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

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

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).

Coxeter, H. S. M. "Star Polygons." §2.8 in Introduction
to Geometry, 2nd ed. New York: Wiley, pp. 36-38, 1969.Coxeter,
H. S. M. Regular
Polytopes, 3rd ed. New York: Dover, pp. 93-94, 1973.Fejes
Tóth, L. Regular
Figures. Oxford, England: Pergamon Press, pp. 102-103, 1964.Frederickson,
G. "Stardom." Ch. 16 in Dissections:
Plane and Fancy. New York: Cambridge University Press, pp. 172-186,
1997.Gerbracht, E. H.-A. "On the Unit Distance Embeddability
of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Magdeburg,
Germany. Nov. 15, 2008.Savio, D. Y. and Suryanaroyan, E. R.
"Chebyshev Polynomials and Regular Polygons." Amer. Math. Monthly100,
657-661, 1993.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 211 and 259-260, 1999.Williams,
R. The
Geometrical Foundation of Natural Structure: A Source Book of Design. New
York: Dover, p. 32, 1979.