Star Polygon

A star polygon , with positive integers, is a figure formed by connecting with straight lines every th point out of regularly spaced points lying on a circumference. The number is called the polygon density of the star polygon. Without loss of generality, take . The star polygons were first systematically studied by Thomas Bradwardine.

The circumradius of a star polygon with and unit edge lengths is given by

(1)

and its characteristic polynomial is a factor of the resultant with respect to of the polynomials

(2) (3)

where is a Chebyshev polynomial of the first kind (Gerbracht 2008).

The usual definition (Coxeter 1969) requires and to be relatively prime. However, the star polygon can also be generalized to the star figure (or "improper" star polygon) when and share a common divisor (Savio and Suryanaroyan 1993). For such a figure, if all points are not connected after the first pass, i.e., if , then start with the first unconnected point and repeat the procedure. Repeat until all points are connected. For , the symbol can be factored as

(4)

where

(5) (6)

to give figures, each rotated by radians, or .

If , a regular polygon is obtained. Special cases of include (the pentagram), (the hexagram, or star of David), (the star of Lakshmi), (the octagram), (the decagram), and (the dodecagram).

Superposing all distinct star polygons for a given gives beautiful patterns such as those illustrated above. These figures can also be obtained by wrapping thread around nails spaced equally around the circumference of a circle (Steinhaus 1999, pp. 259-260).