6-simplex | |
---|---|

Type | uniform polypeton |

Schläfli symbol | {3^{5}} |

Coxeter diagrams | |

Elements |
f = 21, _{4}C = 35, F = 35, E = 21, V = 7(χ=0) ## Contents |

Coxeter group | A_{6}, [3^{5}], order 5040 |

Bowers name and (acronym) | Heptapeton (hop) |

Vertex figure | 5-simplex |

Circumradius | 0.645497 |

Properties | convex, isogonal self-dual |

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°.

It can also be called a **heptapeton**, or **hepta-6-tope**, as a 7-facetted polytope in 6-dimensions. The name *heptapeton* is derived from *hepta* for seven facets in Greek and *-peta* for having five-dimensional facets, and *-on*. Jonathan Bowers gives a heptapeton the acronym **hop**.^{ [1] }

This configuration matrix represents the 6-simplex. 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-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.^{ [2] }^{ [3] }

The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:

The vertices of the *6-simplex* can be more simply positioned in 7-space as permutations of:

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

This construction is based on facets of the 7-orthoplex.

A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [7] | [6] | [5] |

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

Graph | |||

Dihedral symmetry | [4] | [3] |

The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.

- ↑ Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o3o — hop".
- ↑ Coxeter 1973 , §1.8 Configurations
- ↑ Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.

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

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

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 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 **truncated 6-simplex** is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

In six-dimensional geometry, a **runcinated 6-simplex** is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.

- 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. - (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II".
*Math. Zeit*.**188**: 559–591. doi:10.1007/BF01161657. - (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III".
*Math. Zeit*.**200**: 3–45. doi:10.1007/BF01161745.

- (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).Cite journal requires
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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).

- Olshevsky, George. "Simplex".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007.Cite has empty unknown parameter:`|1=`

(help) - Polytopes of Various Dimensions
- Multi-dimensional Glossary

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