Use this window to help you complete the following exercises.

2.- How many intersections of the sides of the polygon (M, N) are there?

3.- How many times do we have to move round the parent polygon to construct the polygon (M, N)?

4.- Are (M, N) and (N-M, N) the same polygon? What is the difference between them? Could all star-shaped polygons be constructed if M<N/2?

5.- Use an algorithm to explain the steps needed to be taken to work out the greatest common divisor, as described in the link  gcd .

7.- Create an algorithm to construct an N-sided regular polygon.

We can classify the stars produced as obtuse or acute depending on whether the points of the stars have an angle of greater than or less than 90º.

2.- You can see that when N is fixed the star changes shape as we increase the value of M. It becomes more pointed or less pointed. Try and use the values of M and N to describe its behaviour. You can see an example in the window for a fixed value of N, where M changes. Note that the behaviour of the shape is cyclical. Can you explain why this is so?

Of all the geometric shapes we have seen so far there are still some types of 'stars' which we haven't focused on yet, e.g. the Star of David. Some of them are illustrated in the following window. Remember to click on the 'CLEAR' button after each example. Note that we cannot always produce a star for all pairs of values of M and N. Why does the message M does not divide into N! appear in such cases?

2.- Can they be defined using a pair of numbers (M, N)? What does each value represent?

3.- Work out the angle of the point of the star referring to the values of M and N.

4.- You can see that when N is fixed the star changes shape as we increase the value of M. It becomes more pointed or less pointed. Try and use the values of M and N to describe its behaviour. You can see an example in the window for a fixed value of N, where M changes. Note that the behaviour of the shape is cyclical. Can you explain why this is so?