Pentellated 8-simplexes


8-simplex

Pentellated 8-simplex

Bipentitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.

There are two unique pentellations of the 8-simplex. Including truncations, canetellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, [38] has order 9 factorial symmetry, or 362880. The bipentalled form is symmetrically ringed, doubling the symmetry order to 725760, and is represented the double-bracketed group [[3<sup>8</sup>]]. The A8 Coxeter plane projection shows order [9] symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to [18] symmetry.

Pentellated 8-simplex

Pentellated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t0,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges5040
Vertices504
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bipentellated 8-simplex

Bipentellated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t1,6{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-facest0,5{3,3,3,3,3,3}
6-faces
5-faces
4-faces
Cells
Faces
Edges7560
Vertices756
Vertex figure
Coxeter groupA8×2, [[3<sup>7</sup>]], order 725760
Propertiesconvex, facet-transitive

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Related polytopes

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

Notes

    References

    External links

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