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Runcinated 8-simplexes

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.

Runcinated 8-simplex

Runcinated 8-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

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

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

Biruncinated 8-simplex

Biruncinated 8-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

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

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

Triruncinated 8-simplex

Triruncinated 8-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

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

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

Runcitruncated 8-simplex

Acronym: potane (Jonathan Bowers)

Images

Biruncitruncated 8-simplex

Acronym: biptene (Jonathan Bowers)

Images

Triruncitruncated 8-simplex

Acronym: toprane (Jonathan Bowers)

Images

Runcicantellated 8-simplex

Acronym: prene (Jonathan Bowers)

Images

Biruncicantellated 8-simplex

Acronym: biprene (Jonathan Bowers)

Images

Runcicantitruncated 8-simplex

Acronym: gapene (Jonathan Bowers)

Images

Biruncicantitruncated 8-simplex

Acronym: gabpene (Jonathan Bowers)

Images

Triruncicantitruncated 8-simplex

Acronym: gatpeb (Jonathan Bowers)

Images

Notes

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, wiley.com,
      • (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.
  • x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb, x3x3o3x3o3o3o3o - potane, o3x3x3o3x3o3o3o3 - biptene, o3o3x3x3o3x3o3o - toprane, x3o3x3x3o3o3o3o - prene, o3x3o3x3x3o3o3o - biprene, x3x3x3x3o3o3o3o3 - gapene, o3x3x3x3x3o3o3o - gabpene, o3o3x3x3x3x3o3o - gatpeb

References

  1. Klitzing (o3x3o3o3x3o3o3o - sabpene)
  2. Klitzing (x3x3o3x3o3o3o3o - potane)
  3. Klitzing (o3x3x3o3x3o3o3o - biptene)
  4. Klitzing (o3o3x3x3o3x3o3o - toprane)
  5. Klitzing (x3o3x3x3o3o3o3o - prene)
  6. Klitzing (o3x3o3x3x3o3o3o - biprene)
  7. Klitzing (x3x3x3x3o3o3o3o - gapene)
  8. Klitzing (o3x3x3x3x3o3o3o - gabpene)
  9. Klitzing (o3o3x3x3x3x3o3o - gatpeb)
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