Fractal rep-tiles of the plane via reflections and integer matrices

The fractal rep-tiles of the Euclidean plane considered in this article are examples of rep-tiles (tilings) with fractal boundaries. Several new examples of fractal rep-tiles are constructed using reflection transformations and integer matrices. A new class of foldable fractal rep-tiles based on general reflection mappings is introduced, and it is shown that these rep-tiles tile the plane using similitudes, including reflections, if the iterated function system (IFS) satisfies lattice tiling conditions. We prove the existence of foldable fractal 2-rep and 4-rep tiles that exhibit novel properties (chirality and aperiodicity) caused by reflection mappings. Fractal variations of foldable rep-tiles are also constructed. The fractal rep-tiles and the foldable rep-tiles presented here are in one-to-one correspondence with finite reflection groups, and this novel class of foldable rep-tiles can be lifted to construct new classes of fractal rep-tiles with roots in classical reflection groups. The images of rep-tiles are rendered using the random iteration algorithm, which is one of the popular iterative methods to generate self-similar fractals and tilings.