8-orthoplex Octacross | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 8-polytope |

Family | orthoplex |

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

Coxeter-Dynkin diagrams | |

7-faces | 256 {3^{6}} |

6-faces | 1024 {3^{5}} |

5-faces | 1792 {3^{4}} |

4-faces | 1792 {3^{3}} |

Cells | 1120 {3,3} |

Faces | 448 {3} |

Edges | 112 |

Vertices | 16 |

Vertex figure | 7-orthoplex |

Petrie polygon | hexadecagon |

Coxeter groups | C_{8}, [3^{6},4]D _{8}, [3^{5,1,1}] |

Dual | 8-cube |

Properties | convex, Hanner polytope |

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*.

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

It has two constructive forms, the first being regular with Schläfli symbol {3^{6},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3^{1,1}} or Coxeter symbol **5 _{11}**.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is an 8-hypercube, or octeract.

**Octacross**, derived from combining the family name*cross polytope*with*oct*for eight (dimensions) in Greek**Diacosipentacontahexazetton**as a 256-facetted 8-polytope (polyzetton)

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [1] }^{ [2] }

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors. ^{ [3] }

B_{8} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | k-figure | notes | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

B_{7} | ( ) | f_{0} | 16 | 14 | 84 | 280 | 560 | 672 | 448 | 128 | {3,3,3,3,3,4} | B_{8}/B_{7} = 2^8*8!/2^7/7! = 16 | |

A_{1}B_{6} | { } | f_{1} | 2 | 112 | 12 | 60 | 160 | 240 | 192 | 64 | {3,3,3,3,4} | B_{8}/A_{1}B_{6} = 2^8*8!/2/2^6/6! = 112 | |

A_{2}B_{5} | {3} | f_{2} | 3 | 3 | 448 | 10 | 40 | 80 | 80 | 32 | {3,3,3,4} | B_{8}/A_{2}B_{5} = 2^8*8!/3!/2^5/5! = 448 | |

A_{3}B_{4} | {3,3} | f_{3} | 4 | 6 | 4 | 1120 | 8 | 24 | 32 | 16 | {3,3,4} | B_{8}/A_{3}B_{4} = 2^8*8!/4!/2^4/4! = 1120 | |

A_{4}B_{3} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 1792 | 6 | 12 | 8 | {3,4} | B_{8}/A_{4}B_{3} = 2^8*8!/5!/8/3! = 1792 | |

A_{5}B_{2} | {3,3,3,3} | f_{5} | 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4 | {4} | B_{8}/A_{5}B_{2} = 2^8*8!/6!/4/2 = 1792 | |

A_{6}A_{1} | {3,3,3,3,3} | f_{6} | 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2 | { } | B_{8}/A_{6}A_{1} = 2^8*8!/7!/2 = 1024 | |

A_{7} | {3,3,3,3,3,3} | f_{7} | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256 | ( ) | B_{8}/A_{7} = 2^8*8!/8! = 256 |

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C_{8} or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D_{8} or [3^{5,1,1}] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an **8-fusil**.

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

regular 8-orthoplex | {3,3,3,3,3,3,4} | [3,3,3,3,3,3,4] | 10321920 | ||

Quasiregular 8-orthoplex | {3,3,3,3,3,3^{1,1}} | [3,3,3,3,3,3^{1,1}] | 5160960 | ||

8-fusil | 8{} | [2^{7}] | 256 |

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are

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

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

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

[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] |

It is used in its alternated form **5 _{11}** with the 8-simplex to form the

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 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, 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 **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, 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*.

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 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 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-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 five-dimensional geometry, a **rectified 5-orthoplex** is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

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.

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

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 **runcinated 7-simplex** is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

- ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ Klitzing, Richard. "x3o3o3o3o3o3o4o - ek".

- 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. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o4o - ek".

- 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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