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

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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.

Document Structure

This documentation is organized into seven parts:

Part I: Foundations

Educational chapters providing mathematical and historical background:

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

Proposed experimental research framework:

  • Hypothesis 1: DLA geometry in lichens and algae
  • Hypothesis 2: Monofractal vs multifractal branching in plants
  • Hypothesis 3: Canopy gap dynamics and Zeta distributions

Part VI: Applications

Interactive tools and demonstrations:

Part VII: Notebooks

Jupyter notebooks for hands-on exploration:

Quick Start

Authors

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

Jon D Pelletier ORCID Department of Geosciences, University of Arizona

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

Key Concepts

Self-Similarity vs Self-Affinity

Property Self-Similar Self-Affine
Scaling Isotropic (same in all directions) Anisotropic (different by direction)
Example Koch snowflake Tree branches
Dimension Single value Direction-dependent
In biology Rare Common

The \( \frac{3}{4} \) Scaling Law

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

\[ B \propto M^{3/4} \]

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'. Here we examine papers where fractal-like self-similarity is incorrectly described. We also link why the hierarchical branching networks in vascular organisms are 'self-affine' rather than self-similar by linking metabolic scaling theory to these structural traits. Last, we propose a mechanistic theory of fractal dimensions for single cell through multicellular life forms.

Our results demonstrate:

  1. Dimensional analysis with appropriate self-affine mass dimension shows that many reported fractal dimensions in ecology are either incorrect or inappropriately reported.

  2. A technique for testable predictions, including a mechanistic explanation for how individual branching networks grow and fill space and how communities of organisms emerge with fractal dimensions based on MST predictions.

These results may help reveal when communities of individuals have maximized their potential to cycle energy through an ecosystem or when they have been disturbed by exogenous forces.

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.