FRACTAL PATTERNS IN DISCRETE LAPLACIANS: ITERATIVE CONSTRUCTION ON 2D SQUARE LATTICES

In this paper, we investigate the iterative construction of discrete Laplacians on 2D square lattices, revealing emergent fractal-like patterns shaped by modular arithmetic. While classical 2222-style iterations reproduce known structures such as the Sierpiński triangle, our alternating binary–ternary (2322-style) process produces a novel class of aperiodic figures. These display low density variance, minimal connectivity loss, and non-repetitive organization reminiscent of Dekking’s sequences. Fourier and autocorrelation analyses confirm their quasi-periodic nature, suggesting structural analogies with self-assembly processes and fractal-based design. [Formula: see text]The results highlight the structural richness of modular Laplacian dynamics and motivate further study of non-periodic discrete systems.