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Runcinated 8-simplexes
8-simplexRuncinated 8-simplexBiruncinated 8-simplexTriruncinated 8-simplex
Runcitruncated 8-simplexBiruncitruncated 8-simplexTriruncitruncated 8-simplexRuncicantellated 8-simplex
Biruncicantellated 8-simplexRuncicantitruncated 8-simplexBiruncicantitruncated 8-simplexTriruncicantitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.

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Runcinated 8-simplex

Runcinated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt0,3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges4536
Vertices504
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Runcinated enneazetton
  • Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)1

Coordinates

The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[9][8][7][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[5][4][3]

Biruncinated 8-simplex

Biruncinated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges11340
Vertices1260
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Biruncinated enneazetton
  • Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)2

Coordinates

The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[9][8][7][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[5][4][3]

Triruncinated 8-simplex

Triruncinated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges15120
Vertices1680
Vertex figure
Coxeter groupA8×2, [[37]], order 725760
Propertiesconvex

Alternate names

  • Triruncinated enneazetton
  • Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)3

Coordinates

The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,2,2,2). This construction is based on facets of the triruncinated 9-orthoplex.

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Runcitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Biruncitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Triruncitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Runcicantellated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Biruncicantellated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Runcicantitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Biruncicantitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

Triruncicantitruncated 8-simplex

Images

orthographic projections
Ak Coxeter planeA8A7A6A5
Graph
Dihedral symmetry[[9]] = [18][8][[7]] = [14][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]] = [10][4][[3]] = [6]

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

A8 polytopes
t0t1t2t3t01t02t12t03t13t23t04t14t24t34t05
t15t25t06t16t07t012t013t023t123t014t024t124t034t134t234
t015t025t125t035t135t235t045t145t016t026t126t036t136t046t056
t017t027t037t0123t0124t0134t0234t1234t0125t0135t0235t1235t0145t0245t1245
t0345t1345t2345t0126t0136t0236t1236t0146t0246t1246t0346t1346t0156t0256t1256
t0356t0456t0127t0137t0237t0147t0247t0347t0157t0257t0167t01234t01235t01245t01345
t02345t12345t01236t01246t01346t02346t12346t01256t01356t02356t12356t01456t02456t03456t01237
t01247t01347t02347t01257t01357t02357t01457t01267t01367t012345t012346t012356t012456t013456t023456
t123456t012347t012357t012457t013457t023457t012367t012467t013467t012567t0123456t0123457t0123467t0123567t01234567

Notes

  • 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. "8D uniform polytopes (polyzetta)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb
  • v
  • t
  • e
Fundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

References

  1. Klitzing (x3o3o3x3o3o3o3o - spene) ↩

  2. Klitzing (o3x3o3o3x3o3o3o - sabpene) ↩

  3. Klitzing (o3o3x3o3o3x3o3o - satpeb) ↩