标题： 离散拉普拉斯算子中的分形模式：二维方形格点上的迭代构造 显示英文标题

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

摘要： 我们研究了二维方形网格上离散拉普拉斯算子的迭代构造，揭示了由模算术塑造的出现的类似分形的模式。 虽然经典的2222式迭代再现了已知的结构，如谢尔宾斯基三角形，但我们的交替二进制-三进制（2322式）过程产生了一类新的非周期图形。 这些图形显示出低密度变化、最小的连通性损失和类似于德克金序列的非重复组织。 傅里叶分析和自相关分析证实了它们的准周期性质，表明它们在自组装、传感器网络和生物建模中的应用。 这些发现为离散系统中的结构随机性和分形动力学开辟了新途径。 这些发现还为探索更高维度的拉普拉斯算子构造及其在准晶、非周期平铺和随机过程中的影响提供了方向。 显示英文摘要

摘要： 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.