Quote from Don Zagier (Mathematicians: An Outer View of the Inner World):
" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful."
The recursion described here is a(n+1)=(1+a(n))/a(n-1) with a(0)=1, a(1)=3. It's fairly straightforward to show that for every positive a(0),a(1) the sequence repeats after 5 steps (and a(0)=a(1)=golden ratio is a fixed point). One can also think of this recursion as the planar map f(x,y)=((1+x)/y,x), and every point in the plane is a period-5 point of this map.
So what is the connection with the deep topics listed in the quote?