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In Defense of Fibonacci

The Fibonacci sequence is rather close to my heart. My grandfather showed it to me when I was very young, and it has proven a veritable fountain of discovery over the years. It is no coincidence that it was the subject of my first blog post.

However, there are those in this world who do not share my fondness. One such person is Matt Parker1. I’m a big fan of Matt’s, with his appearances in various math communication things and fantastic youtube channel, but I have one major point of contention: his position on the apparent superiority of Lucas numbers.

The issues start with this video he hosted for numberphile2:

 

After showing that (rounded) successive powers of phi yield Lucas numbers, Matt says

These are the numbers it generates if you start with [phi] and then go to the sequence. That’s why I think the Lucas numbers are vastly superior.

And I have a problem with this. Why? Because there is another, in my opinion, far more elegant way to use successive powers of phi to produce a sequence. Take a guess as to what sequence that is.

As Matt would say, let's do some working out.

We start with the first power of phi. Anything to the first power is itself, but unlike Matt, I won’t be using the decimal expansion:

Next up, we have phi squared. I demonstrated this in my other post, but…

If you don’t believe me, plug it into a calculator. Phi is roughly 1.618. This also a fairly important identity, so I will use (1) to indicate it when I use it in the future.

Now, how do we go about solving for phi cubed? Well just multiply both sides of the previous equation by phi:

From (1) we replace the phi squared with phi plus one:

Now phi to the fourth. Again we multiply by phi:

Again, we use (1):

Phi to the fifth? Multiply:

(1):

Hopefully, you’ve see what I’m doing and could calculate the next couple powers by hand.

But so what? What’s the significance here? List the powers to find out:

Look at that! Phi’s coefficient of phi to the n is equal to the nth Fibonacci number! In other words, the number after the equal sign is always a fibonacci number, and they pop up in order if you list the powers of phi in order. Is that not super cool?3 We started with powers of phi, and ended up with a sequence!

Hang on why did Matt Parker think his Lucas numbers were so cool?

These are the numbers it generates if you start with [phi] and then go to the sequence.

Oh come on.

Not only does the Fibonacci sequence arise directly from powers of phi, it does so because of its core property: its square is equal to itself plus one. This property is so fundamental that it is sometimes used as the definition of phi.

Because of this, I think it’s fair to say that the Fibonacci sequence has a relationship with phi deeper than the relationship shared by other sequences of the same form. Sure the Lucas numbers are cool and interesting for a variety of reasons, but I think “superior” is far too strong a word.


1Who has written a fantastic book that all math lovers should buy right now

2A fantastic youtube channel in it's own right

3Not to mention the constants, which are equal to F(n-1)