  Symmetry is at the heart of much of mathematics, physics, and art.
Traditional geometric symmetry groups are defined in terms of isometries of the
ambient space of a shape or pattern. If we slightly generalize this notion to
allow the isometries to operate on overlapping but non-identical metric spaces,
we obtain what we call compound symmetry groups. A natural example is that of
the groups generated by discrete rotations of overlapping disks in the plane.
Investigation of these groups reveals a new family of fractals, as well as a
rich structure that is intriguing both mathematically and artistically. We
report on our initial investigations.
