Recursively generated geometric structures are usually something like a fractal: generally in the limit they will be true fractals. At least one recursive function, which is actually very simple, can generate a wide range of non-fractal cyclical structures with various visual characters. There are infinitely many such rosette structures, and it is by no means certain that they have all been identified. (“Rosette” is used here in the general sense of a circular rose-like pattern, rather than referring to classical rosette curves, which are related to epicycloids.) After an analysis of the mathematical structure of the recursive function, methods of searching for the rosettes that it generates are discussed.