It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
This configuration matrix represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
Cartesian coordinatesedit
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.
Projectionsedit
This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1.
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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, ISBN978-0-471-01003-6[1]
(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]
cube, hepteractorthogonal, projectioninside, petrie, polygonthe, central, orange, vertex, doubledtype, regular, polytopefamily, hypercubeschläfli, symbol, coxeter, dynkin, diagrams6, faces, faces, faces, cells, faces, edges, 448vertices, 128vertex, figure, sim. 7 cubeHepteractOrthogonal projectioninside Petrie polygonThe central orange vertex is doubledType Regular 7 polytopeFamily hypercubeSchlafli symbol 4 35 Coxeter Dynkin diagrams6 faces 14 4 34 5 faces 84 4 33 4 faces 280 4 3 3 Cells 560 4 3 Faces 672 4 Edges 448Vertices 128Vertex figure 6 simplexPetrie polygon tetradecagonCoxeter group C7 35 4 Dual 7 orthoplexProperties convex Hanner polytopeIn geometry a 7 cube is a seven dimensional hypercube with 128 vertices 448 edges 672 square faces 560 cubic cells 280 tesseract 4 faces 84 penteract 5 faces and 14 hexeract 6 faces It can be named by its Schlafli symbol 4 35 being composed of 3 6 cubes around each 5 face It can be called a hepteract a portmanteau of tesseract the 4 cube and hepta for seven dimensions in Greek It can also be called a regular tetradeca 7 tope or tetradecaexon being a 7 dimensional polytope constructed from 14 regular facets Contents 1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Projections 5 References 6 External linksRelated polytopes editThe 7 cube is 7th in a series of hypercube Petrie polygon orthographic projections nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Line segment Square Cube 4 cube 5 cube 6 cube 7 cube 8 cubeThe dual of a 7 cube is called a 7 orthoplex and is a part of the infinite family of cross polytopes Applying an alternation operation deleting alternating vertices of the hepteract creates another uniform polytope called a demihepteract part of an infinite family called demihypercubes which has 14 demihexeractic and 64 6 simplex 6 faces As a configuration editThis configuration matrix represents the 7 cube The rows and columns correspond to vertices edges faces cells 4 faces 5 faces and 6 faces The diagonal numbers say how many of each element occur in the whole 7 cube The nondiagonal numbers say how many of the column s element occur in or at the row s element 1 2 128 7 21 35 35 21 7 2 448 6 15 20 15 6 4 4 672 5 10 10 5 8 12 6 560 4 6 4 16 32 24 8 280 3 3 32 80 80 40 10 84 2 64 192 240 160 60 12 14 displaystyle begin bmatrix begin matrix 128 amp 7 amp 21 amp 35 amp 35 amp 21 amp 7 2 amp 448 amp 6 amp 15 amp 20 amp 15 amp 6 4 amp 4 amp 672 amp 5 amp 10 amp 10 amp 5 8 amp 12 amp 6 amp 560 amp 4 amp 6 amp 4 16 amp 32 amp 24 amp 8 amp 280 amp 3 amp 3 32 amp 80 amp 80 amp 40 amp 10 amp 84 amp 2 64 amp 192 amp 240 amp 160 amp 60 amp 12 amp 14 end matrix end bmatrix nbsp Cartesian coordinates editCartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are 1 1 1 1 1 1 1 while the interior of the same consists of all points x0 x1 x2 x3 x4 x5 x6 with 1 lt xi lt 1 Projections edit nbsp This hypercube graph is an orthogonal projection This orientation shows columns of vertices positioned a vertex edge vertex distance from one vertex on the left to one vertex on the right and edges attaching adjacent columns of vertices The number of vertices in each column represents rows in Pascal s triangle being 1 7 21 35 35 21 7 1 orthographic projections Coxeter plane B7 A6 B6 D7 B5 D6 A4Graph nbsp nbsp nbsp Dihedral symmetry 14 12 10 Coxeter plane B4 D5 B3 D4 A2 B2 D3Graph nbsp nbsp nbsp Dihedral symmetry 8 6 4 Coxeter plane A5 A3Graph nbsp nbsp Dihedral symmetry 6 4 References edit Coxeter Regular Polytopes sec 1 8 Configurations Coxeter Complex Regular Polytopes p 117 H S M Coxeter Coxeter Regular Polytopes 3rd edition 1973 Dover edition ISBN 0 486 61480 8 p 296 Table I iii Regular Polytopes three regular polytopes in n dimensions n 5 H S M Coxeter Regular Polytopes 3rd Edition Dover New York 1973 p 296 Table I iii Regular Polytopes three regular polytopes in n dimensions n 5 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 ISBN 978 0 471 01003 6 1 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 1966 Klitzing Richard 7D uniform polytopes polyexa o3o3o3o3o3o4x hept External links editWeisstein Eric W Hypercube MathWorld Weisstein Eric W Hypercube graph MathWorld Olshevsky George Measure polytope Glossary for Hyperspace Archived from the original on 4 February 2007 Multi dimensional Glossary hypercube Garrett Jones Rotation of 7D Cube www 4d screen devteFundamental convex regular and uniform polytopes in dimensions 2 10Family An Bn I2 p Dn E6 E7 E8 F4 G2 HnRegular polygon Triangle Square p gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron IcosahedronUniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cellUniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicubeUniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicubeUniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicubeUniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytopeTopics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title 7 cube amp oldid 1122323336, wikipedia, wiki, book, books, library,
