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Orthogonal projections in A₆ Coxeter plane
6-simplex	Rectified 6-simplex	Birectified 6-simplex

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.

Rectified 6-simplex

Rectified 6-simplex
Type	uniform polypeton
Schläfli symbol	t₁{3⁵} r{3⁵} = {3^4,1} or $\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$
Coxeter diagrams
Elements	f₅ = 14, f₄ = 63, C = 140, F = 175, E = 105, V = 21 (χ=0)
Coxeter group	A₆, [3⁵], order 5040
Bowers name and (acronym)	Rectified heptapeton (ril)
Vertex figure	5-cell prism
Circumradius	0.845154
Properties	convex, isogonal

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S¹
₆. It is also called 0_4,1 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

Coordinates

The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Birectified 6-simplex

Birectified 6-simplex
Type	uniform 6-polytope
Class	A6 polytope
Schläfli symbol	t₂{3,3,3,3,3} 2r{3⁵} = {3^3,2} or $\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$
Coxeter symbol	0₃₂
Coxeter diagrams
5-faces	14 total: 7 t₁{3,3,3,3} 7 t₂{3,3,3,3}
4-faces	84
Cells	245
Faces	350
Edges	210
Vertices	35
Vertex figure	{3}x{3,3}
Petrie polygon	Heptagon
Coxeter groups	A₆, [3,3,3,3,3]
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S²
₆. It is also called 0_3,2 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

Coordinates

The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Related uniform 6-polytopes

The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2₄₁ polytope.

These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A₆ Coxeter plane orthographic projections.

Notes

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.
Klitzing, Richard. "6D uniform polytopes (polypeta)". o3x3o3o3o3o - ril, o3x3o3o3o3o - bril

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