Rectified 7-simplexes


7-simplex
Rectified 7-simplex

Birectified 7-simplex
Trirectified 7-simplex
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.

Rectified 7-simplex

Rectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 051
Schläfli symbol r{36} = {35,1}
or
Coxeter diagrams
Or
6-faces 16
5-faces 84
4-faces 224
Cells 350
Faces 336
Edges 168
Vertices 28
Vertex figure 6-simplex prism
Petrie polygon Octagon
Coxeter group A7, [36], order 40320
Properties convex

The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as .

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
7.

Alternate names

  • Rectified octaexon (Acronym: roc) (Jonathan Bowers)

Coordinates

The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Birectified 7-simplex

Birectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 042
Schläfli symbol 2r{3,3,3,3,3,3} = {34,2}
or
Coxeter diagrams
Or
6-faces 16:
8 r{35}
8 2r{35}
5-faces 112:
28 {34}
56 r{34}
28 2r{34}
4-faces 392:
168 {33}
(56+168) r{33}
Cells 770:
(420+70) {3,3}
280 {3,4}
Faces 840:
(280+560) {3}
Edges 420
Vertices 56
Vertex figure {3}x{3,3,3}
Coxeter group A7, [36], order 40320
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
7. It is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

  • Birectified octaexon (Acronym: broc) (Jonathan Bowers)

Coordinates

The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Trirectified 7-simplex

Trirectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 033
Schläfli symbol 3r{36} = {33,3}
or
Coxeter diagrams
Or
6-faces 16 2r{35}
5-faces 112
4-faces 448
Cells 980
Faces 1120
Edges 560
Vertices 70
Vertex figure {3,3}x{3,3}
Coxeter group A7×2, [[36]], order 80640
Properties convex, isotopic

The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
7.

This polytope is the vertex figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

  • Hexadecaexon (Acronym: he) (Jonathan Bowers)

Coordinates

The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.

The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter 		Hexagon
=
t{3} = {6} 		Octahedron
=
r{3,3} = {31,1} = {3,4}
 Decachoron

2t{33} 		Dodecateron

2r{34} = {32,2}
 Tetradecapeton

3t{35} 		Hexadecaexon

3r{36} = {33,3}
 Octadecazetton

4t{37}
Images
Vertex figure ( )∨( )
{ }×{ }
{ }∨{ }
{3}×{3}
{3}∨{3} 		{3,3}×{3,3}
{3,3}∨{3,3}
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩ 		∩ ∩ ∩

These polytopes are three of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes

t0
t1
t2
t3
t0,1
t0,2
t1,2
t0,3

t1,3
t2,3
t0,4
t1,4
t2,4
t0,5
t1,5
t0,6

t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4

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

t0,1,6
t0,2,6
t0,3,6
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
t1,2,3,5
t0,1,4,5
t0,2,4,5
t1,2,4,5
t0,3,4,5

t0,1,2,6
t0,1,3,6
t0,2,3,6
t0,1,4,6
t0,2,4,6
t0,1,5,6
t0,1,2,3,4
t0,1,2,3,5

t0,1,2,4,5
t0,1,3,4,5
t0,2,3,4,5
t1,2,3,4,5
t0,1,2,3,6
t0,1,2,4,6
t0,1,3,4,6
t0,2,3,4,6

t0,1,2,5,6
t0,1,3,5,6
t0,1,2,3,4,5
t0,1,2,3,4,6
t0,1,2,3,5,6
t0,1,2,4,5,6
t0,1,2,3,4,5,6

See also

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.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he
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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