The Da Vinci Code and Fibonacci

So I saw The Da Vinci Code a few days ago. If you saw it, you know that they talk about Fibonacci sequences and stuff. And if you read the book, because they don't mention in it in the movie, it also talks about the "Golden Ratio" , and goes on to claim that the Fibonacci sequence and are related in a subtle way.


[Side Note: The book The Da Vinci Code actually says that exactly, which isn't true. You can see that phi is actually an irrational number.]


First off, let me tell you what the Fibonacci sequence is in case you don't know. The Fibonacci sequence is defined as: , , and for integers . That is, to get the next number in the sequence, you simply add the two previous numbers. The sequence looks something like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Now according to The Da Vinci Code, the ratio of the Fibonacci sequence approaches as n gets large, which is true.

But is there a proof? When I got home from the movies I searched the web for a proof that this ratio converges to . I found a few, but they were messy and I couldn't follow the logic. The proofs seemed flawed. I even looked at Wikipedia, and their proof just doesn't make sense to me.

So here's an easy proof:


Proof:

We want to show that

,

where is defined above.


Let:


Then:
(since for and n is large)





Now notice that:

(think about this and convince yourself that it is true)


So we get:

(since )





This now a simple quadratic that we can solve.


You might want to note the quadratic formula:
If then


In our case we get:



We have now two solutions because of the plus-or-minus. You can see that the minus version yields a negative number. But we know thatx must be positive. So we must take the positive solution:



Which is exactly , so we are done. ■


So what's the big deal around ? Good question. I have no clue. Well actually, I kinda sorta know, but not really. First of all, comes up in a lot in geometry. The diagonal of a regular pentagon, for example, is times its side length. There are also some suggestions that the human body has proportions equal to . And also comes up a lot patterns in nature (the shape of flowers for example). There are probably countless other places where comes up. But who knows. At least for sure, is a beautiful number.


P.S. No one has solved the problem in the "Euler's Formula" post. It's not that hard.