8-simplexPentellated 8-simplexBipentellated 8-simplexOrthogonal 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, cantellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, 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 []. The A8 Coxeter plane projection shows order symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to symmetry.
Pentellated 8-simplex Pentellated 8-simplexTypeuniform 8-polytopeSchläfli symbol t0,5{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges5040 Vertices504 Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alernate names Acronym: sotane (Jonathan Bowers)Klitzing, (x3o3o3o3o3x3o3o – sotane)
Coordinates The Cartesian coordinates of the vertices of the pentellated 8-simplex can be most simply positioned in 9-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
Bipentellated 8-simplex Bipentellated 8-simplexTypeuniform 8-polytopeSchläfli symbol t1,6{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-facest0,5{3,3,3,3,3,3}6-faces5-faces4-facesCellsFacesEdges7560 Vertices756 Vertex figureCoxeter groupA8×2, [], order 725760Propertiesconvex, facet-transitive
Alternate names Small biterated bienneazetton Bipentellated enneazetton (Acronym: sobteb) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the bipentellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,1,1,1,1,1,2,2). This construction is based on facets of the bipentellated 9-orthoplex.
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Related polytopes
Pentellated 8-simplex and dipentellated 8-simplex are two polytopes selected from 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 Ivić 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. x3o3o3o3o3x3o3o – sotane, o3x3o3o3o3o3x3o – sobteb
External links Polytopes of Various Dimensions Multi-dimensional Glossary