In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.
The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called an omnitruncated 7-simplex with all of the nodes ringed.
7-simplex Hexicated 7-simplex Hexitruncated 7-simplex Hexicantellated 7-simplex Hexiruncinated 7-simplex Hexicantitruncated 7-simplex Hexiruncitruncated 7-simplex Hexiruncicantellated 7-simplex Hexisteritruncated 7-simplex Hexistericantellated 7-simplex Hexipentitruncated 7-simplex Hexiruncicantitruncated 7-simplex Hexistericantitruncated 7-simplex Hexisteriruncitruncated 7-simplex Hexisteriruncicantellated 7-simplex Hexipenticantitruncated 7-simplex Hexipentiruncitruncated 7-simplex Hexisteriruncicantitruncated 7-simplex Hexipentiruncicantitruncated 7-simplex Hexipentistericantitruncated 7-simplex Hexipentisteriruncicantitruncated 7-simplex(Omnitruncated 7-simplex)Orthogonal projections in A7 Coxeter plane
Hexicated 7-simplex Hexicated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,6{36}Coxeter-Dynkin diagrams6-faces254:8+8 {35} 28+28 {}x{34}56+56 {3}x{3,3,3}70 {3,3}x{3,3}5-faces4-facesCellsFacesEdges336 Vertices56 Vertex figure5-simplex antiprismCoxeter groupA7×2, [], order 80640Propertiesconvex In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.
Root vectors Its 56 vertices represent the root vectors of the simple Lie group A7.
Alternate names Expanded 7-simplex Small petated hexadecaexon (Acronym: suph) (Jonathan Bowers)
Coordinates The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .
A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0,0,0)
Images
Hexitruncated 7-simplex Hexitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges1848 Vertices336 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petitruncated octaexon (Acronym: puto) (Jonathan Bowers)Klitzing, (x3x3o3o3o3o3x - puto)
Coordinates The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .
Images
Hexicantellated 7-simplex Hexicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges5880 Vertices840 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petirhombated octaexon (Acronym: puro) (Jonathan Bowers)Klitzing, (x3o3x3o3o3o3x - puro)
Coordinates The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .
Images
Hexiruncinated 7-simplex Hexiruncinated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,3,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges8400 Vertices1120 Vertex figureCoxeter groupA7×2, [], order 80640Propertiesconvex
Alternate names Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers)
Coordinates The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
Images
Hexicantitruncated 7-simplex Hexicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges8400 Vertices1680 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petigreatorhombated octaexon (Acronym: pugro) (Jonathan Bowers)Klitzing, (x3o3o3o3x3o3x - pugro)
Coordinates The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .
Images
Hexiruncitruncated 7-simplex Hexiruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges20160 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petiprismatotruncated octaexon (Acronym: pupato) (Jonathan Bowers)Klitzing, (x3x3x3o3o3o3x - pupato)
Coordinates The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .
Images
Hexiruncicantellated 7-simplex Hexiruncicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,3,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges16800 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.
Alternate names Petiprismatorhombated octaexon (Acronym: pupro) (Jonathan Bowers)Klitzing, (x3o3x3x3o3o3x - pupro)
Coordinates The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .
Images
Hexisteritruncated 7-simplex Hexisteritruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,4,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges20160 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Peticellitruncated octaexon (Acronym: pucto) (Jonathan Bowers)Klitzing, (x3x3o3o3x3o3x - pucto)
Coordinates The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .
Images
Hexistericantellated 7-simplex Hexistericantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,4,6{36}Coxeter-Dynkin diagrams6-facest0,2,4{3,3,3,3,3} {}xt0,2,4{3,3,3,3} {3}xt0,2{3,3,3} t0,2{3,3}xt0,2{3,3}5-faces4-facesCellsFacesEdges30240 Vertices5040 Vertex figureCoxeter groupA7×2, [], order 80640Propertiesconvex
Alternate names Peticellirhombihexadecaexon (Acronym: pucroh) (Jonathan Bowers)Klitzing, (x3o3x3o3x3o3x - pucroh)
Coordinates The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .
Images
Hexipentitruncated 7-simplex Hexipentitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,5,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges8400 Vertices1680 Vertex figureCoxeter groupA7×2, [], order 80640Propertiesconvex
Alternate names Petiteritruncated hexadecaexon (Acronym: putath) (Jonathan Bowers)Klitzing, (x3x3o3o3o3x3x - putath)
Coordinates The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .
Images
Hexiruncicantitruncated 7-simplex Hexiruncicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges30240 Vertices6720 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petigreatoprismated octaexon (Acronym: pugopo) (Jonathan Bowers)Klitzing, (x3x3x3x3o3o3x - pugopo)
Coordinates The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .
Images
Hexistericantitruncated 7-simplex Hexistericantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,4,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges50400 Vertices10080 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Peticelligreatorhombated octaexon (Acronym: pucagro) (Jonathan Bowers)Klitzing, (x3x3x3o3x3o3x - pucagro)
Coordinates The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .
Images
Hexisteriruncitruncated 7-simplex Hexisteriruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,4,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges45360 Vertices10080 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Peticelliprismatotruncated octaexon (Acronym: pucpato) (Jonathan Bowers)Klitzing, (x3x3o3x3x3o3x - pucpato)
Coordinates The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
Images
Hexisteriruncicantellated 7-simplex Hexisteriruncicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,3,4,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges45360 Vertices10080 Vertex figureCoxeter groupA7×2, [], order 80640Propertiesconvex
Alternate names Peticelliprismatorhombihexadecaexon (Acronym: pucproh) (Jonathan Bowers)Klitzing, (x3o3x3x3x3o3x - pucproh)
Coordinates The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
Images
Hexipenticantitruncated 7-simplex Hexipenticantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,5,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges30240 Vertices6720 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Petiterigreatorhombated octaexon (Acronym: putagro) (Jonathan Bowers)Klitzing, (x3x3x3o3o3x3x - putagro)
Coordinates The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .
Images
Hexipentiruncitruncated 7-simplex Hexipentiruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,5,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdgesVertices10080Vertex figureCoxeter groupA7×2, [], order 80640Propertiesconvex
Alternate names Petiteriprismatotruncated hexadecaexon (Acronym: putpath) (Jonathan Bowers)Klitzing, (x3x3o3x3o3x3x - putpath)
Coordinates The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .
Images
Hexisteriruncicantitruncated 7-simplex Hexisteriruncicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,4,6{36}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges80640 Vertices20160 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names