Regular Star Polygons

3. The Interior Structure of a Star

Notice that in the {7/3} star,

we see a {7/2} star inscribed in the circle. The {7/2} star is in the orientation which is commonly used by law enforcement agencies.

This illustrates the following result.

Theorem 3.1: (Coxeter [4]) If 1 < k < n/2, then inside an {n/k} star is an {n/k-1} star.

Theorem 3.2: (Coxeter [4] ) If 1 < k < n/2, then for all 1 < j < k there will be an {n/j} star inside {n/k}.

Example: In this example we look at the stars inside the {15,7} star.

While there are no other indiscrete stars inside an n-gon each {n/j} star where j is between 1 and k has an n-gon inside of it, so the n-gons play a role of central importance in this theory. If the discrete star and the asterisk are corresponding degenerate cases at opposite ends of the spectrum, then the star which corresponds to the n-gon at the opposite end of the spectrum would be the {n/[(n-1)/2]} star in that the n-gon is contained in every indiscrete n pointed star except for the asterisk, and the {n/[(n-1)/2]} star contains every n pointed star except for the asterisk.

Remark: If n is prime, then all n pointed stars are simple. But if n is not prime then one will generally find composite stars inside simple stars and simple stars inside composite stars.

Noticing that stars are made up of concentric rings of points will make it easier for us to analyze their structure.

Theorem 3.3: If 1 < k < n/2, then, in the {n/k} star, the lines intersect at nk points.

After determining how many points there are in the figures, it seems reasonable to ask how many regions there are and what types of figures these regions would be.

Theorem 3.4: If 1 < k < n/2, then in an {n/k} star there are 1 + nk - n regions. They consist of

1 regular convex n-gon, and

if k > 1, there will be n triangles, and

n(k - 2) quadrilaterals.

Now that we know how many points and regions are in the figure, the next question which we might ask is how many edges are there to these regions.

Theorem 3.5: There are 2n(k - 1) + n edges in a {n/k} star.

Section: 4. Angles and Distances