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.


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

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
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
Circumradius0.845154
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
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.

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.
    • Klitzing, Richard. "6D uniform polytopes (polypeta)". o3x3o3o3o3o - ril, o3x3o3o3o3o - bril
    Fundamental convex regular and uniform polytopes in dimensions 2–10
    Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
    Regular polygon Triangle Square p-gon Hexagon Pentagon
    Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
    Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
    Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
    Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
    Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
    Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
    Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
    Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
    Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
    Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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