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Star Polygon

\begin{figure}\begin{center}\BoxedEPSF{Star_Polygons.epsf scaled 650}\end{center}\end{figure}

A star polygon $\{p/q\}$, with $p,q$ Positive Integers, is a figure formed by connecting with straight lines every $q$th point out of $p$ regularly spaced points lying on a Circumference. The number $q$ is called the Density of the star polygon. Without loss of generality, take $q<p/2$.

The usual definition (Coxeter 1969) requires $p$ and $q$ to be Relatively Prime. However, the star polygon can also be generalized to the Star Figure (or ``improper'' star polygon) when $p$ and $q$ 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 $(p,q)\not=1$, then start with the first unconnected point and repeat the procedure. Repeat until all points are connected. For $(p,q)\not=1$, the $\{p/q\}$ symbol can be factored as

\left\{p\over q\right\}=n\left\{p'\over q'\right\},
\end{displaymath} (1)

$\displaystyle p'$ $\textstyle \equiv$ $\displaystyle {p\over n}$ (2)
$\displaystyle q'$ $\textstyle \equiv$ $\displaystyle {q\over n},$ (3)

to give $n$ $\{p'/q'\}$ figures, each rotated by $2\pi/p$ radians, or $360^\circ/p$.

If $q=1$, a Regular Polygon $\{p\}$ is obtained. Special cases of $\{p/q\}$ include $\{5/2\}$ (the Pentagram), $\{6/2\}$ (the Hexagram, or Star of David), $\{8/2\}$ (the Star of Lakshmi), $\{8/3\}$ (the Octagram), $\{10/3\}$ (the Decagram), and $\{12/5\}$ (the Dodecagram).

The star polygons were first systematically studied by Thomas Bradwardine.

See also Decagram, Hexagram, Nonagram, Octagram, Pentagram, Regular Polygon, Star of Lakshmi, Stellated Polyhedron


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.

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© 1996-9 Eric W. Weisstein