### 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