Regular 5-orthoplex (pentacross) | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 5-polytope |

Family | orthoplex |

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

Coxeter-Dynkin diagrams | |

4-faces | 32 {3^{3}} |

Cells | 80 {3,3} |

Faces | 80 {3} |

Edges | 40 |

Vertices | 10 |

Vertex figure | 16-cell |

Petrie polygon | decagon |

Coxeter groups | BC_{5}, [3,3,3,4]D _{5}, [3^{2,1,1}] |

Dual | 5-cube |

Properties | convex, Hanner polytope |

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.

- Alternate names
- As a configuration
- Cartesian coordinates
- Construction
- Other images
- Related polytopes and honeycombs
- References
- External links

It has two constructed forms, the first being regular with Schläfli symbol {3^{3},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3^{1,1}} or Coxeter symbol **2 _{11}**.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 5-hypercube or 5-cube.

**pentacross**, derived from combining the family name*cross polytope*with*pente*for five (dimensions) in Greek.**Triacontaditeron**(or*triacontakaiditeron*) - as a 32-facetted 5-polytope (polyteron).

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

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

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

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C_{5} or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of *5-cell* facets, alternating, with the D_{5} or [3^{2,1,1}] Coxeter group, and the final one as a dual 5-orthotope, called a **5-fusil** which can have a variety of subsymmetries.

Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure(s) |
---|---|---|---|---|---|

regular 5-orthoplex | {3,3,3,4} | [3,3,3,4] | 3840 | ||

Quasiregular 5-orthoplex | {3,3,3^{1,1}} | [3,3,3^{1,1}] | 1920 | ||

5-fusil | |||||

{3,3,3,4} | [4,3,3,3] | 3840 | |||

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

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

{3,4}+2{} | [4,3,2,2] | 192 | |||

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

{4}+3{} | [4,2,2,2] | 64 | |||

5{} | [2,2,2,2] | 32 |

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

Graph | |||

Dihedral symmetry | [10] | [8] | [6] |

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

Graph | |||

Dihedral symmetry | [4] | [4] |

The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection. |

2_{k1} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [[3^{1,2,1}]] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |

This polytope is one of 31 uniform 5-polytopes generated from the B_{5} Coxeter plane, including the regular 5-cube and 5-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 **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

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, 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 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*.

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 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. "5D uniform polytopes (polytera) x3o3o3o4o - tac".

- 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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