Part II: Metabolic Scaling Theory and Biological Fractals¶

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

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

Much of the ecological literature describes the hierarchical branching networks of vascular organisms as self-similar fractals and measures them with self-similar dimension estimators. This manuscript reviews that literature, argues from Metabolic Scaling Theory (MST) that such networks are self-affine because branch radii and lengths shrink at different rates (\( \xi = n^{-1/2} \) and \( \gamma = n^{-1/3} \)), and tests the resulting prediction of a mass dimension \( d = 3/2 \) with differential box-counting on synthetic L-system trees, x-ray images of leaves, branches and roots, and airborne LiDAR canopy height models. It closes by proposing a mechanistic theory of fractal dimension, from single-celled colonies to forests, in which entropy maximisation through mass transfer drives self-affine geometry.

Table of Contents¶

This Part is organized as a manuscript. Each section links to its page; the Methods and Results sections are additionally supported by executable Jupyter notebooks that reproduce every equation, figure, and table.

Abstract¶

• Key Findings

Introduction¶

• Differentiating Between Self-Similarity and Self-Affinity
• Table 1: Reported fractal dimensions and techniques

Methods¶

• Metabolic Scaling Theory  ·  notebook: MST & Allometric Relationships
• Allometric Relationships  ·  notebook: MST & Allometric Relationships
• Branching Ratios  ·  notebook: Branching Ratios
• Volume and Mass Dimension Predictions  ·  notebook: Volume & Mass Dimension
• Differential Box-Counting Methodology  ·  notebooks: Volume & Mass Dimension, DBC & FracLac
• FracLac Analysis Procedures  ·  notebook: DBC & FracLac

Results¶

• Table 1: Synthetic Fractal Dimensions  ·  notebook: Synthetic Dimensions
• Table 2: Observed Leaf Mass Dimensions  ·  notebook: Observed Leaf, Branch & Root Dimensions
• Table 3: Observed Branch/Root Mass Dimensions  ·  notebook: Observed Leaf, Branch & Root Dimensions
• Table 4: Forest Canopy Height Model Dimensions  ·  notebook: Canopy Dimensions & Statistical Validation
• Statistical Validation  ·  notebook: Canopy Dimensions & Statistical Validation

Discussion¶

• Closing in on Self-Affinity from the Metabolic Scaling Theory
• Demonstrations of Self-Affine Fractals
• A Fractal Theory of Ecology
• Fractal Dimensions of Non-Vascular Organisms
• Extended Applications: Explaining Why Vascular Life Forms Dominate
• Toward a Unified Theory

Conclusion¶

• Validation of Metabolic Scaling Theory
• Implications for Ecology
• Future Directions

References¶

• Key Papers on Metabolic Scaling Theory
• Fractal Geometry and Mandelbrot
• Allometry and Historical Background
• Fractal Analysis Methods
• Ecological Applications
• Thermodynamics and Entropy

Supporting Notebooks¶

Methods

1. MST & Allometric Relationships — Kleiber's law \(B \propto M^{3/4}\) and the power-law allometric form.
2. Branching Ratios — \(\xi = n^{-1/2}\), \(\gamma = n^{-1/3}\), and the space-filling \(D=3\) result.
3. Volume & Mass Dimension — Equations 11–18 and the \(d=3/2\) network mass dimension.
4. DBC & FracLac — differential box-counting validated against fBm of known dimension.

Results

1. Synthetic Dimensions (Table 1) — generated fractal trees, box-counting, DBC, and lacunarity.
2. Observed Leaf, Branch & Root Dimensions (Tables 2–3) — reproduction of published mass dimensions with the DBC estimator on synthetic proxies.
3. Canopy Dimensions & Statistical Validation (Table 4) — forest CHM dimensions and a TOST equivalence test against \(d=3/2\).

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