Regular pentacontagon | |
---|---|

A regular pentacontagon | |

Type | Regular polygon |

Edges and vertices | 50 |

Schläfli symbol | , t |

Coxeter diagram | |

Symmetry group | Dihedral (D_{50}), order 2×50 |

Internal angle (degrees) | 172.8° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In geometry, a **pentacontagon** or **pentecontagon** or 50-gon is a fifty-sided polygon.^{[1]}^{[2]} The sum of any pentacontagon's interior angles is 8640 degrees.

A *regular pentacontagon* is represented by Schläfli symbol and can be constructed as a quasiregular truncated icosipentagon, t, which alternates two types of edges.

## Regular pentacontagon properties[edit]

One interior angle in a regular pentacontagon is 172^{4}⁄_{5}°, meaning that one exterior angle would be 7^{1}⁄_{5}°.

The area of a regular pentacontagon is (with *t* = edge length)

and its inradius is

The circumradius of a regular pentacontagon is

Since 50 = 2 × 5^{2}, a regular pentacontagon is not constructible using a compass and straightedge,^{[3]} and is not constructible even if the use of an angle trisector is allowed.^{[4]}

## Symmetry[edit]

The *regular pentacontagon* has Dih_{50} dihedral symmetry, order 100, represented by 50 lines of reflection. Dih_{50} has 5 dihedral subgroups: Dih_{25}, (Dih_{10}, Dih_{5}), and (Dih_{2}, Dih_{1}). It also has 6 more cyclic symmetries as subgroups: (Z_{50}, Z_{25}), (Z_{10}, Z_{5}), and (Z_{2}, Z_{1}), with Z_{n} representing π/*n* radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[5]} He gives **d** (diagonal) with mirror lines through vertices, **p** with mirror lines through edges (perpendicular), **i** with mirror lines through both vertices and edges, and **g** for rotational symmetry. **a1** labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular pentacontagons. Only the **g50** subgroup has no degrees of freedom but can seen as directed edges.

## Dissection[edit]

Coxeter states that every zonogon (a 2*m*-gon whose opposite sides are parallel and of equal length) can be dissected into *m*(*m*-1)/2 parallelograms.^{[6]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular pentacontagon*, *m*=25, it can be divided into 300: 12 sets of 25 rhombs. This decomposition is based on a Petrie polygon projection of a 25-cube.

## Pentacontagram[edit]

A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols , , , , , , , , and , as well as 16 compound star figures with the same vertex configuration.

Picture | { ^{50}⁄_{3}} | { ^{50}⁄_{7}} | { ^{50}⁄_{9}} | { ^{50}⁄_{11}} | ^{50}⁄_{13} |
---|---|---|---|---|---|

Interior angle | 158.4° | 129.6° | 115.2° | 100.8° | 86.4° |

Picture | { ^{50}⁄_{17} } | { ^{50}⁄_{19} } | { ^{50}⁄_{21} } | { ^{50}⁄_{23} } | |

Interior angle | 57.6° | 43.2° | 28.8° | 14.4° |

## References[edit]

**^**Gorini, Catherine A. (2009),*The Facts on File Geometry Handbook*, Infobase Publishing, p. 120, ISBN 9781438109572.**^***The New Elements of Mathematics: Algebra and Geometry**by Charles Sanders Peirce (1976), p.298***^**Constructible Polygon**^**"Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.CS1 maint: archived copy as title (link)**^****The Symmetries of Things**, Chapter 20**^**Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141