Abstract: Conventional scene graphs use directed acyclic graphs; conventional iterated function systems use infinitely recursive definitions. We investigate scene graphs with recursive cycles for defining graphical scenes. This permits both conventional scene graphs and iterated function systems within the same framework and opens the way for other definitions not possible with either. We explore several mechanisms for limiting the implied recursion in cyclic graphs, including both global and local limits. This approach permits a range of possibilities, including scenes with carefully controlled and locally varying recursive depth. It has applications in art and design.