Compound symmetry groups are defined as groups generated by a set of isometries of subspaces of a metric space. Here the main focus is on groups generated by discrete rotations of overlapping disks in the plane. The simplest case occurs when only two disks are involved, which are called two-disk systems. When the disks are rotated, they are partitioned into pieces. Below a certain radius of the disks, we can label the pieces and express the rotations as permutations, from which we can identify the group structure. However, after a certain radius, the group may become infinite, and we will call this radius the critical radius.

Two-disk systems can be extended to three-disk systems, and generalised to k-disk systems. Estimates of critical radii for three-disk systems are found using a brute-force approach.