In this talk, we will take the Fourier series picture of the geometry of the circle, discussed last week, and disturb it in various ways. We will review the noncommutative torus, built from badly behaved dynamics on the circle. Although its topology is very different from an ordinary torus, its geometry is essentially the same. We will also see how the group algebra of the free group on 2 generators can be given the geometry of a ‘fractal circle’, fibering over infinitely many circles. For this purpose, trees and words will make an appearance. Throughout, we will explore the idea that derivatives are, in some sense, dual to distance + direction.