6-simplexRuncinated 6-simplexBiruncinated 6-simplexRuncitruncated 6-simplexBiruncitruncated 6-simplexRuncicantellated 6-simplexRuncicantitruncated 6-simplexBiruncicantitruncated 6-simplexOrthogonal projections in A6 Coxeter plane
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.
Runcinated 6-simplex Runcinated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,3{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces704-faces455Cells1330Faces1610Edges840Vertices140Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)
Coordinates The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
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Biruncinated 6-simplex Biruncinated 6-simplexTypeuniform 6-polytopeSchläfli symbol t1,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces844-faces714Cells2100Faces2520Edges1260Vertices210Vertex figureCoxeter groupA6, [], order 10080Propertiesconvex
Alternate names Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)
Coordinates The vertices of the biruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
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Runcitruncated 6-simplex Runcitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,3{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces70 4-faces560 Cells1820 Faces2800 Edges1890 Vertices420 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)
Coordinates The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
Images
Biruncitruncated 6-simplex Biruncitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t1,2,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces84 4-faces714 Cells2310 Faces3570 Edges2520 Vertices630 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)
Coordinates The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
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Runcicantellated 6-simplex Runcicantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,2,3{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces70 4-faces455 Cells1295 Faces1960 Edges1470 Vertices420 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)
Coordinates The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
Images
Runcicantitruncated 6-simplex Runcicantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,3{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces70 4-faces560 Cells1820 Faces3010 Edges2520 Vertices840 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Runcicantitruncated heptapeton Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)
Coordinates The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.
Images
Biruncicantitruncated 6-simplex Biruncicantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t1,2,3,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces844-faces714Cells2520Faces4410Edges3780Vertices1260Vertex figureCoxeter groupA6, [], order 10080Propertiesconvex
Alternate names Biruncicantitruncated heptapeton Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)
Coordinates The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.
Images
Related uniform 6-polytopes The runcinated 6-simplexes are in a set of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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. x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof
External links Polytopes of Various Dimensions Multi-dimensional Glossary