Orthogonal projections in B4 Coxeter plane7-orthoplexHexicated 7-orthoplexHexicated 7-cubeHexi-truncated 7-orthoplexHexi-cantellated 7-orthoplexHexicanti-truncated 7-orthoplexHexirunci-truncated 7-orthoplexHexirunci-cantellated 7-orthoplexHexisteri-truncated 7-orthoplexHexiruncicanti-truncated 7-orthoplexHexistericanti-truncated 7-orthoplexHexisterirunci-truncated 7-orthoplexHexipenticanti-truncated 7-orthoplexHexisteriruncicanti-truncated 7-orthoplexHexipentiruncicanti-truncated 7-orthoplex In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.
There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.
Hexitruncated 7-orthoplex Hexitruncated 7-orthoplexTypeUniform 7-polytopeSchläfli symbol t0,1,6{35,4Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges29568Vertices5376Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petitruncated heptacross
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Hexicantellated 7-orthoplex Hexicantellated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,2,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges94080Vertices13440Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petirhombated heptacross
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Hexicantitruncated 7-orthoplex Hexicantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges134400Vertices26880Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petigreatorhombated heptacross
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Hexiruncitruncated 7-orthoplex Hexiruncitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,6{35,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges322560Vertices53760Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petiprismatotruncated heptacross
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Hexiruncicantellated 7-orthoplex
Hexiruncicantellated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,2,3,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges268800Vertices53760Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope. Alternate names Petiprismatorhombated heptacross
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Hexisteritruncated 7-orthoplex hexisteritruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,4,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges322560Vertices53760Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Peticellitruncated heptacross
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Hexiruncicantitruncated 7-orthoplex Hexiruncicantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges483840Vertices107520Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petigreatoprismated heptacross
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Hexistericantitruncated 7-orthoplex Hexistericantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,4,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges806400Vertices161280Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Peticelligreatorhombated heptacross
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Hexisteriruncitruncated 7-orthoplex Hexisteriruncitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,4,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges725760Vertices161280Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Peticelliprismatotruncated heptacross
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Hexipenticantitruncated 7-orthoplex hexipenticantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,5,6{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges483840Vertices107520Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petiterigreatorhombated heptacross
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Hexisteriruncicantitruncated 7-orthoplex
Hexisteriruncicantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,4,6{4,35}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges1290240Vertices322560Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Great petacellated heptacross
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Hexipentiruncicantitruncated 7-orthoplex Hexipentiruncicantitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,5,6{35,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges1290240Vertices322560Vertex figureCoxeter groupsB7, [4,35]Propertiesconvex Alternate names Petiterigreatoprismated heptacross
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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 Ivic Weiss, Wiley-Interscience Publication, 1995, (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, PhD (1966)
External links Polytopes of Various Dimensions Multi-dimensional Glossary