Fibonacci Scarf and Prime Factorization Hat

What do you knit for a mathematician that is aesthetically pleasing and also intellectually interesting? Voila! For Ari's birthday I made him a Fibonacci Scarf and a Prime Factorization Hat. I gave it to him and told him that he had to find the math in each gift, and he was so excited and impressed when he figured it out!

The idea of the Fibbonaci scarf is simple: stripes according the the Fibbonaci sequence, which is a famous mathematical sequence found by adding the two previous terms to get the next term. 

ie. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. This scarf is really four short sequences, each half contains an increasing and a decreasing sequence overlapped with each other. 

The only math-y part of the prime factorization hat is the band around the bottom. I got the idea from this amazing prime factorization sweater pattern. The prime factorization of a number is simply any number broken down into it's prime factors. For example: the prime factorization of 16 is 2x2x2x2, and the prime factorization of 15 is 3x5. The band starts at one (the grey square, since one is special and isn't really prime) and goes around. Each new prime gets it's own color, and each composite (not prime) number is represented by it's prime factors. So clever. It only took him a few minutes to figure it out though.