Automatic generation of symmetric IFSs contracted in the hyperbolic plane

We present an effective method to automatically construct IFS which can be used to generate symmetric fractal in a hyperbolic plane [ p , q ] + . First, the basic IFS, which was composed of i contraction affine transforms (where i takes one of {2, 3, 4 and 5}), was randomly constructed according to three inequalities. Then the symmetric IFS was constructed by applying the rotational symmetry group Z n to the basic IFS. By the Q contraction or Q 1 contraction in the central lattice of a hyperbolic plane [ p , q ] + (where Q or Q 1 is a vertex or a middle point of a border of the central lattice, respectively), the n -fold symmetry fractals from the symmetric IFS were limited near the central lattice of the hyperbolic plane. The fractal was randomly transformed into the unit circle by isometries of hyperbolic geometry. All the work was automatically done. Many patterns with hyperbolic symmetry have been produced by this method.