6-orthoplex Hexacross | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 6-polytope |

Family | orthoplex |

Schläfli symbols | {3,3,3,3,4} {3,3,3,3 ^{1,1}} |

Coxeter-Dynkin diagrams | = |

5-faces | 64 {3^{4}} |

4-faces | 192 {3^{3}} |

Cells | 240 {3,3} |

Faces | 160 {3} |

Edges | 60 |

Vertices | 12 |

Vertex figure | 5-orthoplex |

Petrie polygon | dodecagon |

Coxeter groups | B_{6}, [4,3^{4}]D _{6}, [3^{3,1,1}] |

Dual | 6-cube |

Properties | convex, Hanner polytope |

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*.

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

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

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 6-hypercube, or hexeract.

**Hexacross**, derived from combining the family name cross polytope with*hex*for six (dimensions) in Greek.**Hexacontitetrapeton**as a 64-facetted 6-polytope.

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

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C_{6} or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D_{6} or [3^{3,1,1}] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a **6-fusil**.

Name | Coxeter | Schläfli | Symmetry | Order |
---|---|---|---|---|

Regular 6-orthoplex | {3,3,3,3,4} | [4,3,3,3,3] | 46080 | |

Quasiregular 6-orthoplex | {3,3,3,3^{1,1}} | [3,3,3,3^{1,1}] | 23040 | |

6-fusil | {3,3,3,4}+{} | [4,3,3,3,3] | 7680 | |

{3,3,4}+{4} | [4,3,3,2,4] | 3072 | ||

2{3,4} | [4,3,2,4,3] | 2304 | ||

{3,3,4}+2{} | [4,3,3,2,2] | 1536 | ||

{3,4}+{4}+{} | [4,3,2,4,2] | 768 | ||

3{4} | [4,2,4,2,4] | 512 | ||

{3,4}+3{} | [4,3,2,2,2] | 384 | ||

2{4}+2{} | [4,2,4,2,2] | 256 | ||

{4}+4{} | [4,2,2,2,2] | 128 | ||

6{} | [2,2,2,2,2] | 64 |

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

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

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

Coxeter plane | B_{6} | B_{5} | B_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [12] | [10] | [8] |

Coxeter plane | B_{3} | B_{2} | |

Graph | |||

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

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

Graph | |||

Dihedral symmetry | [6] | [4] |

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.^{ [3] }

2D | 3D | ||
---|---|---|---|

Icosahedron {3,5} = H _{3} Coxeter plane | 6-orthoplex {3,3,3,3 ^{1,1}} = D _{6} Coxeter plane | Icosahedron | 6-orthoplex |

This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D_{6} to H_{3} Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron. |

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 | 9 |

Coxeter group | A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |

Coxeter diagram | ||||||

Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [[3^{1,3,1}]]= [4,3,3,3,3] | [3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |

Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | |

Graph | - | - | ||||

Name | 3_{1,-1} | 3_{10} | 3_{11} | 3_{21} | 3_{31} | 3_{41} |

This polytope is one of 63 uniform 6-polytopes generated from the B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.

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-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-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, 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 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 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 **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 five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-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 six-dimensional geometry, a **rectified 6-orthoplex** is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

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

- 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. "6D uniform polytopes (polypeta) x3o3o3o3o4o - gee".

- Specific

- ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑
*Quasicrystals and Geometry*, Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1*The I*_{6}crystal

- 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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