# Tridecagon

Regular tridecagon | |
---|---|

A regular tridecagon | |

Type | Regular polygon |

Edges and vertices | 13 |

Schläfli symbol | {13} |

Coxeter diagram | |

Symmetry group |
Dihedral (D_{13}), order 2×13 |

Internal angle (degrees) | ≈152.308° |

Dual polygon | Self |

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

In geometry, a **tridecagon** or **triskaidecagon** or 13-gon is a thirteen-sided polygon.

## Regular tridecagon

A *regular tridecagon* is represented by Schläfli symbol {13}.

The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length *a* is given by

## Construction

As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.

The following is an animation from a *neusis construction* of a regular tridecagon with radius of circumcircle according to Andrew M. Gleason,^{[1]} based on the angle trisection by means of the Tomahawk (light blue).

An approximate construction of a regular tridecagon using straightedge and compass is shown here.

Another possible animation of an approximate construction, also possible with using straightedge and compass.

GeoGebra: BME_{1} = 27.692307692307764°

GeoGebra: 360° ÷ 13 = 27.69230769230769°

Absolute error of BME_{1} ≈ 7.4E-14°

Example to illustrate the error:

At a circumscribed circle radius *r = 1 billion km* (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be approximately *1 mm*.

For details, see: Wikibooks: Tridecagon, construction description (German)

## Symmetry

The *regular tridecagon* has Dih_{13} symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{13}, and Z_{1}.

These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order.^{[2]} Full symmetry of the regular form is **r26** 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 **g13** subgroup has no degrees of freedom but can seen as directed edges.

## Numismatic use

The regular tridecagon is used as the shape of the Czech 20 korun coin.^{[3]}

## Related polygons

A **tridecagram** is a 13-sided star polygon. There are 5 regular forms given by Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}.

Picture | {13/2} |
{13/3} |
{13/4} |
{13/5} |
{13/6} |
---|---|---|---|---|---|

Internal angle | ≈124.615° | ≈96.9231° | ≈69.2308° | ≈41.5385° | ≈13.8462° |

### Petrie polygons

The regular tridecagon is the Petrie polygon 12-simplex:

A_{12} |
---|

12-simplex |

## References

- ↑ Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 192–194 (p. 193 Fig.4)" (PDF).
*The American Mathematical Monthly*.**95**(3): 186–194. Archived from the original (PDF) on 2015-12-19. Retrieved 24 December 2015. - ↑ 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)
- ↑ Colin R. Bruce, II, George Cuhaj, and Thomas Michael,
*2007 Standard Catalog of World Coins*, Krause Publications, 2006, ISBN 0896894290, p. 81.