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 polyexon
Coxeter symbol 051
Schläfli symbol r{3,3,3,3,3,3}
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.

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 polyexon
Coxeter symbol 042
Schläfli symbol 2r{3,3,3,3,3,3}
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

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 polyexon
Coxeter symbol 033
Schläfli symbol 3r{36}
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

This polytope is the vertex figure of the 133 honeycomb.

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 t{3}
Hexagon
r{3,3}
Octahedron
2t{3,3,3}
Decachoron
2r{3,3,3,3}
Dodecateron
3t{3,3,3,3,3}
Tetradecapeton
3r{3,3,3,3,3,3}
Hexadecaexon
4t{3,3,3,3,3,3,3}
Octadecazetton
Coxeter
diagram
CDel branch 11.png CDel node 1.pngCDel split1.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
Images Truncated triangle.png 3-simplex t1.svgUniform polyhedron-33-t1.png 4-simplex t12.svgSchlegel half-solid bitruncated 5-cell.png 5-simplex t2.svg5-simplex t2 A3.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

Related polytopes[edit]

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

7-simplex t0.svg
t0
7-simplex t1.svg
t1
7-simplex t2.svg
t2
7-simplex t3.svg
t3
7-simplex t01.svg
t0,1
7-simplex t02.svg
t0,2
7-simplex t12.svg
t1,2
7-simplex t03.svg
t0,3
7-simplex t13.svg
t1,3
7-simplex t23.svg
t2,3
7-simplex t04.svg
t0,4
7-simplex t14.svg
t1,4
7-simplex t24.svg
t2,4
7-simplex t05.svg
t0,5
7-simplex t15.svg
t1,5
7-simplex t06.svg
t0,6
7-simplex t012.svg
t0,1,2
7-simplex t013.svg
t0,1,3
7-simplex t023.svg
t0,2,3
7-simplex t123.svg
t1,2,3
7-simplex t014.svg
t0,1,4
7-simplex t024.svg
t0,2,4
7-simplex t124.svg
t1,2,4
7-simplex t034.svg
t0,3,4
7-simplex t134.svg
t1,3,4
7-simplex t234.svg
t2,3,4
7-simplex t015.svg
t0,1,5
7-simplex t025.svg
t0,2,5
7-simplex t125.svg
t1,2,5
7-simplex t035.svg
t0,3,5
7-simplex t135.svg
t1,3,5
7-simplex t045.svg
t0,4,5
7-simplex t016.svg
t0,1,6
7-simplex t026.svg
t0,2,6
7-simplex t036.svg
t0,3,6
7-simplex t0123.svg
t0,1,2,3
7-simplex t0124.svg
t0,1,2,4
7-simplex t0134.svg
t0,1,3,4
7-simplex t0234.svg
t0,2,3,4
7-simplex t1234.svg
t1,2,3,4
7-simplex t0125.svg
t0,1,2,5
7-simplex t0135.svg
t0,1,3,5
7-simplex t0235.svg
t0,2,3,5
7-simplex t1235.svg
t1,2,3,5
7-simplex t0145.svg
t0,1,4,5
7-simplex t0245.svg
t0,2,4,5
7-simplex t1245.svg
t1,2,4,5
7-simplex t0345.svg
t0,3,4,5
7-simplex t0126.svg
t0,1,2,6
7-simplex t0136.svg
t0,1,3,6
7-simplex t0236.svg
t0,2,3,6
7-simplex t0146.svg
t0,1,4,6
7-simplex t0246.svg
t0,2,4,6
7-simplex t0156.svg
t0,1,5,6
7-simplex t01234.svg
t0,1,2,3,4
7-simplex t01235.svg
t0,1,2,3,5
7-simplex t01245.svg
t0,1,2,4,5
7-simplex t01345.svg
t0,1,3,4,5
7-simplex t02345.svg
t0,2,3,4,5
7-simplex t12345.svg
t1,2,3,4,5
7-simplex t01236.svg
t0,1,2,3,6
7-simplex t01246.svg
t0,1,2,4,6
7-simplex t01346.svg
t0,1,3,4,6
7-simplex t02346.svg
t0,2,3,4,6
7-simplex t01256.svg
t0,1,2,5,6
7-simplex t01356.svg
t0,1,3,5,6
7-simplex t012345.svg
t0,1,2,3,4,5
7-simplex t012346.svg
t0,1,2,3,4,6
7-simplex t012356.svg
t0,1,2,3,5,6
7-simplex t012456.svg
t0,1,2,4,5,6
7-simplex t0123456.svg
t0,1,2,3,4,5,6

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

External links[edit]