AI 中文总结
研究三角格子上模拉普拉斯自动机，发现掩模几何决定长期形态，稀疏三元干预可稳定二元动力学，仅需三次干预即可引导演化至更密的自相似结构。
AI 中文摘要
我们研究了三角格子上的模拉普拉斯自动机，其演化由二元和三元模数控制。扩展了之前在方格上的研究，我们考察了格子几何如何影响长期增长、密度、碎片化以及自相似结构的出现。我们进一步研究了稀疏的三元干预是否能稳定主要的二元动力学。实验表明，掩模几何是大尺度形态的主要决定因素。完整的六边形掩模会产生周期性的密度危机和碎片化，而三角形掩模则支持持续增长，并揭示了一个由增长能力核控制的阈值现象。尽管种子对称性影响瞬态行为，但渐近形态主要继承自掩模。为了控制二元碎片化，我们研究了稀疏的发育性三元扰动，其中在原本二元的序列中插入少量精心定时的模3事件。蒙特卡洛优化表明，仅需三次干预就足以将后续的二元演化引导至显著更密的地毯状构型。该策略的有效性主要取决于干预的时机而非数量。对干预后动力学的分析表明，三元塑造并未取代二元演化，而是产生了更密的自相似结构，显著降低了危机深度，并重置了二元危机时钟的相位。结果表明，几何决定了可容许形态的族，而稀疏的发育性扰动则在该族内选择有利的长期轨迹。
英文摘要
We investigate modular Laplacian automata on triangular lattices with evolution governed by binary and ternary moduli. Extending previous studies on square lattices, we examine how lattice geometry influences long-term growth, density, fragmentation, and the emergence of self-similar structures. We further investigate whether sparse ternary interventions can stabilize predominantly binary dynamics. The experiments reveal that mask geometry is the primary determinant of large-scale morphology. Full hexagonal masks generate recurrent density crises and fragmentation, whereas triangular masks support persistent growth and reveal a threshold phenomenon governed by growth-capable nuclei. Although seed symmetry influences transient behaviour, the asymptotic morphology is inherited mainly from the mask. To control binary fragmentation, we investigate sparse developmental ternary perturbations in which a small number of carefully timed occurrences of modulus 3 are inserted into an otherwise binary sequence. A Monte Carlo optimization demonstrates that as few as three interventions are sufficient to redirect the subsequent binary evolution toward substantially denser carpet-like configurations. The effectiveness of this strategy depends primarily on the timing of the interventions rather than on their number. Analysis of the post-intervention dynamics shows that ternary shaping does not replace binary evolution. Instead, it produces denser self-similar structures, substantially reduces crisis depth, and resets the phase of the binary crisis clock. The results suggest that geometry determines the family of admissible morphologies, whereas sparse developmental perturbations select favourable long-term trajectories within that family.
Comments21 pages, 10 figures. Submitted to "Fractals"
