Rectified 6-simplexes

Rectified 6-simplex
Type	uniform polypeton
Schläfli symbol	t1{35} r{35} = {34,1} or
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

Birectified 6-simplex
Type	uniform 6-polytope
Class	A6 polytope
Schläfli symbol	t2{3,3,3,3,3} 2r{35} = {33,2} or
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

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.