Fractal Patterns in Nature: Self-Affinity vs Self-Similarity¶

This work is licensed under a Creative Commons Attribution 4.0 International License.

Welcome to the Fractal Notebooks project—an interdisciplinary exploration of fractal geometry in biological systems, from the mathematics of Mandelbrot to the metabolic scaling theory of vascular organisms.

About This Project¶

Much of the scientific literature describes the fractal-like hierarchical branching networks of vascular organisms as "self-similar." This project examines this terminology and demonstrates why these structures are more accurately described as self-affine—scaling differently in different spatial directions.

We link metabolic scaling theory to the structural traits of organisms and propose mechanistic theories for fractal dimensions from single cells to ecosystems.

Site Structure¶

The site is organised into seven tabs. The original Part numbers are kept in page titles so that cross-references and citations into the manuscripts stay stable.

Foundations (Part I)¶

Educational chapters providing mathematical and historical background:

• History of Fractals — From Cantor to Mandelbrot to modern complex dimensions
• Mathematical Foundations — Hausdorff dimension, box-counting, Hurst exponent
• Fractal Dimensionality — 1D through 4D fractals with examples
• Glossary & Equations — Quick reference for terms and formulas

Research (Parts II–V)¶

Four research manuscripts, each with a landing page and sections from Abstract to References; Part II also has supporting notebooks. They are unpublished drafts.

Part II: Metabolic Scaling Theory¶

The core research paper on self-affinity in vascular organisms:

• Distinction between self-similarity and self-affinity
• West-Brown-Enquist (WBE) model derivations
• Empirical validation using differential box-counting
• Tables of fractal dimensions for leaves, branches, and forest canopies

Part III: Spectral Geometry¶

Advanced mathematical theory connecting fractals to the Riemann zeta function:

• Fractal strings and geometric zeta functions
• Complex dimensions and log-periodic oscillations
• Applications to algorithmic sorting and spatial indexing
• The "Fractal Riemann Hypothesis"

Part IV: Biological Geometry¶

Applications of self-affine geometry to biological systems:

• DLA models for lichen, algae, and kelp
• IFS models for ferns and plant architecture
• Crown shyness and Apollonian packing in forest canopies
• Root foraging and spectral dimensions

Part V: Research Hypotheses¶

Preregistration framework for testing whether old-growth forest canopies, read from LiDAR canopy height models, carry the signatures of self-organised criticality:

• Five primary hypotheses: Optimal Filling (canopy fractal dimension), Scale Invariance (power-law lacunarity), Zeta Distribution of Gaps (gap sizes with exponent near 2), Universal Repulsion (Wigner-Dyson/GUE spacing of dominant trees) and Biotic Decoupling (weaker coupling to topography)
• Four spatial-distribution hypotheses from Lapidus's fractal strings: transect gap lengths, GUE pair correlation of canopy emergents, log-periodic lacunarity oscillations and "Riemann gas" tree density
• Methods and expected results stated before any data are analysed, with the statistical tests that will decide each hypothesis

Notebooks¶

Jupyter notebooks rendered as pages with their stored outputs: the four Part I chapter notebooks, the Part II methods and results notebooks, and four stand-alone fractal demonstrations — diffusion-limited aggregation, self-similar fractals, self-affine generators and Riemann zeta space. The notebook index also lists the research notebooks that are kept on GitHub only.

Apps (Part VI)¶

Interactive tools and demonstrations:

• React web applications — GPU-accelerated fractal visualizations
• Streamlit apps — Python-based interactive tools
• Differential box-counting — Fractal dimension estimation
• 3D visualization — Branching network rendering
• Blender DLA — 3D point-cloud generator with Geometry Nodes

Knowledge¶

Machine-readable knowledge bundles in Open Knowledge Format (OKF) v0.2: the concept ontology exported from the Fractal RAG seed, the verified proofs behind Chapter 1, and the Fractal RAG stack that indexes this corpus for retrieval agents.

For AI Agents¶

How agents and harnesses should read this site: llms.txt, a Markdown twin of every page, the OKF frontmatter on each page, and the MCP retrieval tools.

Quick Start¶

Authors¶

Tyson Lee Swetnam Institute for Computation and Data-enabled Insight, University of Arizona

Jon D Pelletier Department of Geosciences, University of Arizona

Brian J. Enquist Department of Ecology and Evolutionary Biology, University of Arizona

Key Concepts¶

Self-Similarity vs Self-Affinity¶

The \( \frac{3}{4} \) Scaling Law¶

Metabolic Scaling Theory predicts that metabolic rate \( B \) scales with body mass \( M \) as:

This quarter-power scaling emerges from the fractal geometry of resource distribution networks, where:

• Branch radius scales as \( \xi = n^{-1/2} \)
• Branch length scales as \( \gamma = n^{-1/3} \)

Since \( \xi \neq \gamma \), these networks are self-affine.

Abstract¶

Much of the scientific literature describes the fractal-like hierarchical branching networks of vascular organisms as self-similar; Part II links metabolic scaling theory to measured structural traits to show that they are self-affine, and that many fractal dimensions reported in ecology are therefore estimated with inappropriate self-similar methods. The full abstract, key findings and manuscript outline are on the Part II: Metabolic Scaling Theory landing page.

Citation¶

If you use this work, please cite:

@misc{swetnam2026fractals,
  author = {Swetnam, Tyson Lee and Pelletier, Jon D and Enquist, Brian J},
  title = {Fractal Patterns in Nature: Self-Affinity vs Self-Similarity},
  year = {2026},
  publisher = {GitHub},
  url = {https://github.com/tyson-swetnam/fractal-notebooks}
}

License¶

This project is released under the Creative Commons Attribution 4.0 International License (CC BY 4.0). See LICENSE for details.