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Peer-reviewedCC BY 4.0Fractal and Fractional

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

  • Gang Gyoo Jin1
  • Kwangcheol Shin2
  • Jongkap Ahn3
  1. 1Department of IT, Huree College of ICT, Ulaanbaatar 16061, Mongolia
  2. 2Department of Computer Science, Huree College of ICT, Ulaanbaatar 16061, Mongolia
  3. 3Training Ship Operation Center, Gyeongsang National University, Tongyeong-si 53064, Republic of Korea
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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.

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Cite

BibTeXRIS
BibTeX
@article{jin2026fractal,
  title = {{Fractal Dimension Analysis of Constants π and e Using Pseudo-Optimized Maximum Window Selection for the Higuchi Algorithm}},
  author = {Gang Gyoo Jin and Kwangcheol Shin and Jongkap Ahn},
  journal = {Fractal and Fractional},
  year = {2026},
  month = oct,
  doi = {10.3390/fractalfract10100724},
  url = {https://www.mdpi.com/2504-3110/10/10/724},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Fractal Dimension Analysis of Constants π and e Using Pseudo-Optimized Maximum Window Selection for the Higuchi Algorithm
AU  - Gang Gyoo Jin
AU  - Kwangcheol Shin
AU  - Jongkap Ahn
PY  - 2026
DA  - 2026/10/15
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10100724
UR  - https://www.mdpi.com/2504-3110/10/10/724
AB  - 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.
SN  - 2504-3110
ER  -