8-simplexStericated 8-simplexBistericated 8-simplexSteritruncated 8-simplexBisteritruncated 8-simplexStericantellated 8-simplexBistericantellated 8-simplexStericantitruncated 8-simplexBistericantitruncated 8-simplexSteriruncinated 8-simplexBisteriruncinated 8-simplexSteriruncitruncated 8-simplexBisteriruncitruncated 8-simplexSterirunci-cantellated 8-simplexBisterirunci-cantellated 8-simplexSteriruncicanti-truncated 8-simplexBisteriruncicanti-truncated 8-simplexOrthogonal projections in A8 Coxeter plane
In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.
Stericated 8-simplex Stericated 8-simplexTypeuniform 8-polytopeSchläfli symbol t0,4{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges6300 Vertices630 Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Acronym: secane (Jonathan Bowers)Klitzing, (x3o3o3o3x3o3o3o - secane)
Coordinates The Cartesian coordinates of the vertices of the stericated 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 stericated 9-orthoplex.
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Bistericated 8-simplex Bistericated 8-simplexTypeuniform 8-polytopeSchläfli symbol t1,5{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges12600Vertices1260Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Acronym: sobcane (Jonathan Bowers)Klitzing, (o3x3o3o3o3x3o3o - sobcane)
Coordinates The Cartesian coordinates of the vertices of the bistericated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 9-orthoplex.
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Steritruncated 8-simplex Steritruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol t0,1,4{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA8, , order 362880Propertiesconvex
Acronym: catene (Jonathan Bowers)Klitzing, (x3x3o3o3x3o3o3o - catene)
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Bisteritruncated 8-simplex Bisteritruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol t1,2,5{3,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA8, , order 362880Propertiesconvex
Acronym: bictane (Jonathan Bowers)Klitzing, (o3x3x3o3o3x3o3o - bictane)
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Stericantellated 8-simplex
Acronym: crane (Jonathan Bowers)Klitzing, (x3o3x3o3x3o3o3o - crane)
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Bistericantellated 8-simplex
Acronym: bocrane (Jonathan Bowers)Klitzing, (o3x3o3x3o3x3o3o - bocrane)
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Stericantitruncated 8-simplex
Acronym: cograne (Jonathan Bowers)Klitzing, (x3x3x3o3x3o3o3o - cograne)
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Bistericantitruncated 8-simplex
Acronym: bocagrane (Jonathan Bowers)Klitzing, (o3x3x3x3ox3o3o3 - bocagrane)
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Steriruncinated 8-simplex
Acronym: capene (Jonathan Bowers)Klitzing, (x3o3o3x3x3o3o3o - capene)
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Bisteriruncinated 8-simplex
Acronym: bacpane (Jonathan Bowers)Klitzing, (o3x3o3o3x3x3o3o - bacpane)
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Steriruncitruncated 8-simplex
Acronym: coptane (Jonathan Bowers)Klitzing, (x3x3o3x3x3o3o3o - coptane)
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Bisteriruncitruncated 8-simplex
Acronym: bicpotane (Jonathan Bowers)Klitzing, (o3x3x3o3x3x3o3o - bicpotane)
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Steriruncicantellated 8-simplex
Acronym: coprene (Jonathan Bowers)Klitzing, (x3o3x3x3x3o3o3o - coprene)
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Bisteriruncicantellated 8-simplex
Acronym: bicprene (Jonathan Bowers)Klitzing, (o3x3o3x3x3x3o3o - bicprene)
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Steriruncicantitruncated 8-simplex
Acronym: gacene (Jonathan Bowers)Klitzing, (x3x3x3x3x3o3o3o - gacene)
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Bisteriruncicantitruncated 8-simplex
Acronym: gobcane (Jonathan Bowers)Klitzing, (o3x3x3x3x3x3o3o - gobcane)
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Related polytopes The 16 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. x3o3o3o3x3o3o3o - secane, o3x3o3o3o3x3o3o - sobcane, x3x3o3o3x3o3o3o - catene, o3x3x3o3o3x3o3o - bictane, x3o3x3o3x3o3o3o - crane, o3x3o3x3o3x3o3o - bocrane, x3x3x3o3x3o3o3o - cograne, o3x3x3x3ox3o3o3 - bocagrane, x3o3o3x3x3o3o3o - capene, o3x3o3o3x3x3o3o - bacpane, x3x3o3x3x3o3o3o - coptane, o3x3x3o3x3x3o3o - bicpotane, x3o3x3x3x3o3o3o - coprene, o3x3o3x3x3x3o3o - bicprene, x3x3x3x3x3o3o3o - gacene, o3x3x3x3x3x3o3o - gobcane
External links Polytopes of Various Dimensions Multi-dimensional Glossary