6-simplexStericated 6-simplexSteritruncated 6-simplexStericantellated 6-simplexStericantitruncated 6-simplexSteriruncinated 6-simplexSteriruncitruncated 6-simplexSteriruncicantellated 6-simplexSteriruncicantitruncated 6-simplexOrthogonal projections in A6 Coxeter plane In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.
Stericated 6-simplex Stericated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces700 Cells1470 Faces1400 Edges630 Vertices105 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)
Coordinates The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.
Images
Steritruncated 6-simplex Steritruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces945 Cells2940 Faces3780 Edges2100 Vertices420 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)
Coordinates The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.
Images
Stericantellated 6-simplex Stericantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,2,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces1050 Cells3465Faces 5040 Edges3150 Vertices630 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)
Coordinates The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.
Images
Stericantitruncated 6-simplex Stericantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces1155 Cells4410 Faces7140 Edges5040 Vertices1260 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)
Coordinates The vertices of the stericanttruncated 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 stericantitruncated 7-orthoplex.
Images
Steriruncinated 6-simplex Steriruncinated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,3,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces700 Cells1995 Faces2660Edges1680 Vertices420 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)
Coordinates The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.
Images
Steriruncitruncated 6-simplex Steriruncitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,3,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces945 Cells3360 Faces5670 Edges4410 Vertices1260 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)
Coordinates The vertices of the steriruncittruncated 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 steriruncitruncated 7-orthoplex.
Images
Steriruncicantellated 6-simplex Steriruncicantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,2,3,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces1050 Cells3675 Faces5880Edges4410 Vertices1260 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3} Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)
Coordinates The vertices of the steriruncitcantellated 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 steriruncicantellated 7-orthoplex.
Images
Steriruncicantitruncated 6-simplex Steriuncicantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,3,4{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces105 4-faces1155 Cells4620 Faces8610 Edges7560Vertices 2520 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)
Coordinates The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.
Images
Related uniform 6-polytopes The stericated 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.
External links Polytopes of Various Dimensions Multi-dimensional Glossary