8-simplexRuncinated 8-simplexBiruncinated 8-simplexTriruncinated 8-simplexRuncitruncated 8-simplexBiruncitruncated 8-simplexTriruncitruncated 8-simplexRuncicantellated 8-simplexBiruncicantellated 8-simplexRuncicantitruncated 8-simplexBiruncicantitruncated 8-simplexTriruncicantitruncated 8-simplexOrthogonal 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 order reflectional symmetry in the A8 Coxeter plane.
Runcinated 8-simplex Runcinated 8-simplexTypeuniform 8-polytopeSchläfli symbol t0,3{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges4536 Vertices504 Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
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.
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Biruncinated 8-simplex Biruncinated 8-simplexTypeuniform 8-polytopeSchläfli symbol t1,4{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdges11340Vertices1260Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Biruncinated enneazetton Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)Klitzing (o3x3o3o3x3o3o3o - sabpene)
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.
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Triruncinated 8-simplex Triruncinated 8-simplexTypeuniform 8-polytopeSchläfli symbol t2,5{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges15120 Vertices1680 Vertex figureCoxeter groupA8×2, [], order 725760 Propertiesconvex
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.
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Runcitruncated 8-simplex
Acronym: potane (Jonathan Bowers)Klitzing (x3x3o3x3o3o3o3o - potane)
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Biruncitruncated 8-simplex
Acronym: biptene (Jonathan Bowers)Klitzing (o3x3x3o3x3o3o3o - biptene)
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Triruncitruncated 8-simplex
Acronym: toprane (Jonathan Bowers)Klitzing (o3o3x3x3o3x3o3o - toprane)
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Runcicantellated 8-simplex
Acronym: prene (Jonathan Bowers)Klitzing (x3o3x3x3o3o3o3o - prene)
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Biruncicantellated 8-simplex
Acronym: biprene (Jonathan Bowers)Klitzing (o3x3o3x3x3o3o3o - biprene)
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Runcicantitruncated 8-simplex
Acronym: gapene (Jonathan Bowers)Klitzing (x3x3x3x3o3o3o3o - gapene)
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Biruncicantitruncated 8-simplex
Acronym: gabpene (Jonathan Bowers)Klitzing (o3x3x3x3x3o3o3o - gabpene)
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Triruncicantitruncated 8-simplex
Acronym: gatpeb (Jonathan Bowers)Klitzing (o3o3x3x3x3x3o3o - gatpeb)
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Related polytopes The 11 presented polytopes are in the family 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 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. x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb, x3x3o3x3o3o3o3o - potane, o3x3x3o3x3o3o3o - biptene, o3o3x3x3o3x3o3o - toprane, x3o3x3x3o3o3o3o - prene, o3x3o3x3x3o3o3o - biprene, x3x3x3x3o3o3o3o - gapene, o3x3x3x3x3o3o3o - gabpene, o3o3x3x3x3x3o3o - gatpeb
External links Polytopes of Various Dimensions Multi-dimensional Glossary