Rumî is the name of a motif which can be seen in traditional Turkish decorative arts, such as tilings, window lattices, locks, book bindings, plates, rugs, and fabrics. There are many different composition styles in which rumî motifs are used. We focus on a special style called continuous rumî compositions. First, we introduce a historical background of the motif, explaining its zoomorphic and shamanistic roots, and the fundamental transformations of the post-Islamic era. Then, we determine the geometric underpinnings and construction steps of abstract rumî compositions. Finally, we present a computer program which is used to generate new designs based on our studies.