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).
