We have seen the appearance of the "Golden Ratio" in nature and in architecture. We have seen how the ancient Greeks used the proportion of the golden ratio in their architecture and their sculpture and paintings. Also, we have observed how the spirals of the nautilus also follow this proportion. But, what is this ratio? This activity will show how to find the ratio, and how it fits into other patterns in nature.

Part A:

1. To find the ratio, begin with a rectangle. Label the length (the longer side) L, and the width (or shorter side) W. Now, create a second rectangle, inside the first, so that the length of the second rectangle is equal to the width of the first. In terms of L and W, what is the width of the second rectangle?

2. Write a proportion comparing the lengths and widths of the two rectangles.

   or       or   

What is the problem with trying to solve this equation ?

3. Since we are looking for a ratio, what might be a convenient value for W ?

4. If W = 1, the equation becomes . Solve this equation. You will need the quadratic formula. Show your work.

5. Solving a quadratic yields two answers. Can one be eliminated here ?

6. This result is the "Golden Ratio". Express the ratio in radical and decimal form.

Part B:

7. Another pattern that occurs in nature is called the Fibonacci sequence.

This sequence is 1, 2, 3, 5, 8, 13, 21, 34, ....

Generate a recurrence relation to find the nth term of this sequence.

8. If you were asked to find the 100th term of this sequence, why would that be a problem ?

9. It would be much easier to find the 100th term if we had an explicit formula. Use the method of finite differences to make a prediction about the nature of the explicit formula.

10. It becomes apparent that no common difference exists. What does this tell you about the explicit formula?

11. Look instead at the ratio between consecutive terms. What appears to be happening?

12. What conclusions can you draw from the 2 parts of this activity?

The Discrete Mathematics Project