# Rectified 6-simplexes

(Redirected from Rectified 6-simplex)
 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}
Coxeter-Dynkin 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
Properties convex, isogonal

### 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
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
Schläfli symbol t2{3,3,3,3,3}
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

### 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.

 t0 t1 t2 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t0,5 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t0,1,5 t0,2,5 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t0,1,4,5 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,2,3,4,5

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