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 Density of the star polygon. Without loss of generality, take .

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

(1) |

(2) | |||

(3) |

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).

The star polygons were first systematically studied by Thomas Bradwardine.

**References**

Coxeter, H. S. M. ``Star Polygons.'' §2.8 in *Introduction to Geometry, 2nd ed.* New York: Wiley, pp. 36-38, 1969.

Frederickson, G. ``Stardom.'' Ch. 16 in *Dissections: Plane and Fancy.* New York: Cambridge University Press,
pp. 172-186, 1997.

Savio, D. Y. and Suryanaroyan, E. R. ``Chebyshev Polynomials and Regular Polygons.'' *Amer. Math. Monthly* **100**, 657-661, 1993.

© 1996-9

1999-05-26