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Monday, April 7, 2014

Golden Ratios in Fibonacci

The magical and mysterious number: Phi
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By now we've seen some interesting and fairly unexpected places where values in the Fibonacci sequence can be found. I confess that I could spend hours dissecting even more intricate applications and examples of these numbers (don't worry, I probably still will). But for now, let's get back to the sequence itself. Here are the first 12 values in the sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Eventually, even the most ardent of mathematicians get a little bored looking at number sequences and they start playing around with patterns. Because the sequence always grows (after the first two numbers), we could try and find the ratio of any value to the one just before it. 

You may be familiar with the idea of a GEOMETRIC SEQUENCE, or one in which the ratio of consecutive elements is always the same. We call it a 'common ratio'. Below is an example of a geometric sequence in which the common ratio is 2:

1, 2, 4, 8, 16, 32, 64, 128, ...

So what happens when we look at ratios in Fibonacci? Well, at first, nothing really exciting. Here are the first few:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.67

See? Nothing really exciting ...yet. But let's look at little more closely at these values. The first ratio (1) is fairly small but the second one gets larger (2). Then the third (1.5) is smaller, followed by the fourth one (1.67) which is larger. It's as if the pattern of ratios is bouncing around, up, then down, then up a little less, then down a little less....

Let's see what happens when we keep going from the previous ratio, which was 1.67
8/5 = 1.6 (smaller)
13/8 = 1.625 (larger)
(and here I'm going to start using approximations because the ratios will not come out to be nice, clean integers, fractions, or decimals).
21/13 = 1.615 (smaller)
34/21 = 1.619 (larger)

See? The pattern continues - up, then down, then up, then down..but something else is happening at the same time. Can you see it? Not only are the values bouncing up and down, but they seem to be getting closer together. That is, each time they jump up, they don't jump up as much as they did the previous time. And each time they jump down, that jump isn't as big as the previous jump down. Here are a few more:

55/34 = 1.6176 (smaller)
89/55 = 1.61818 (larger)
144/89 = 1.617977 (smaller)

The pattern of ups and down continues, but by now, we are really closing in on what seems to be a magic number. But what is that number?

Ratios of consecutive Fibonacci numbers
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The ancient Greeks figured out what it was. It's 'phi'! If you keep finding ratios of consecutive Fibonacci numbers (and by all means, feel free to do so...), you will find that the ratios continue both bouncing around (slightly above and slightly below) and getting closer to this number, phi.

The value of the number is approximately 1.618. I say approximately, because phi is a number like pi, in that neither decimal ever ends or repeats. One interesting difference between phi and pi (besides the fact that they are different letters in the Greek alphabet) is that pi (approximately, 3.14159) is TRANSCENDENTAL. That means it is not the solution to any polynomial equation you could ever come up with. But strangely, phi is such a solution. So phi is not considered transcendental. That one might win you Final Jeopardy - might want to tuck it away for future reference. You're welcome.

What does one do with phi? Oh, we're just getting started answering that question. Stay tuned. For now, just know that it's out there, waiting for you. You can't ever actually find it, but you can get ever so close to it - kind of like true love, right?

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