Hyperbolic geometry can be modelled in many different ways. We will focus here on the Poincaré disk model, developed by Henri Poincaré (1854-1912) in around 1880. Poincaré did remarkable work in mathematics, though he was never actually a professor of mathematics. He was particularly interested in the relationship between mathematics, physics, and psychology. He began studying non-Euclidean geometry in detail after it appeared in his study of two apparently unrelated disciplines: differential equations and number theory. 4 See Arthur Miller's chapter in  for a discussion of Poincaré's diverse interests. Poincaré took Klein's view that geometries are generated by sets and groups of transformations on them. We consider a second model of hyperbolic geometry, the upper half-plane model, in Section 5.5.