A consolidation of mature mathematical subjects like geometry, linear algebra, group theory, complex analysis and Galois theory into one source, this simple, easy-to-follow text develops deep connections between seemingly unrelated areas in mathematics. It updates Felix Klein's "Lectures on the Icosahedron and Equations of the Fifth Degree", and Peter Doyle's and Curt McMullen's "Solving the quintic by iteration." It provides an active approach to learning, and presents familiar subjects in a nonredundant, forward looking fashion.