties	convex

Alternate names

Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)^[4]

Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₁₀	A₉	A₈
Graph
Dihedral symmetry	[11]	[10]	[9]
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Notes

^ Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
^ Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
^ Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
^ Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)

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. "10D uniform polytopes (polyxenna)". x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru

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, (o3x3o3o3o3o3o3o3o3o - ru)

[2] Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)

[3] Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)

[4] Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)

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