Robert Hearn
In the process of investigating two-dimensional twisty puzzles, I
stumbled onto a new kind of geometric symmetry, generalizing the
traditional notion of symmetry group. A "compound symmetry group" is a
group generated by multiple isometries whose domains overlap. A simple
example is the group generated by $n$-fold rotations within one disk,
and $m$-fold rotations within an overlapping disk. In developing this
notion and investigating its properties, I have created software which
lets me explore the space of compound symmetries, and create
mathematical artwork based on it. This artwork can also be thought of
as images from a new kind of kaleidoscope, which renders symmetries
that are traditionally forbidden.
This piece shows a portion of the image of a compound symmetry
group combining three-fold and five-fold symmetries. Both
symmetries are present here, as well as some multiples (e.g
15-fold symmetries). Necessarily, the local symmetries become
broken at larger scales — there are no repeating patterns in the
plane with these symmetries. The parameters here are chosen to be
just short of the transition to a fractal, displaying a large but
finite complexity. The colors are chosen to complement the
structure, in this case suggesting a floral pattern. The
individual regions may also be thought of as puzzle pieces in an
immensely complex two-dimensional twisty puzzle. Pieces with the
same colors may be permuted by this puzzle.