this paper is to explain these phenomena. In Section 3, we prove that G  is periodic if and only if  is rational, and find its shortest period. In Section 4, we describe the geometrical structure of G  . In Section 5, we explain why secondary systems of rings arise, and where they are located. In Section 6, we show that one can find a rational approximation of  by visual inspection of G  and an elementary calculation. Section 7 concludes with some generalizations. Dewdney [1986] has discussed similar patterns, but to my knowledge there has been no mathematical treatment of them. 2. NOTATION AND CONVENTION