The star polygon Star(n,d) is the geometric figure formed by connecting with line segments every dth point out of n evenly spaced points on a circle. If the points aren't all connected after the first pass, then start with the first unconnected point and repeat the procedure. Star(8,2) and Star(8,3) are illustrated below.

This project is challenging! Start early and ask many questions.

1. Obtain a copy of Winlab and experiment with star polygons. Construct Star(n,1) and Star(n,n-1) for several values of n. What is the common name given to the type of geometric figure you obtain?

2. The first person to systematically study star polygons was Thomas Bradwardine. Write a short biography of him.

3. There are several star polygons that commonly appear in art and on decorations. Find a few examples, sketch them, and name them using the notation Star(n,d).

4. Sketch the following pairs of star polygons.

   1. Star(8,3) and Star(8,5)
   2. Star(12,5) and Star(12,7)
   3. Star(15,7) and Star(15,8)
   Make a conjecture about Star(n,s) and Star(n,r) when s+r = n. Write your conjecture as a complete sentence in if-then form.

5. Each star polygon in the problem above could be completed by beginning at any point and drawing one continuous path (i.e. by never picking up your pencil). This is not the case for Star(8,2); it required 2 different paths (see the illustration above).

   1. Determine the number of paths required to complete each of the following.

      1. Star(10,2)
      2. Star(6,3)
      3. Star(16,4)
      4. Star(n,s) (Your answer will depend on n and s.)
   2. What conditions must n and s satisfy for Star(n,s) to be formed with one continuous path? Test this conjecture as you complete the next part of this problem.

   3. Let S(n) denote the number of different star polygons with n points that can be constructed with one continuous path. Complete the following table.

      n	S(n)
      3	1
      4	1
      5	2
      6	1
      7
      8
      9
      10
      11
      12
      13
      14
      15

   4. Find a simple formula for S(n) in the case when n is prime. (Hint: Double S(n) and look for a pattern.)

   5. A special function called Euler's totient function, f(n), often arises when studying divisors. Find out a little bit about this function and how it is defined. How are S(n) and f(n) related?

6. We will use the word orbit to describe a complete trip around a circle. (Do not to confuse the words orbit and path. They mean different things.) For example, we needed 1 orbit to complete a single path of Star(8,2) and 3 orbits to complete a single path of Star(8,3).

   1. Explain how the concept of least common multiple can be used to determine the number of orbits required to complete a single path of Star(n,s).
   2. Test your conjecture on Star(15,6) and Star(24,8).

7. Re-analyze your conjecture in Problem 6 after constructing Star(15,9) and Star(24,16).