At this stage we take a radical new view of the theory, turning from purely algebraic methods to techniques inspired by geometry. This approach requires a different attitude of mind from the reader, in which formal ideas are built on a visual foundation. We begin with basic properties of lattices: subsets of Rn which in some sense generalize the way Z is embedded in R. We characterize lattices topologically as the discrete subgroups of Rn. We introduce the fundamental domain and quotient torus corresponding to a lattice and relate the two concepts. Finally we define a concept of volume for subsets of the quotient torus.