7-simplexRuncinated 7-simplexBiruncinated 7-simplexRuncitruncated 7-simplexBiruncitruncated 7-simplexRuncicantellated 7-simplexBiruncicantellated 7-simplexRuncicantitruncated 7-simplexBiruncicantitruncated 7-simplexOrthogonal projections in A7 Coxeter plane
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.
There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.
Runcinated 7-simplex Runcinated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,3{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges2100 Vertices280 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Small prismated octaexon (acronym: spo) (Jonathan Bowers)
Coordinates The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.
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Biruncinated 7-simplex Biruncinated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges4200 Vertices560 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Small biprismated octaexon (sibpo) (Jonathan Bowers)
Coordinates The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.
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Runcitruncated 7-simplex Runcitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,3{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges4620 Vertices840 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)Klitzing, (x3x3o3x3o3o3o - patto)
Coordinates The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.
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Biruncitruncated 7-simplex Biruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges8400 Vertices1680 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)Klitzing, (o3x3x3o3x3o3o - bipto)
Coordinates The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.
Images
Runcicantellated 7-simplex Runcicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,3{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges3360 Vertices840 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)Klitzing, (x3o3x3x3o3o3o - paro)
Coordinates The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.
Images
Biruncicantellated 7-simplex Biruncicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)
Coordinates The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.
Images
Runcicantitruncated 7-simplex Runcicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges5880 Vertices1680 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Great prismated octaexon (acronym: gapo) (Jonathan Bowers)Klitzing, (x3x3x3x3o3o3o - gapo)
Coordinates The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.
Images
Biruncicantitruncated 7-simplex Biruncicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges11760 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)Klitzing, (o3x3x3x3x3o3o- gibpo)
Coordinates The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.
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Related polytopes These polytopes are among 71 uniform 7-polytopes with A7 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. x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o - gibpo
External links Polytopes of Various Dimensions Multi-dimensional Glossary