A physical process accounts for the origin and widespread occurrence of two-armed spiral patterns with a high degree of symmetry extending from the nucleus to the outermost part of a galaxy. Self-gravitating systems, starting from a wide variety of initial conditions, settle down into twofold symmetries through a sequence of forms that were identified in numerical experiments. Twofold symmetries rarely develop directly. Instead, the systems pass through a sequence of more complicated shapes. The computer experiments and the sequence leading to twofold symmetries are described. At the end of this process, a two-armed spiral density wave with a 'grand design' has been set up in the galaxy, and the stage has been set for some mechanism, such as that of Lin and Shu (1971), to take over to assure a long lifetime for the pattern.