Trirectified 8-simplex

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

Rectified 8-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0_6,1 for its branching Coxeter-Dynkin diagram, shown as .

Coordinates

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

Images

Birectified 8-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0_5,2 for its branching Coxeter-Dynkin diagram, shown as .

The ''birectified 8-simplex'' is the vertex figure of the 1₅₂ honeycomb.

Coordinates

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

Images

Trirectified 8-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0_4,3 for its branching Coxeter-Dynkin diagram, shown as .

Coordinates

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

Images

Related polytopes

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 2₆₁ honeycomb.

It is also one of 135 uniform 8-polytopes with A₈ 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,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene

External links

Polytopes of Various Dimensions


{{Polytopes

8-polytopes