Star Polygon


A star polygon {p/q}, with p,q positive integers, is a figure formed by connecting with straight lines every qth point out of p regularly spaced points lying on a circumference. The number q is called the polygon density of the star polygon. Without loss of generality, take q<p/2. The star polygons were first systematically studied by Thomas Bradwardine.

The circumradius of a star polygon {p/q} with (p,q)=1 and unit edge lengths is given by


and its characteristic polynomial is a factor of the resultant with respect to z of the polynomials


where T_m(z) is a Chebyshev polynomial of the first kind (Gerbracht 2008).

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)!=1, then start with the first unconnected point and repeat the procedure. Repeat until all points are connected. For (p,q)!=1, the {p/q} symbol can be factored as




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


Superposing all distinct star polygons {p/q} for a given p gives beautiful patterns such as those illustrated above. These figures can also be obtained by wrapping thread around p nails spaced equally around the circumference of a circle (Steinhaus 1999, pp. 259-260).

See also

Decagram, Hexagram, Nonagram, Octagram, Pentagram, Regular Polygon, Star of Lakshmi, Star Polyhedron, Stellation

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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. Monthly 100, 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.

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Star Polygon

Cite this as:

Weisstein, Eric W. "Star Polygon." From MathWorld--A Wolfram Web Resource.

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