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Phi and the Fibonaccis

In which I distract y'all with math.

(Posted 2015-10-02 14:40:56 -0400)

Yesterday was a bad, bad day for Roseburg, Oregon, and to a only-slightly-lesser extent, for all of the United States. To give you something else to think about, here’s some math.

You may have heard of the Fibonacci sequence. It’s a sequence of integers where each entry in the sequence is the sum of the previous two. (To “seed” the sequence, the first and second numbers in the sequence are both 1.) So the sequence goes:

1, 1, 2, 3, 5, 8, 13, 21, …

and so on.

In general, we designate the first number of the sequence as \(F_1\), the second as \(F_2\), and so on. So we can say that \(F_n = F_{n-1} + F_{n-2}\).

We’ll get back to Fibonacci.For now, let’s talk about \(\phi\) (pronounced “fee”).

\(\phi\) is the Golden Ratio, supposedly the most pleasing ratio to the eye. (Seriously, it comes up a lot in architecture.) Mathematically, \(\phi = \frac{1+\sqrt{5}}{2}\).

Now, something interesting happens when you multiply \(\phi\) by itself:

\[\phi^2 = (\frac{1+\sqrt{5}}{2})^2\]

\[= (\frac{1+\sqrt{5}}{2})(\frac{1+\sqrt{5}}{2})\]

\[= \frac{1 + 2\sqrt{5} + 5}{4}\]

\[= \frac{2 + 2\sqrt{5}}{4} + \frac{4}{4}\]

\[= \phi + 1\]

So \(\phi^2 = \phi + 1\). This means:

\[\phi^3 = \phi(\phi^2) = \phi(\phi + 1) = \phi^2 + \phi = 2\phi + 1\]

I’ll spare you the remaining calculations, but you can show for yourself that:

\[\phi^4 = 3\phi + 2\]

\[\phi^5 = 5\phi + 3\]

and so on.

Do those coefficients look familiar? They all belong to the Fibonacci sequence. In fact, in general:

\[\phi^n = F_n\phi + F_{n-1}\]

In fact, if we define \(F_0 = 0\), then this holds for all \(n \ge 0\).

And one more connection: Let’s look at the ratio of each number in the Fibonacci sequence to the one just before it, starting with (F_3):

\[\frac{3}{2} = 1.5\] \[\frac{5}{3} = 1.66666…\] \[\frac{8}{5} = 1.6\] \[\frac{13}{8} = 1.625\] \[\frac{21}{13} = 1.61538…\] \[\frac{34}{21} = 1.61904…\]

and so on. Now, \(\phi = 1.61803…\), and in fact the ratio of adjacent Fibonacci numbers approaches \(\phi\) as we go to infinity. Or in mathematical terms:

\[\lim_{n\to\infty} \frac{F_n}{F_{n-1}} = \phi\]

Cool, huh?