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Fractal rep-tiles of the plane via reflections and integer matrices
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DOI：10.3389/fphy.2026.1699796.png)
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Abstract
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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.
Keywords:
algorithm
foldable rep-tiles
fractal rep-tiles
integer matrices
iterated function system
iterative methods
reflections
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