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

TL;DR

The paper investigates iterative constructions of discrete Laplacians on 2D square lattices, introducing a novel alternating binary-ternary (2322-style) scheme that yields quasi-aperiodic, fractal-like figures with low density variance and strong connectivity. By contrasting 2322-style growth with classical binary (2222-style) evolution, it demonstrates richer, non-repetitive organization and links to Dekking’s non-repetitive sequences, supported by Fourier and autocorrelation analyses. The work characterizes the statistical and geometric properties of these patterns, including density measures, Sierpiński-like triangles at specific iterations, and fractal dimensions, highlighting potential applications in self-assembly, sensor networks, and biological modeling. It further discusses implications for quasicrystals, aperiodic tilings, and higher-dimensional extensions, suggesting a unifying framework for structured randomness in discrete dynamical systems.

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.

Figures (12)

• Figure 1: Examples of seed configurations.
• Figure 2: Examples of neighborhoods.
• Figure 3: Binary iterations from a Sierpiński seed and Neumann neighborhood.
• Figure 4: Sierpinski-like triangle constructions obtained as 2222-style figures at iterations $i = 2^{k+1}-1$, where $k=0,1,2,\ldots$. The same neighborhood applied to different seeds produces structurally similar results, though the fractal dimension (via box-counting) varies depending on the seed.
• Figure 5: Comparison of 2222-style and 2322-style constructions from the same seeds and neighborhoods at the iteration sequence $i = 2^{k+1}-1$, where $k=0,1,\ldots ,5$. The 2322-style construction exhibits more irregularity and disorder compared to the structured Sierpinski-like growth of 2222-style figures.