Modular and fractal patterns in Pascal’s Triangle provide insights into age-related developmental cycles and aging: an observational study

Sacco, Rob G.*

Fibonacci LifeChart, Toronto, ON, Canada

*Correspondence to: Rob G. Sacco, PhD, robgsacco@gmail.com

Abstract

This study explored the hypothesis that modular and fractal patterns in Pascal’s Triangle correspond to human age-related developmental milestones. Pascal’s Triangle, known for its applications in combinatorics, reveals self-similar and fractal patterns, especially under modular transformations such as modulus 2 (forming the Sierpiński triangle). Previous research has linked these patterns to biological rhythms and developmental processes. Therefore, the present research investigated whether these mathematical symmetries align with key human developmental stages, such as growth and cognitive transitions. The study focused on (1) analyzing modular patterns under moduli 2 through 9, (2) calculating entropy and classifying modular sequences by complexity, (3) mapping these patterns to developmental stages, and (4) exploring correlations between entropy levels and stable or transitional phases. These findings suggest that the modular and fractal structures in Pascal’s Triangle may provide insights into developmental transitions and aging, with potential applications in predicting biological changes.

帕斯卡三角形中的模块和分形模式及其与年龄相关的发育周期的关系

摘要

帕斯卡三角形因其在组合学中的应用而闻名，它揭示了自相似和分形模式，尤其是在模数 2 等模数转换下（形成Sierpiński三角形）。以往的研究已将这些模式与生物节律和发育过程联系起来。在此基础上，此研究探讨了这些数学对称性是否与人类的关键发展阶段（如成长和认知转变）相一致。研究的重点是 (1) 分析模数 2 到 9 下的模数模式，(2) 计算熵并根据复杂程度对模数序列进行分类，(3) 将这些模式映射到发展阶段，以及 (4) 探索熵水平与稳定或过渡阶段之间的相关性。研究结果表明，帕斯卡三角形中的模数和分形结构可为发展过渡和衰老提供启示，并有可能应用于预测生物变化。