Rectified 7-simplexes

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7-simplex t0.svg
7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t1.svg
Rectified 7-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t2.svg
Birectified 7-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t3.svg
Trirectified 7-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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[edit]

Rectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 051
Schläfli symbol r{36} = {35,1}
or
Coxeter diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t1.svg 7-simplex t1 A6.svg 7-simplex t1 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t1 A4.svg 7-simplex t1 A3.svg 7-simplex t1 A2.svg
Dihedral symmetry [5] [4] [3]

Birectified 7-simplex[edit]

Birectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 042
Schläfli symbol 2r{3,3,3,3,3,3} = {34,2}
or
Coxeter diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-faces 16:
8 r{35} 6-simplex t1.svg
8 2r{35} 6-simplex t2.svg
5-faces 112:
28 {34} 5-simplex t0.svg
56 r{34} Rectified 5-simplex.png
28 2r{34} 5-simplex t2.svg
4-faces 392:
168 {33} 4-simplex t0.svg
(56+168) r{33} 5-simplex t1.svg
Cells 770:
(420+70) {3,3} 3-simplex t0.svg
280 {3,4} 3-simplex t1.svg
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 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t2.svg 7-simplex t2 A6.svg 7-simplex t2 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t2 A4.svg 7-simplex t2 A3.svg 7-simplex t2 A2.svg
Dihedral symmetry [5] [4] [3]

Trirectified 7-simplex[edit]

Trirectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 033
Schläfli symbol 3r{36} = {33,3}
or
Coxeter diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
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 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Alternate names[edit]

  • Hexadecaexon (Acronym: he) (Jonathan Bowers)

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t3.svg 7-simplex t3 A6.svg 7-simplex t3 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t3 A4.svg 7-simplex t3 A3.svg 7-simplex t3 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Related polytopes[edit]

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
CDel branch 11.png = CDel node 1.pngCDel 6.pngCDel node.png
t{3} = {6}
Octahedron
CDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{3,3} = {31,1} = {3,4}
Decachoron
CDel branch 11.pngCDel 3ab.pngCDel nodes.png
2t{33}
Dodecateron
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
2r{34} = {32,2}
Tetradecapeton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3t{35}
Hexadecaexon
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3r{36} = {33,3}
Octadecazetton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
4t{37}
Images Truncated triangle.png 3-cube t2.svgUniform polyhedron-33-t1.png 4-simplex t12.svgSchlegel half-solid bitruncated 5-cell.png 5-simplex t2.svg5-simplex t2 A4.svg 6-simplex t23.svg6-simplex t23 A5.svg 7-simplex t3.svg7-simplex t3 A5.svg 8-simplex t34.svg8-simplex t34 A7.svg
Facets {3} Regular polygon 3 annotated.svg t{3,3} Uniform polyhedron-33-t01.png r{3,3,3} Schlegel half-solid rectified 5-cell.png 2t{3,3,3,3} 5-simplex t12.svg 2r{3,3,3,3,3} 6-simplex t2.svg 3t{3,3,3,3,3,3} 7-simplex t23.svg
As
intersecting
dual
simplexes
Regular hexagon as intersection of two triangles.png
CDel branch 10.pngCDel branch 01.png
Stellated octahedron A4 A5 skew.png
CDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png
Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png
CDel branch.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes 01l.png
Dual 5-simplex intersection graph a5.pngDual 5-simplex intersection graph a4.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

Related polytopes[edit]

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

See also[edit]

References[edit]

  • 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

External links[edit]

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds