The Star Show

1
The Star Show

• From
• Robert S. Wilson
• Sonoma State University

2
Definitions and Conventions

• Definition
• A star is formed by taking N points equally
  spaced on a circle and connecting every kth point
• Denoted N/K
• Here is 8/3
• Definition If two stars produce similar figures,
  they are the same star. So 6/2 6/4, 7/3
  7/4.
• Theorem 1 n/k n/n-k.
• Theorem 2
• n/k k is an integer, 0 lt k lt n/2
• is the set of distinct n pointed stars.

3
Curious Stars

• The discrete star n/0 consists of n
  unconnected points.
• Asterisks if N is even, n/(n/2) is an n
  pointed asterisk. 6/3 is a six pointed
  asterisk, and 8/4 is an eight pointed asterisk.
  
• n/1 is a n-gon. Thus 8 evenly spaced points on
  a circle with all connected is an octagon.

4
Sample Stars
5
Observations

• Theorem 3 The angle at the points of an n/k
  star is 180 - 360k/n for 0 lt k lt n/2
• So the interior angles of 5/2 are 36 degrees

6
Yet More

• Definition 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
• Theorem 4 n/k is a simple star if and only if
  n and k are relatively prime.

7
Some Stars By Category

• Simple Stars
• 5/2, 7/2, 7/3, 8/3, 9/4, 11/5, 12/5
  
• Composite Stars
• (6/2, 6/3, 8/4, 9/3, 10/4

8
Yet More

• Definition
• For n gt 1, F(n) is the number of positive
  integers not exceeding n that are relatively
  prime to n
• So, F(5) 4
• 1, 2, 3, 4
• Theorem 5
• If n gt 2 then there are F(n)/2 simple n pointed
  stars
• Example n 5. So, F(5)/2 4/2 2.
• Recall Theorem 2 n/k where k is an integer,
  0 lt k lt n/2, is the set of distinct n pointed
  stars.
• So 5/0, 5/1, 5/2 are in the set of stars
• But only 5/1 and 5/2 are simple stars because
  only in these cases are n and k relatively prime.