Regular 9-orthoplex Ennecross | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 9-polytope |

Family | orthoplex |

Schläfli symbol | {3^{7},4}{3 ^{6},3^{1,1}} |

Coxeter-Dynkin diagrams | |

8-faces | 512 {3^{7}} |

7-faces | 2304 {3^{6}} |

6-faces | 4608 {3^{5}} |

5-faces | 5376 {3^{4}} |

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

Cells | 2016 {3,3} |

Faces | 672 {3} |

Edges | 144 |

Vertices | 18 |

Vertex figure | Octacross |

Petrie polygon | Octadecagon |

Coxeter groups | C_{9}, [3^{7},4]D _{9}, [3^{6,1,1}] |

Dual | 9-cube |

Properties | convex, Hanner polytope |

In geometry, a **9-orthoplex** or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells *4-faces*, 5376 5-simplex *5-faces*, 4608 6-simplex *6-faces*, 2304 7-simplex *7-faces*, and 512 8-simplex *8-faces*.

It has two constructed forms, the first being regular with Schläfli symbol {3^{7},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3^{6},3^{1,1}} or Coxeter symbol **6 _{11}**.

It is one of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 9-hypercube or enneract.

**Enneacross**, derived from combining the family name*cross polytope*with*ennea*for nine (dimensions) in Greek**Pentacosidodecayotton**as a 512-facetted 9-polytope (polyyotton)

There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C_{9} or [4,3^{7}] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D_{9} or [3^{6,1,1}] symmetry group.

Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are

- (±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

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] |

In geometry, a **five-dimensional polytope** is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In five-dimensional geometry, a **5-orthoplex**, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In geometry, a **6-orthoplex**, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell *4-faces*, and 64 *5-faces*.

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.

In geometry, a **7-orthoplex**, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells *4-faces*, 448 *5-faces*, and 128 *6-faces*.

In geometry, an **8-orthoplex** or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells *4-faces*, 1792 *5-faces*, 1024 *6-faces*, and 256 *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 ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

In geometry, a **10-orthoplex** or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells *4-faces*, 13440 *5-faces*, 15360 *6-faces*, 11520 *7-faces*, 5120 *8-faces*, and 1024 *9-faces*.

In seven-dimensional geometry, a **rectified 7-orthoplex** is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

In six-dimensional geometry, a **six-dimensional polytope** or **6-polytope** is a polytope, bounded by 5-polytope facets.

In six-dimensional geometry, a **truncated 6-simplex** is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

In seven-dimensional geometry, a **truncated 7-simplex** is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

In seven-dimensional geometry, a **rectified 7-simplex** is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In eight-dimensional geometry, a **rectified 8-simplex** is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In eight-dimensional geometry, a **heptellated 8-simplex** is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

In eight-dimensional geometry, a **truncated 8-simplex** is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.

In eight-dimensional geometry, a **runcinated 8-simplex** is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

In eight-dimensional geometry, a **stericated 8-simplex** is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

In eight-dimensional geometry, a **pentellated 8-simplex** is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.

- H.S.M. Coxeter:
- H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973 **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, 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,

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

- N.W. Johnson:
- Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee".

- Olshevsky, George. "Cross polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Polytopes of Various Dimensions
- Multi-dimensional Glossary

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