6-simplexPentellated 6-simplexPentitruncated 6-simplexPenticantellated 6-simplex Penticantitruncated 6-simplex Pentiruncitruncated 6-simplex Pentiruncicantellated 6-simplex Pentiruncicantitruncated 6-simplex Pentisteritruncated 6-simplexPentistericantitruncated 6-simplex Pentisteriruncicantitruncated 6-simplex(Omnitruncated 6-simplex)Orthogonal projections in A6 Coxeter plane In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.
There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed.
Pentellated 6-simplex Pentellated 6-simplexTypeUniform 6-polytopeSchläfli symbol t0,5{3,3,3,3,3}Coxeter-Dynkin diagram5-faces126:7+7 {34} 21+21 {}×{3,3,3}35+35 {3}×{3,3}4-faces434Cells630Faces490Edges210Vertices42Vertex figure5-cell antiprismCoxeter group A6×2, , order 10080Propertiesconvex
Alternate names Expanded 6-simplex Small terated tetradecapeton (Acronym: ) (Jonathan Bowers)
Cross-sections The maximal cross-section of the pentellated 6-simplex with a 5-dimensional hyperplane is a stericated hexateron. This cross-section divides the pentellated 6-simplex into two hexateral hypercupolas consisting of 7 5-simplexes, 21 5-cell prisms and 35 Tetrahedral-Triangular duoprisms each.
Coordinates The vertices of the pentellated 6-simplex can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex.
A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0,0)
Root vectors Its 42 vertices represent the root vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb.
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Configuration This configuration matrix represents the expanded 6-simplex, with 12 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Elementfkf0f1f2f3f4f5f042102020206010403021020f1221044618416121510f233280*3336313444*21006066026f34640210*2201216923*420022013f45101005084**1108168624*210*0119186906**140002f561520015060014**102520101010250*42*12301618318034**70
Pentitruncated 6-simplex Pentitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces826 Cells1785 Faces1820 Edges945 Vertices210 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)
Coordinates The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
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Penticantellated 6-simplex Penticantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,2,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1246 Cells3570 Faces4340 Edges2310 Vertices420 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)
Coordinates The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the penticantellated 7-orthoplex.
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Penticantitruncated 6-simplex Penticantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1351 Cells4095 Faces5390 Edges3360 Vertices840 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)
Coordinates The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 7-orthoplex.
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Pentiruncitruncated 6-simplex Pentiruncitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,3,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1491 Cells5565 Faces8610 Edges5670 Vertices1260 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)
Coordinates The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the pentiruncitruncated 7-orthoplex.
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Pentiruncicantellated 6-simplex Pentiruncicantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,2,3,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1596 Cells5250 Faces7560 Edges5040 Vertices1260 Vertex figureCoxeter groupA6, , order 10080Propertiesconvex
Alternate names Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)
Coordinates The vertices of the pentiruncicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 7-orthoplex.
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Pentiruncicantitruncated 6-simplex Pentiruncicantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,3,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1701 Cells6825 Faces11550 Edges8820 Vertices2520 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)
Coordinates The vertices of the pentiruncicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 7-orthoplex.
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Pentisteritruncated 6-simplex Pentisteritruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,4,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1176 Cells3780 Faces5250Edges3360 Vertices840 Vertex figureCoxeter groupA6, , order 10080Propertiesconvex
Alternate names Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)
Coordinates The vertices of the pentisteritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,2,3,4). This construction is based on facets of the pentisteritruncated 7-orthoplex.
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Pentistericantitruncated 6-simplex Pentistericantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol t0,1,2,4,5{3,3,3,3,3}Coxeter-Dynkin diagrams5-faces126 4-faces1596 Cells6510 Faces11340Edges8820 Vertices2520 Vertex figureCoxeter groupA6, [3,3,3,3,3], order 5040Propertiesconvex
Alternate names Great teracellirhombated heptapeton (Acronym: tacogral) (Jonathan Bowers)
Coordinates The vertices of the pentistericantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 7-orthoplex.
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Omnitruncated 6-simplex Omnitruncated 6-simplexTypeUniform 6-polytopeSchläfli symbol t0,1,2,3,4,5{35}Coxeter-Dynkin diagrams5-faces126:14 t0,1,2,3,4{34} 42 {}×t0,1,2,3{33} ×70 {6}×t0,1,2{3,3} ×4-faces1806Cells8400Faces16800:4200 {6} 1260 {4} Edges15120Vertices5040Vertex figureirregular 5-simplexCoxeter groupA6, [], order 10080Propertiesconvex, isogonal, zonotope The omnitruncated 6-simplex has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex.
Alternate names Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes) Omnitruncated heptapeton Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)
Permutohedron and related tessellation The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. It has Coxeter-Dynkin diagram of .
Coordinates The vertices of the omnitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 7-orthoplex, t0,1,2,3,4,5{35,4}, .
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Configuration This configuration matrix represents the omnitruncated 6-simplex, with 35 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Elementfkf0f1f2f3f4f5f050402222222122112222222222222221211222f125040**11111000011121121001121211101222*5040*10010111011210101121212111012122**50400110011011100211121211110211221f263301680********11110000001111110001124202*2520*******10001110001110101101214202**2520******01001011001011101101214220***2520*****00110101000111111001124400****1260****00020020000020210100226033*****1680***10000001111111001012114022******2520**01000100111101101012114040*******1260*00200000020202010012026006********8400000200020200000211220f324121212460004000420*********11100000011112666203000300*840********100110000111126120200300030**840*******010101000102121260200330000***840******001011000012126012033000002****840*****1000001101208444020200200*****1260****0100101001118804022020000******1260***00101001002112666003302000*******840**0011001001112401224000004604********420*100000101210120126000002330*********840010100001201f4120606012020303000203002051000100005084********1104824482481201208121202040060004*210*******1014848242481212121280002004006400**210******011361836186099069900330000303***280*****101242412124666606000202033000****420****0113636360120018900900066000000*****140***0024824244801212120812080000460420******210**1102424024012120600040000406000*******210*02012001201200000040603020000000001020********42200f57203607207201201801801800240360180120306060060900606012061502000150614**240240120240401201206060406004010200204030602010020501005100*42*144144144724836367236243636061224240181812012033464000**70
Full snub 6-simplex The full snub 6-simplex or omnisnub 6-simplex, defined as an alternation of the omnitruncated 6-simplex is not uniform, but it can be given Coxeter diagram and symmetry +, and constructed from 14 snub 5-simplexes, 42 snub 5-cell antiprisms, 70 3-s{3,4} duoantiprisms, and 2520 irregular 5-simplexes filling the gaps at the deleted vertices.
Related uniform 6-polytopes The pentellated 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]