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Runcinated 8-simplex

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 A<sub>8</sub> Coxeter plane.

Runcinated 8-simplex

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

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

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

Related polytopes

The 11 presented polytopes are in the family of 135 uniform 8-polytopes with A<sub>8</sub> symmetry.

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

External links