Regular Star Polygons

2. Simple and Composite Stars

 

It will prove helpful to adopt some conventions and notations from number theory and group theory.

Notation: Let Zn denote the integers mod n, Zn * denote the multiplicative group of units in Zn, and let E(n) denote the Euler phi-function of n, the number of positive integers less than n which are relatively prime to n. If k is an element of Zn then (k) denotes the additive cyclic subgroup generated by k.

If you compare the familiar 5 pointed star {5/2} and the Star of David {6/2} we will see that there is a difference between these two stars. If you put your pencil at any of the points in the 5 pointed star and trace the lines from point to point, you will eventually touch all the points on the star. However, if you try this in the Star of David, you will return to where you started after tracing only 3 lines. You will touch 3 of the points and not touch the other 3.

Definition 2.1: If you can get to all the points in a star by tracing the lines without taking your pencil off the paper, we will call the star a simple star. If a star is not simple, we will call the star a composite star.

The first question which can be asked is, which stars are simple and which stars are composite?

Theorem 2.2: (Coxeter [4]) {n/k} is a simple star if and only if n and k are relatively prime.

The next question is, how many simple stars are there?

Theorem 2.3: If n > 2 then there are E(n)/2 simple n pointed stars. Moreover, there is a one to one correspondence between the simple n pointed stars and the elements of the group Zn */{1, -1}.

This technique allows us to define a group structure on the set of simple n pointed stars.

Theorem 2.4: The simple n pointed stars from a group under the operation {n/a}*{n/b} = {n/ab}.

Notice that the n-gon is the identity element in the group.

It is possible to define a semigroup structure on the set of all n pointed stars using this operation. The reader can verify that the operation is well defined. However, if n is not prime, then in order for the set of stars to be closed under multiplication, we need to include the discrete star. If n is not prime then there will be two indiscrete stars whose product is discrete.

In our examples, {5/2}, {7/2}, {7/3}, {8/3}, {9/4}, {11/5}, and {12/5} are all simple stars. (6/2}, {6/3}, {8/4}, {9/3}, and {10/4} are not

Before dealing with the general case, let us agree on some more notation.

Notation: In the following discussion, n and k will be fixed but arbitrary. Let d = GCF(n, k). Let n' = n/d, and let k' = k/d.

Theorem 2.5: {n/k} consists of d {n'/k'} stars.

This is to say that a composite star can be formed by putting together simple stars.

If you trace out the star starting at the point labeled 1, you will touch all the points labeled by elements of the coset 1 + (k). The figure will be the figure you would get by rotating the star generated by the cyclic subgroup generated by k through an angle of 360/n degrees which would be another simple {n'/k'} star. Each coset in Zn /(k) will produce another simple star, and there are d cosets.

We now see that it is natural to use fractions in our notation for stars. Given an {n/k} star, if we reduce the fraction n/k to lowest terms, we get the fraction which corresponds to the simple stars which make up {n/k}. In our formula for the angles at the points of the stars, we see that if the fraction n/k reduces to the same thing as n'/k' then the {n/k} star will have the same angles as the {n'/k'} star.

Asterisks are never simple except for the {2/1} star, or regular convex 2-gon, or line segment. Theorem 2.5 tells us that an n pointed asterisk will consist of n/2 line segments. The only simple discrete star is the {1/0} star or regular convex 1-gon or point. Theorem 2.5 tells us that a discrete n pointed star consists of n points.

These ideas can be used to illustrate some ideas in number theory and abstract algebra using beautiful and interesting visualizations. First, we can illustrate the cyclic subgroups of Zn generated by k by tracing out the star {n/k}. In the process, the student can see that the GCF(n, k) will be the smallest integer of the form sk - tn for positive integers s and t. Finally, the other simple stars will illustrate cosets and help in the discussion of Lagrange's theorem.

The fact that the greatest common factor of k and n is the smallest positive integer of the form ak + bn where a and b are integers is illustrated in regular star polygons by the fact that the greatest common factor is the number of points to get from the starting point to the closest point in the cyclic subgroup generated by k modulo n. Since regular star polygons are symmetric, if there is a solution to d = ak + bn where a is positive and b is negative, there will be another solution where a is negative and b is positive.

Since regular star polygons are useful in illustrating multiplication tables in clock arithmetic, they are useful in solving bucket problems.

Section 3: The Interior Structure of a star.