I'm sure it's hiding somewhere in Concrete Mathematics (knuth/graham/patashnik), and I'm sure there are more elegant proofs, but it's not too hard to work out t

I'm sure it's hiding somewhere in Concrete Mathematics (knuth/graham/patashnik), and I'm sure there are more elegant proofs, but it's not too hard to work out this proof yourself. Here is a hint: Suppose you were at a point in the mediant algorithm where the two terms were a/b and c/d. Taking the left segment transforms this to [a/b, (a+c)/(b+d)] and taking the right segment transforms this to [(a+c)/(b+d), c/d]. Represent this as a matrix of the form [[a,b],[c,d]]. In this form, what does "take the left segment" look like in terms of a matrix operation? what does "take the right segment" look like? Can you find any useful properties (for example, how do you repeat) A convenient root matrix is [[0,1],[1,0]]