Regular enneazetton (8-simplex) | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 8-polytope |

Family | simplex |

Schläfli symbol | {3,3,3,3,3,3,3} |

Coxeter-Dynkin diagram | |

7-faces | 9 7-simplex |

6-faces | 36 6-simplex |

5-faces | 84 5-simplex |

4-faces | 126 5-cell |

Cells | 126 tetrahedron |

Faces | 84 triangle |

Edges | 36 |

Vertices | 9 |

Vertex figure | 7-simplex |

Petrie polygon | enneagon |

Coxeter group | A_{8} [3,3,3,3,3,3,3] |

Dual | Self-dual |

Properties | convex |

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°.

It can also be called an **enneazetton**, or **ennea-8-tope**, as a 9-facetted polytope in eight-dimensions. The name *enneazetton* is derived from *ennea* for nine facets in Greek and *-zetta* for having seven-dimensional facets, and *-on*.

This configuration matrix represents the 8-simplex. 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-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.^{ [1] }^{ [2] }

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

More simply, the vertices of the *8-simplex* can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
---|---|---|---|---|

Graph | ||||

Dihedral symmetry | [9] | [8] | [7] | [6] |

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

Graph | ||||

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

This polytope is a facet in the uniform tessellations: 2_{51}, and 5_{21} with respective Coxeter-Dynkin diagrams:

- ,

This polytope is one of 135 uniform 8-polytopes with A_{8} symmetry.

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-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 **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 **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, 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°.

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 7-dimensional geometry, **2 _{31}** is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{31} honeycomb** is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3

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 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.

In geometry, the **simplectic honeycomb** is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3^{[n+1]}}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of *n+1* nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an *n-simplex honeycomb* is an expanded n-simplex.

- ↑ Coxeter 1973 , §1.8 Configurations
- ↑ Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.

- Coxeter, H.S.M.:
- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)".
*Regular Polytopes*(3rd ed.). Dover. pp. 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. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o3o — ene".

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

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