# Enneadecagon

Regular enneadecagon | |
---|---|

A regular enneadecagon | |

Type | Regular polygon |

Edges and vertices | 19 |

Schläfli symbol | {19} |

Coxeter diagram | |

Symmetry group |
Dihedral (D_{19}), order 2×19 |

Internal angle (degrees) | ≈161.052° |

Dual polygon | Self |

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

In geometry an **enneadecagon** or 19-gon is a nineteen-sided polygon.^{[1]} It is also known as an **enneakaidecagon** or a **nonadecagon**.^{[2]}

## Regular form

A *regular enneadecagon* is represented by Schläfli symbol {19}.

The radius of the circumcircle of the *regular enneadecagon* with side length *t* is
(angle in degrees). The area, where *t* is the edge length, is

### Construction

As 19 is a Pierpont prime but not a Fermat prime, the regular enneadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.

Another animation of an approximate construction.

Based on the unit circle r = 1 [unit of length]

- Constructed side lenght of the enneadecagon in GeoGebra = 0.329189180561468... [unit of length]
- Side lenght of the enneadecagon = = 0.329189180561467788... [unit of length]
- Absolute error of the constructed side lenght = 2.12...E-16 [unit of length]
- Constructed central angle of the enneadecagon in GeoGebra = 18.94736842105263...°
- Central angle of the enneadecagon = = 18.947368421052631578...°
- Absolute error of the constructed central angle = -1.578...E-15°

Example to illustrate the error

- At a radius r = 1 billion km (the light would need about 55 min for this distance) the absolute error of the side length constructed would be approx. 0.21 mm.

## Symmetry

The *regular enneadecagon* has Dih_{19} symmetry, order 38. Since 19 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{19}, and Z_{1}.

These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon. John Conway labels these by a letter and group order.^{[3]} Full symmetry of the regular form is **r38** and no symmetry is labeled **a1**. The dihedral symmetries are divided depending on whether they pass through vertices (**d** for diagonal) or edges (**p** for perpendiculars), and **i** when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as **g** for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the **g19** subgroup has no degrees of freedom but can seen as directed edges.

## Related polygons

A enneadecagram is a 19-sided star polygon. There are nine regular forms given by Schläfli symbols: {19/2}, {19/3}, {19/4}, {19/5}, {19/6}, {19/7}, {19/8}, and {19/9}.

Picture | {19/2} |
{19/3} |
{19/4} |
{19/5} |
---|---|---|---|---|

Interior angle | ≈142.105° | ≈123.158° | ≈104.211° | ≈85.2632° |

Picture | {19/6} |
{19/7} |
{19/8} |
{19/9} |

Interior angle | ≈66.3158° | ≈47.3684° | ≈28.4211° | ≈9.47368° |

### Petrie polygons

The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in a skew orthogonal projection:

18-simplex (18D) |

## References

- ↑ Borges, Samantha; Morgan, Matthew (2012),
*Children's Miscellany: Useless Information That's Essential to Know*, Chronicle Books, p. 110, ISBN 9781452119731. - ↑ McKinney, Sueanne; Hinton, KaaVonia (2010),
*Mathematics in the K-8 Classroom and Library*, ABC-CLIO, p. 67, ISBN 9781586835224. - ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)