Tridecagon

In geometry, a tridecagon or triskaidecagon (also called a 13-gon) is a shape with thirteen sides and thirteen angles. Such shapes are part of the study of polygons, which are flat figures made by joining straight lines end to end.

Polygons like the tridecagon appear in many areas of life, from architecture to computer graphics. While a tridecagon may not be as common as a square or triangle, understanding it helps mathematicians and designers work with more complex shapes.

The word “triskaidecagon” comes from ancient Greek, where “trikaideka” means thirteen. This shows how old the study of shapes really is, linking us to ideas from thousands of years ago. Whether you see it in art, nature, or math class, the tridecagon is an interesting example of how shapes can fascinate and challenge our minds.

Regular tridecagon

A regular tridecagon is a shape with thirteen sides that are all the same length and all angles the same size. Each angle inside a regular tridecagon is about 152.308 degrees. We can find the area, or size, of a regular tridecagon if we know the length of one side, called a. The way to calculate the area is a bit complicated, but it tells us the area is about 13.1858 times a squared.

Construction

A tridecagon, or 13-sided polygon, cannot be made perfectly using just a compass and straightedge because 13 is a special kind of number called a Pierpont prime. However, people can create a tridecagon using a method called neusis, which uses special tools to measure angles more precisely. This way, mathematicians have found clever ways to draw this shape.

One famous method was shown by Andrew M. Gleason, who used a tool called the Tomahawk to help divide angles into three equal parts, making the construction of a tridecagon possible.

Symmetry

The regular tridecagon has Dih₁₃ symmetry, which means it has 26 ways to look the same. Because 13 is a prime number, it has one subgroup with dihedral symmetry: Dih₁, and two cyclic group symmetries: Z₁₃, and Z₁.

These symmetries can be seen in four ways on the tridecagon. John Conway gave these symmetries letters and group orders. The full symmetry is labeled r26, and no symmetry is labeled a1. The dihedral symmetries are split depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines pass through both edges and vertices. Cyclic symmetries are labeled as g for their central gyration orders. Only the g13 subgroup has no degrees of freedom but can be seen as directed edges.

Numismatic use

The regular tridecagon is used as the shape of the Czech 20 korun coin. This means a coin from the Czech Republic has a thirteen-sided shape, which is unusual for coins.

Related polygons

A tridecagram is a 13-sided star polygon. There are 5 regular shapes, shown with special symbols called Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is a prime number, these star shapes cannot be made from simpler shapes.

These 13-sided star shapes appear in old drawings, like the Topkapı Scroll. The regular tridecagon is also linked to a special shape called a Petrie polygon for a 12-dimensional shape called the 12-simplex.