6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

.
7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

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7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

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l> 7-cube

 6-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-simplex 6-simplex Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube 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 Schläfli symbol 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.Contents1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External linksRelated polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube 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-simplex 6-faces. As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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&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&14end matrix end bmatrix Cartesian coordinates Cartesian coordinates 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.ImagesThis 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.Play mediaHepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.orthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]References^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.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 linksWeisstein, Eric W. "Hypercube". MathWorld.  Weisstein, Eric W. " Hypercube 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 Cube www.4d-screen.dev t eFundamental 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 Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

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