# 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 Parker ^{1}. 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 numberphile ^{2}:**

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

**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?**

**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.**

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

^{2}A fantastic youtube channel in it's own right

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