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The Info List - 7-cube


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6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube
HOME
The Info List - 7-cube


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6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube
HOME
The Info List - 7-cube


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6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube
HOME
The Info List - 7-cube


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6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.
l> 7-cube


--- Advertisement ---



6-faces 14 4,34

5-faces 84 4,33

4-faces 280 4,3,3

Cells 560 4,3

Faces 672 4

Edges 448

Vertices 128

Vertex figure 6-simplex
6-simplex

Petrie polygon tetradecagon

Coxeter group C7, [35,4]

Dual 7-orthoplex

Properties convex

In 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.

Contents

1 Related polytopes 2 As a configuration 3 Cartesian coordinates 4 Images 5 References 6 External links

Related polytopes[edit] 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[edit] 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[edit] 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.

Images[edit]

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.

Play media

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

orthographic projections

Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4

Graph

Dihedral symmetry [14] [12] [10]

Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane A5 A3

Graph

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[edit]

Weisstein, 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.de

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.

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