# Leonardo of Pisa: Bunny Rabbits to Bull Markets - Golden Ratio

Author(s):
Sandra Monteferrante

If you actually divide successive terms in the Fibonacci sequence, the quotients settle down to a specific value as you can see in the table below.

 1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.666... 8/5 = 1.6 13/8 = 1.625 21/13 = 1.61538... 34/21 = 1.61904 55/34 = 1.61764 89/55 = 1.61818 144/89 = 1.61798 233/144 = 1.61805 377/233 = 1.61802 610/377 = 1.61803

The quotients approach a number called the Golden Ratio whose value can be calculated algebraically. Using conventional subscript notation, the Fibonacci sequence is defined:

f1 = f2 = 1, fn+1 = fn + fn-1, n = 2, 3, ....

Now, the Golden Ratio is usually called Ø. We say fn+1/fn → Ø as n → ∞. Noting that both fn+1/fn and fn /fn-1 have the same limit Ø as n → ∞, the calculation goes like this:

 Now replace the two terms: fn+1/fn with the variable x and fn-1/fn with 1/x. This quadratic equation has the two solutions:

The first solution is the Golden Ratio Ø, and the second is just - 1/Ø !

Why is this ratio golden? That's really hard to say. It is definitely related to the pleasing nature of the proportion and the uncanny way it pops up in architecture and art; for example, we find the ratio in:

Many people claim that the ratio also appears in the Parthenon, in Athens. Unfortunately, it is not possible to determine the original measurements of the building due to its partial destruction over the centuries.

Lesson Plan 4

Sandra Monteferrante, "Leonardo of Pisa: Bunny Rabbits to Bull Markets - Golden Ratio," Convergence (August 2011)