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Moir\'e lattices are aperiodic systems formed by a superposition of twoperiodic lattices with a relative rotational angle. In optics, the photonicmoir\'e lattice has many appealing properties such as its ability to localizelight, thus attracting much attention on exploring features of such astructure. One fundamental research area for photonic moir\'e lattices is theproperties of eigenstates, particularly the existence of localized eigenstatesand the localization-to-delocalization transition in the energy band structure.Here we propose an accurate algorithm for the eigenproblems of aperiodicsystems by combining plane wave discretization and spectral indicatorvalidation under the higher-dimensional projection, allowing us to exploreenergy bands of fully aperiodic systems. A localization-delocalizationtransition regarding the intensity of the aperiodic potential is observed and anovel Pythagoras superposition principle for localized eigenstates of 2Dmoir\'e lattices is revealed by analyzing the relationship between theaperiodic and its corresponding periodic eigenstates. This principle shedslight on exploring the physics of localizations for moir\'e lattice.

Pythagoras Superposition Principle for Localized Eigenstates of 2D Moir\'e Lattices