# 5-simplex

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°.

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 |

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.*

## Alternate names

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]

## As a configuration

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]

## Regular hexateron cartesian coordinates

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.

## Projected images

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. |

## Lower symmetry forms

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 |

## Compound

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.

## Related uniform 5-polytopes

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 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}] |
[3^{2,3,1}] |
[[3<sup>3,3,1</sup>]] | [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 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<sup>1,3,1</sup>]] = [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} |

## Notes

- Klitzing, (x3o3o3o3o - hix)
- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117

## References

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - H.S.M. Coxeter:
- Coxeter,
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) - H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) **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,

- Coxeter,
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_{n1}) - 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) x3o3o3o3o - hix".

## External links

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