Regular Star Polygons

Robert S. Wilson

Sonoma State University

This paper comprises the results that I have been able to accumulate on regular star polygons. The results have been broken up into five sections. The first section covers some definitions and the preliminary result of how many regular star polygons there are. The second section discusses the difference between simple stars, those which can be traced out going from point to point and contacting all the points before getting back to the starting point, and composite stars, those where if you try that, you will get back to the starting point before you have touched all the points. This section contains applications to subgroups of finite cyclic groups and can be used to illustrate Lagrange's theorem in the case of cyclic groups as well as methods for solving bucket problems. In the third section it is noted that each star contains all the stars with the same number of points with a smaller index. These ideas are illustrated in the examples to which one can link from the table of contents through an examples menu page.

The fourth section provides general formulas for the angles at points in the figures and distances between intersection points on the lines that make up the figure. One of the features of studying regular star polygons which is the most entertaining is the number of ways that these problems can be attacked, and the way that through the magic of trigonometry, the formulas which are obtained by different methods can be shown to be equivalent. This is particularly true in the fifth section on areas and perimeters which culminates in the remarkable fact that the area of a regular star polygon is one half of the perimeter times the length of the apothem. This very important result in the study of regular convex polygons which is used to show that the same number can be used for pi in the circumference of a circle as in the area is also true for the non convex regular star polygons.

The page for each section contains only definitions and statements of theorems with a small amount of discussion. To find the proofs, click on the theorem number. This is a link to the proof. At the end of the proof, there is a link labeled "return to text" which will take you back to the statement of the theorem in the main page for the section. Any place you see a link from "Regular Star Polygons", it will take you back to the Table of Contents where you will find links to all of the sections, this Forward and the example menu page. Many of the calculations are presented using graphics. If you set your browser to a 12 point New York font, the transition between text and graphics will appear to be a more seamless.

I hope you have as much fun with these figures as I have.