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

Rectified 6-simplex
Type	uniform polypeton
Schläfli symbol	t1{3^5} r{3^5} = {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, [3^5], 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

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} 2r{3^5} = {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

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.

A6 polytopes 

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

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
    • (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

External links

• Olshevsky, George, Simplex at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary