In recent years, Roger Penrose has attracted a popular audience with thought-provoking books on physics, consciousness and the theory of computation, beginning with "The Emperor's New Mind" in 1989. But the work that established his status as one of the most remarkable mathematical physicists of our era was done in the 1960s and '70s. Along with Stephen Hawking, he developed mathematical techniques for studying the structure of the geometries that describe black holes and the Big Bang. These techniques allowed a precise characterization of the infinitely dense singularities found at the center...