Rectified 6-simplexes

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

 Orthogonal projections in A6 Coxeter plane 6-simplex Rectified 6-simplex Birectified 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
Typeuniform polypeton
Schläfli symbolt1{35}
r{35} = {34,1}
or ${\displaystyle \left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}}$
Coxeter diagrams
Elements

f5 = 14, f4 = 63, C = 140, F = 175, E = 105, V = 21
(χ=0)

Coxeter groupA6, [35], order 5040
Bowers name
and (acronym)
Rectified heptapeton
(ril)
Vertex figure5-cell prism
Propertiesconvex, isogonal

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
6
. It is also called 04,1 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
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
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbolt2{3,3,3,3,3}
2r{35} = {33,2}
or ${\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}}$
Coxeter symbol032
Coxeter diagrams
5-faces14 total:
7 t1{3,3,3,3}
7 t2{3,3,3,3}
4-faces84
Cells245
Faces350
Edges210
Vertices35
Vertex figure{3}x{3,3}
Petrie polygonHeptagon
Coxeter groupsA6, [3,3,3,3,3]
Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
6
. It is also called 03,2 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]

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
• (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