Regular Star Polygons
A Bit of Theory...
Given equally spaced points on a circle and a step a regular star polygon is obtained by connecting each of the points to the point that is steps ahead in the cyclic ordering, when and are coprime.
Regular polygons can be seen as the special case of regular star polygons.
The notation was introduced by the Swiss mathematician Ludwig Schläfli in the first half of the 19th century.
Some Interesting Properties
Similarity Property
When and are coprime, hence a regular star polygon is generated, the intersection points of consecutive edges form a new regular -gon, that is similar to the initial regular -gon, and concentric with it.
Angles Measure Property
When and are coprime, hence a regular star polygon is generated, each interior (vertex) angle measures .
Interact with the app below to generate the inner -gons and view the measure of the interior angles.
The star pentagon is the first regular star polygon whose sum of interior (vertex) angles is . Can you find more star polygons having this property? What is their common characteristic?