Rectified 6-simplexes

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

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

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S16. It is also called 0_4,1 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

• Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak 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} = {33,2} or [math]\displaystyle{ \left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\} }[/math]
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 S26. 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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak 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.

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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  o3x3o3o3o3o - ril, o3x3o3o3o3o - bril

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary