The GEOMETRY of regular star polygons was first studied systematically by Thomas Bradwardine, and then later by Johannes Kepler. Kepler defined a star polygon as an augmentation of a regular polygon, achieved by extending each side of the regular polygon until it meets a non-adjacent side, forming a vertex.
More precisely, regular star polygons can be created by connecting one vertex of a simple, regular, p-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again - i.e., for integers p and q, a star polygon is constructed by connecting every qth point out of p points, regularly spaced in a circular placement.
from the second vertex to the fourth
vertex , and from the fourth vertex
back to the first vertex . In this
example, p = 5 (5 vertices / full sides)
and q = 2 (every second vertex is
adjoined).
For instance, in a regular pentagon, a
five-pointed star can be obtained by
drawing a line from the first vertex
to the third vertex ⦻, from the third
vertex to the fifth vertex , from the
fifth vertex to the second vertex ,