Regular 7-orthoplex (heptacross) | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 7-polytope |

Family | orthoplex |

Schläfli symbol | {3^{5},4}{3,3,3,3,3 ^{1,1}} |

Coxeter-Dynkin diagrams | |

6-faces | 128 {3^{5}} |

5-faces | 448 {3^{4}} |

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

Cells | 560 {3,3} |

Faces | 280 {3} |

Edges | 84 |

Vertices | 14 |

Vertex figure | 6-orthoplex |

Petrie polygon | tetradecagon |

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

Dual | 7-cube |

Properties | convex, Hanner polytope |

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*.

- Alternate names
- As a configuration
- Images
- Construction
- Cartesian coordinates
- See also
- References
- External links

It has two constructed forms, the first being regular with Schläfli symbol {3^{5},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3^{1,1}} or Coxeter symbol **4 _{11}**.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 7-hypercube, or hepteract.

**Heptacross**, derived from combining the family name*cross polytope*with*hept*for seven (dimensions) in Greek.**Hecatonicosoctaexon**as a 128-facetted 7-polytope (polyexon).

This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [1] }^{ [2] }

Coxeter plane | B_{7} / A_{6} | B_{6} / D_{7} | B_{5} / D_{6} / A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [14] | [12] | [10] |

Coxeter plane | B_{4} / D_{5} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | A_{5} | A_{3} | |

Graph | |||

Dihedral symmetry | [6] | [4] |

There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C_{7} or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D_{7} or [3^{4,1,1}] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a **7-fusil**.

Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure |
---|---|---|---|---|---|

regular 7-orthoplex | {3,3,3,3,3,4} | [3,3,3,3,3,4] | 645120 | ||

Quasiregular 7-orthoplex | {3,3,3,3,3^{1,1}} | [3,3,3,3,3^{1,1}] | 322560 | ||

7-fusil | 7{} | [2^{6}] | 128 |

Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are

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

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

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 **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 five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos^{−1}(1/5), or approximately 78.46°.

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-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 **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, 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 **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 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos^{−1}(1/6), or approximately 80.41°.

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos^{−1}(1/7), or approximately 81.79°.

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, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos^{−1}(1/8), or approximately 82.82°.

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*.

The **7-cubic honeycomb** or **hepteractic honeycomb** is the only regular space-filling tessellation in Euclidean 7-space.

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 8-dimensional geometry, the **2 _{41}** is a uniform 8-polytope, constructed within the symmetry of the E

In six-dimensional geometry, a **pentellated 6-simplex** is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

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

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

- 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. (1966)

- N.W. Johnson:
- Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o4o - zee".

- 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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