5-simplex Hexateron (hix) | ||
---|---|---|

Type | uniform 5-polytope | |

Schläfli symbol | {3^{4}} | |

Coxeter diagram | ||

4-faces | 6 | 6 {3,3,3} |

Cells | 15 | 15 {3,3} |

Faces | 20 | 20 {3} |

Edges | 15 | |

Vertices | 6 | |

Vertex figure | 5-cell | |

Coxeter group | A_{5}, [3^{4}], order 720 | |

Dual | self-dual | |

Base point | (0,0,0,0,0,1) | |

Circumradius | 0.645497 | |

Properties | convex, isogonal regular, self-dual |

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°.

- Alternate names
- As a configuration
- Regular hexateron cartesian coordinates
- Projected images
- Lower symmetry forms
- Compound
- Related uniform 5-polytopes
- See also
- Notes
- References
- External links

The 5-simplex is a solution to the problem: *Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.*

It can also be called a **hexateron**, or **hexa-5-tope**, as a 6-facetted polytope in 5-dimensions. The name *hexateron* is derived from *hexa-* for having six facets and * teron * (with *ter-* being a corruption of * tetra- *) for having four-dimensional facets.

By Jonathan Bowers, a hexateron is given the acronym **hix**.^{ [1] }

This configuration matrix represents the 5-simplex. 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-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 *hexateron* can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.

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

The vertices of the *5-simplex* can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) *or* (0,1,1,1,1,1). These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

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

Graph | ||

Dihedral symmetry | [6] | [5] |

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

Graph | ||

Dihedral symmetry | [4] | [3] |

Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron. |

A lower symmetry form is a *5-cell pyramid* ( )v{3,3,3}, with [3,3,3] symmetry order 120, constructed as a 5-cell base in a 4-space hyperplane, and an apex point *above* the hyperplane. The five *sides* of the pyramid are made of 5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube.

Another form is { }v{3,3}, with [2,3,3] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}v{3}, with [3,2,3] symmetry order 36, and extended symmetry [[3,2,3]], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.

These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.

( )v{3,3,3} | { }v{3,3} | {3}v{3} | ||
---|---|---|---|---|

truncated 6-simplex | truncated 6-cube | bitruncated 6-simplex | bitruncated 6-cube | tritruncated 6-simplex |

The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5-simplex vertices and edges. This compound has [[3,3,3,3]] symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ .

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} 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}] | [3^{2,3,1}] | [[3^{3,3,1}]] | [3^{4,3,1}] |

Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |

Graph | - | - | ||||

Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |

It is first 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 dihedron.

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} |

The 5-simplex, as 2_{20} polytope is first in dimensional series 2_{2k}.

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

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

Coxeter group | A_{2}A_{2} | A_{5} | E_{6} | =E_{6}^{+} | E_{6}^{++} |

Coxeter diagram | |||||

Graph | ∞ | ∞ | |||

Name | 2_{2,-1} | 2_{20} | 2_{21} | 2_{22} | 2_{23} |

The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A_{5} Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

t _{0} | t _{1} | t _{2} | t _{0,1} | t _{0,2} | t _{1,2} | t _{0,3} | |||||

t _{1,3} | t _{0,4} | t _{0,1,2} | t _{0,1,3} | t _{0,2,3} | t _{1,2,3} | t _{0,1,4} | |||||

t _{0,2,4} | t _{0,1,2,3} | t _{0,1,2,4} | t _{0,1,3,4} | t _{0,1,2,3,4} |

- ↑ Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o3o — hix".
- ↑ 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 **24-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

In geometry, the **5-cell** is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a **C _{5}**,

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In four-dimensional geometry, the **rectified 5-cell** is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

In geometry, a **pyramid** is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a *lateral face*. It is a conic solid with polygonal base. A pyramid with an *n*-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. All pyramids are self-dual.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

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, 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 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 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 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 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.

- Gosset, T. (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions".
*Messenger of Mathematics*. Macmillan. pp. 43–. - 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.

- Johnson, N.W. (1966).

- Olshevsky, George. "Simplex".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Polytopes of Various Dimensions, Jonathan Bowers
- Multi-dimensional Glossary

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