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arXiv · 2504.02126

Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices

Małgorzata Nowak-Kępczyk
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Abstract

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 Sierpinski 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 applications in self-assembly, sensor networks, and biological modeling. The findings open new paths toward structured randomness and fractal dynamics in discrete systems. These findings also open avenues for exploring higher-dimensional Laplacian constructions and their implications in quasicrystals, aperiodic tilings, and stochastic processes.

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Małgorzata Nowak-Kępczyk. 2026-03-13. Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices. https://doi.org/10.1142/s0218348x26500714

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