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

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.