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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.02126 |
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| _version_ | 1866914388880916480 |
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| author | Nowak-Kępczyk, Małgorzata |
| author_facet | Nowak-Kępczyk, Małgorzata |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices Nowak-Kępczyk, Małgorzata Dynamical Systems 2020: 05C75, 37B10, 52C23 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. |
| title | Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices |
| topic | Dynamical Systems 2020: 05C75, 37B10, 52C23 |
| url | https://arxiv.org/abs/2504.02126 |