Runcinated 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 t03.svg
Runcinated 7-simplex
CDel node 1.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
7-simplex t14.svg
Biruncinated 7-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t013.svg
Runcitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t124.svg
Biruncitruncated 7-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t023.svg
Runcicantellated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t134.svg
Biruncicantellated 7-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t0123.svg
Runcicantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t1234.svg
Biruncicantitruncated 7-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.

Runcinated 7-simplex[edit]

Runcinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.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
6-faces
5-faces
4-faces
Cells
Faces
Edges 2100
Vertices 280
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Small prismated octaexon (acronym: spo) (Jonathan Bowers)[1]

Coordinates[edit]

The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t03.svg 7-simplex t03 A6.svg 7-simplex t03 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t03 A4.svg 7-simplex t03 A3.svg 7-simplex t03 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncinated 7-simplex[edit]

Biruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 4200
Vertices 560
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Small biprismated octaexon (sibpo) (Jonathan Bowers)[2]

Coordinates[edit]

The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t14.svg 7-simplex t14 A6.svg 7-simplex t14 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t14 A4.svg 7-simplex t14 A3.svg 7-simplex t14 A2.svg
Dihedral symmetry [5] [4] [3]

Runcitruncated 7-simplex[edit]

runcitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 4620
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)[3]

Coordinates[edit]

The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t013.svg 7-simplex t013 A6.svg 7-simplex t013 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t013 A4.svg 7-simplex t013 A3.svg 7-simplex t013 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncitruncated 7-simplex[edit]

Biruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)[4]

Coordinates[edit]

The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t124.svg 7-simplex t124 A6.svg 7-simplex t124 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t124 A4.svg 7-simplex t124 A3.svg 7-simplex t124 A2.svg
Dihedral symmetry [5] [4] [3]

Runcicantellated 7-simplex[edit]

runcicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)[5]

Coordinates[edit]

The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t023.svg 7-simplex t023 A6.svg 7-simplex t023 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t023 A4.svg 7-simplex t023 A3.svg 7-simplex t023 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncicantellated 7-simplex[edit]

biruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

Coordinates[edit]

The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t134.svg 7-simplex t134 A6.svg 7-simplex t134 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t134 A4.svg 7-simplex t134 A3.svg 7-simplex t134 A2.svg
Dihedral symmetry [5] [4] [3]

Runcicantitruncated 7-simplex[edit]

runcicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 5880
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Great prismated octaexon (acronym: gapo) (Jonathan Bowers)[6]

Coordinates[edit]

The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0123.svg 7-simplex t0123 A6.svg 7-simplex t0123 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0123 A4.svg 7-simplex t0123 A3.svg 7-simplex t0123 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncicantitruncated 7-simplex[edit]

biruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 11760
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

  • Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)[7]

Coordinates[edit]

The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t1234.svg 7-simplex t1234 A6.svg 7-simplex t1234 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t1234 A4.svg 7-simplex t1234 A3.svg 7-simplex t1234 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes[edit]

These polytopes are among 71 uniform 7-polytopes with A7 symmetry.

Notes[edit]

  1. ^ Klitzing, (x3o3o3x3o3o3o - spo)
  2. ^ Klitzing, (o3x3o3o3x3o3o - sibpo)
  3. ^ Klitzing, (x3x3o3x3o3o3o - patto)
  4. ^ Klitzing, (o3x3x3o3x3o3o - bipto)
  5. ^ Klitzing, (x3o3x3x3o3o3o - paro)
  6. ^ Klitzing, (x3x3x3x3o3o3o - gapo)
  7. ^ Klitzing, (o3x3x3x3x3o3o- gibpo)

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)".  x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
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Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds