Homeomorphisms from a compact surface to itself were classified by Thurston, and he associated to each such map an algebraic integer, called the dilatation - or the stretching factor - of the map. The question of which positive algebraic integers can be realized as the constant of a pseudo-Anosov surface homeomorphism has a long history. A well-known necessary condition is that the number must be strictly greater in absolute value than all its Galois conjugates. I will describe recent work with Hyungryul Baik, and Chenxi Wu that gives sufficient conditions for an algebraic number to be a pseudo-Anosov dilatation of a compact surface. I’ll describe an explicit construction of the surface and the map when the sufficient condition is met.