Rectified 10-simplexes

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

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

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

Ak Coxeter plane	A10	A9	A8
Graph
Dihedral symmetry	[11]	[10]	[9]
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak 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

Ak Coxeter plane	A10	A9	A8
Graph
Dihedral symmetry	[11]	[10]	[9]
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak 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

Ak Coxeter plane	A10	A9	A8
Graph
Dihedral symmetry	[11]	[10]	[9]
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak 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

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

Notes

1. Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
2. Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
3. Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
4. 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
    • (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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds