Regular Star Polygons

1. Definitions and Preliminary Results

 

Definition 1.1: A star is formed by taking n points equally spaced on a circle and connecting every kth point.

The n pointed star where every kth point is connected will be denoted by {n/k}. This notation was popularized by Coxeter [4] who attributes it to Schläfli [12].

To see some examples, click on this link.

Let us first agree on some conventions. If two stars are similar figures, we will consider them to be the same star. Note that {5/2} is the same figure as {5/3}, {6/2} = {6/4}, and {7/3} = {7/4}. In general, {n/k} = {n/n-k}. Therefore, the {n/k} where k is an integer between 0 and n/2 forms a complete list of distinct n pointed stars.

There are some singular examples in this list. We will call {n/0} the discrete star. It consists of n points evenly spaced in a circle with no lines between them. The discrete stars are easily overlooked because of the modesty of their appearance. At the opposite extreme, if n is even, then {n/(n/2)} is an n pointed asterisk. For example {6/3} is a six pointed asterisk, and {8/4} is an eight pointed asterisk.

The other example, which is often considered to be degenerate, is {n/1} the regular convex n-gon, which we will simply refer to as an n-gon. The n-gon is a singular example possessing much less resplendence than its non degenerate mates and meriting special notice in the ensuing discussion, but it is the discrete star which is the truly degenerate case at the opposite extreme of the spectrum from the asterisk.

The following result was published by Coxeter [4]. Let [x] denote the greatest integer which is less than or equal to x.

Theorem 1.2: (Coxeter [4] ) There are [n/2] indiscrete n pointed stars.

The next result has to do with the angles at the points of a star. All angles will be measured in degrees.

Theorem 1.3: (Coxeter [3]) Let 0 < k < n/2. The angle at the points of an {n/k} star is 180 - 360k/n.

This formula gives us the supplement of the familiar deflection angle in an n-gon. It also correctly tells us that the angles at the points of an asterisk have 0 degrees. We cannot tell if it will give us the correct result for a discrete star because the angles do not appear to exist.

Section 2: Simple and Composite Stars