Abstract
Moiré patterns arise from the superposition of periodic lattices with a relative rotation or translation, producing emergent periodicities at larger scales. From a mathematical perspective, these patterns represent remarkable examples of discrete symmetries in the plane, where the commensurability among two or more lattices gives rise to higher-order supercells. While the case of two-layer systems with square or hexagonal symmetry has been analyzed in previous works, a general recursive description for multilayer configurations has not been systematically established. In this paper, we present a formalism based on difference equations that generates Moiré supercells recursively for an arbitrary number of layers. The method applies to both square and hexagonal Bravais lattices, capturing the exponential growth of the supercell area while preserving the characteristic symmetry axes of each lattice. Numerical examples illustrate how the formalism provides a unified and compact description of commensurability conditions, enabling the calculation of intrinsic structural properties such as atomic density and specific area in two-dimensional materials with fixed bond length. These results highlight the role of discrete symmetry in governing the large-scale organization of Moiré patterns and provide a systematic mathematical framework for their extension to multilayer systems.
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