# Two-Disk Compound Symmetry Groups

## TL;DR

This paper introduces the concept of compound symmetry groups generated by overlapping disks, revealing new fractal structures and complex mathematical properties with potential artistic applications.

## Contribution

It defines and explores a novel class of symmetry groups based on overlapping metric spaces, extending traditional symmetry concepts.

## Key findings

- Discovery of new fractal structures
- Rich mathematical properties of compound groups
- Potential artistic and mathematical applications

## Abstract

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.