Greg and Colin's Golden Ratio Extravaganza

The Golden Ratio!

The Golden Ratio is, for our intensive purposes, a mathematical constant with the purpose of facilitating the calculating of Fibonacci numbers. The Golden Proportion-observed in Fibonacci series- is the following:


But what is a Fibonacci Number? Fibonacci was an Italian mathematician whose name is accredited to the following set of numbers:

1, 1, 2, 3, 5,8, 13, 21, 34, 55, 89, 144, 233...

Notice a pattern? Well, 2 = 1+1 and 3 = I +2 so that F3 = F2 + F I or

F(N)= F(N-1)+F(N-2)

Fibonacci numbers are extremely closely related to the Golden Ratio... Take the following Fibonacci Numbers and divide the first number into the next.

1

1

2

3

5

8

13

21

34

55

1/1=1

1/2=.5

2/3=.667

3/5=.6

5/8=.625

8/13=.615

13/21=.619

21/34=.617

34/55=.618

55/89=.618

You don't know it yet, but these quotients are approaching the constant from the Golden Ratio.

Phi can be derived in the following manner: If a Golden Rectangle is constructed so that ((phi) = l/w) it follows that l = w/2 + r. Using the Pythagorean theorem, we can say that r^2 = w^2/4 + w^2 = 5w^2/4. Combining these steps, l = w/2 + (sqrt(5)(w))/2 = (1 + sqrt(5))w/2. Thus, (phi) = l/w = (1 + sqrt(5))w/2 = (1 + sqrt(5/2)) = 1.61803

= [5(1/2) +1] / 2 which is about 1.61803. For a big expanded form of Phi, click here)

Phi has many special properties one of which is Phi * Phi = Phi + 1 and another is that sqrt(Phi) = Phi - 1

F(n) = [(n - (-)-n )] / [5(1/2)].

This is what is so important about the Golden Ratio and Fibonacci numbers- they are intimately connected. Take a look at the above statement. Without knowing the value of Phi, F(n) cannot be calculated without an extreme amount of labor. And without the Greeks observing nature to determine Phi, we would have no clue as to how Fibonacci numbers could be calculated (or at least no so early in history).

You see that! Isn't that amazing? Fibonacci numbers, in a way, explain why we attach so much importance to the Golden Ratio and at the same time, the Golden Ratio clarifies Fibonacci sequences.

Colin has made a program that will generate as many Fibonacci numbers as you like. It is for any Macintosh, and you can read the code here (It's written in pascal). Click here to download the program (24k).





This image, from Surrey University shows the relationship between Fibonacci numbers and their ratios. Notice the numbers are approaching Phi.