Regular decayotton (9-simplex) | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 9-polytope |

Family | simplex |

Schläfli symbol | {3,3,3,3,3,3,3,3} |

Coxeter-Dynkin diagram | |

8-faces | 10 8-simplex |

7-faces | 45 7-simplex |

6-faces | 120 6-simplex |

5-faces | 210 5-simplex |

4-faces | 252 5-cell |

Cells | 210 tetrahedron |

Faces | 120 triangle |

Edges | 45 |

Vertices | 10 |

Vertex figure | 8-simplex |

Petrie polygon | decagon |

Coxeter group | A_{9} [3,3,3,3,3,3,3,3] |

Dual | Self-dual |

Properties | convex |

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos^{−1}(1/9), or approximately 83.62°.

It can also be called a **decayotton**, or **deca-9-tope**, as a 10-facetted polytope in 9-dimensions.. The name *decayotton* is derived from *deca* for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and *-on*.

The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are:

More simply, the vertices of the *9-simplex* can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 10-orthoplex.

A_{k} Coxeter plane | A_{9} | A_{8} | A_{7} | A_{6} |
---|---|---|---|---|

Graph | ||||

Dihedral symmetry | [10] | [9] | [8] | [7] |

A_{k} Coxeter plane | A_{5} | A_{4} | A_{3} | A_{2} |

Graph | ||||

Dihedral symmetry | [6] | [5] | [4] | [3] |

In geometry, the **5-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,3}. It is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a **C _{5}**,

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-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-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-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 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos^{−1}(1/10), or approximately 84.26°.

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 **stericated 5-simplex** is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

In five-dimensional geometry, a **truncated 5-simplex** is a convex uniform 5-polytope, being a truncation of the regular 5-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.

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

In five-dimensional geometry, a **truncated 5-cube** is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

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

In ten-dimensional geometry, a **rectified 10-orthoplex** is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.

- Coxeter, H.S.M.:
- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)".
*Regular Polytopes*(3rd ed.). Dover. p. 296. ISBN 0-486-61480-8. - Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995).
*Kaleidoscopes: Selected Writings of H.S.M. Coxeter*. Wiley. ISBN 978-0-471-01003-6.- (Paper 22) — (1940). "Regular and Semi Regular Polytopes I".
*Math. Zeit*.**46**: 380–407. doi:10.1007/BF01181449. S2CID 186237114. - (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II".
*Math. Zeit*.**188**(4): 559–591. doi:10.1007/BF01161657. S2CID 120429557. - (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III".
*Math. Zeit*.**200**: 3–45. doi:10.1007/BF01161745. S2CID 186237142.

- (Paper 22) — (1940). "Regular and Semi Regular Polytopes I".

- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)".
- Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1
_{n1}".*The Symmetries of Things*. p. 409. ISBN 978-1-56881-220-5. - Johnson, Norman (1991). "Uniform Polytopes" (Manuscript).
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(help)- Johnson, N.W. (1966).
*The Theory of Uniform Polytopes and Honeycombs*(PhD). University of Toronto. OCLC 258527038.

- Johnson, N.W. (1966).
- Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o3o — day".

- Glossary for hyperspace , George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary

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