Fractal Dimension Analysis of Constants π and e Using Pseudo-Optimized Maximum Window Selection for the Higuchi Algorithm

Abstract

The mathematical constants π and e are irrational and transcendental numbers whose infinite, non-repeating decimal expansions mimic randomness. This study investigates potential underlying structures within these sequences by treating their digits as discrete time-series signals. Using Higuchi’s algorithm, we calculate their fractal dimensions to quantify their inherent complexity. Because the efficacy of Higuchi’s method depends heavily on the maximum window size (tmax), we establish a pseudo-optimal formula for tmax calibrated against fractional Brownian motion of known dimensions. Finally, we enrich our analysis with random walk and power spectral density analysis, uncovering necessary empirical conditions for white-noise behavior and long-range dependence.