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9-cube Enneract | |
---|---|

Orthogonal projection inside Petrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8 | |

Type | Regular 9-polytope |

Family | hypercube |

Schläfli symbol | {4,3^{7}} |

Coxeter-Dynkin diagram | |

8-faces | 18 {4,3^{6}} |

7-faces | 144 {4,3^{5}} |

6-faces | 672 {4,3^{4}} |

5-faces | 2016 {4,3^{3}} |

4-faces | 4032 {4,3,3} |

Cells | 5376 {4,3} |

Faces | 4608 {4} |

Edges | 2304 |

Vertices | 512 |

Vertex figure | 8-simplex |

Petrie polygon | octadecagon |

Coxeter group | C_{9}, [3^{7},4] |

Dual | 9-orthoplex |

Properties | convex, Hanner polytope |

In geometry, a **9-cube** is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

It can be named by its Schläfli symbol {4,3^{7}}, being composed of three 8-cubes around each 7-face. It is also called an **enneract**, a portmanteau of tesseract (the *4-cube*) and *enne* for nine (dimensions) in Greek. It can also be called a regular **octadeca-9-tope** or **octadecayotton**, as a nine-dimensional polytope constructed with 18 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are

- (±1,±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}, *x*_{5}, *x*_{6}, *x*_{7}, *x*_{8}) with −1 < *x*_{i} < 1.

This 9-cube 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 1:9:36:84:126:126:84:36:9:1. |

B_{9} | B_{8} | B_{7} | |||
---|---|---|---|---|---|

[18] | [16] | [14] | |||

B_{6} | B_{5} | ||||

[12] | [10] | ||||

B_{4} | B_{3} | B_{2} | |||

[8] | [6] | [4] | |||

A_{7} | A_{5} | A_{3} | |||

[8] | [6] | [4] |

Applying an * alternation * operation, deleting alternating vertices of the *9-cube*, creates another uniform polytope, called a * 9-demicube *, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets.

In geometry, a **tesseract** is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In four-dimensional geometry, a **cantellated tesseract** is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, the **rectified tesseract**, **rectified 8-cell** is a uniform 4-polytope bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a **runcic tesseract**.

In five-dimensional geometry, a **5-cube** is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In five-dimensional geometry, a **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

In geometry, a **6-cube** is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In geometry, a **6-demicube** or **demihexteract** is a uniform 6-polytope, constructed from a *6-cube* (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a **7-cube** is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

In geometry, a **demihepteract** or **7-demicube** is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a **demiocteract** or **8-demicube** is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, an **8-cube** is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

In geometry, a **10-cube** is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

In six-dimensional geometry, a **rectified 6-cube** is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

In geometry of five dimensions or higher, a **cantic 5-cube**, **cantihalf 5-cube**, **truncated 5-demicube** is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

In six-dimensional geometry, a **runcic 5-cube** or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

In seven-dimensional geometry, a **cantic 7-cube** or **truncated 7-demicube** as a uniform 7-polytope, being a truncation of the 7-demicube.

In eight-dimensional geometry, a **cantic 8-cube** or **truncated 8-demicube** is a uniform 8-polytope, being a truncation of the 8-demicube.

In six-dimensional geometry, a **runcic 6-cube** is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

In seven-dimensional geometry, a **runcic 7-cube** is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

- 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- (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]

- (Paper 22) H.S.M. Coxeter,

- Coxeter,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. (1966)

- N.W. Johnson:
- Klitzing, Richard. "9D uniform polytopes (polyyotta) o3o3o3o3o3o3o3o4x - enne".

- Weisstein, Eric W. "Hypercube".
*MathWorld*. - Olshevsky, George. "Measure polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Multi-dimensional Glossary: hypercube Garrett Jones

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