--- type: Index title: "Part II: Metabolic Scaling Theory and Biological Fractals" description: >- Landing page for Part II, an unpublished manuscript arguing that the branching networks of vascular organisms are self-affine rather than self-similar, with metabolic scaling theory predictions tested by differential box-counting: table of contents, supporting notebooks and authors. resource: https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/ tags: [research, paper, part-ii, metabolic-scaling, self-affinity] paper: title: "Fractal Patterns in Nature: Self-Affinity vs Self-Similarity" part_title: Metabolic Scaling Theory and Biological Fractals landing: https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/ stage: manuscript-draft authors: - name: Tyson Lee Swetnam orcid: https://orcid.org/0000-0002-6639-7181 affiliation: Institute for Computation and Data-enabled Insight, University of Arizona - name: Jon D Pelletier orcid: https://orcid.org/0000-0002-0702-2646 affiliation: Department of Geosciences, University of Arizona - name: Brian J. Enquist orcid: https://orcid.org/0000-0002-6124-7096 affiliation: Department of Ecology and Evolutionary Biology, University of Arizona citation: key: swetnam2026fractals year: 2026 publisher: GitHub url: https://github.com/tyson-swetnam/fractal-notebooks generated: by: claude-code/claude-fable-5-1 at: "2026-10-24T00:00:00Z" status: stable --- # Part II: Metabolic Scaling Theory and Biological Fractals Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. !!! warning "Status: unpublished manuscript draft" Part II is an unpublished manuscript in preparation (Swetnam, Pelletier and Enquist). Several entries in the Introduction's literature table still carry DOI placeholders, and some cross-references to supplementary figures and tables are being finalised. Every equation and table in the Methods and Results sections is reproduced in the executable notebooks listed under [Supporting Notebooks](#supporting-notebooks). **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](abstract.md) - [Key Findings](abstract.md#key-findings) ### [Introduction](introduction.md) - [Differentiating Between Self-Similarity and Self-Affinity](introduction.md#differentiating-between-self-similarity-and-self-affinity) - [Table 1: Reported fractal dimensions and techniques](introduction.md#table-1-reported-fractal-dimensions-and-techniques-measuring-fractal-behavior-in-plants-or-forests) ### [Methods](methods.md) - [Metabolic Scaling Theory](methods.md#metabolic-scaling-theory)  ·  notebook: [MST & Allometric Relationships](../notebooks/metabolic-scaling/methods/1_mst_allometry.md) - [Allometric Relationships](methods.md#allometric-relationships)  ·  notebook: [MST & Allometric Relationships](../notebooks/metabolic-scaling/methods/1_mst_allometry.md) - [Branching Ratios](methods.md#branching-ratios)  ·  notebook: [Branching Ratios](../notebooks/metabolic-scaling/methods/2_branching_ratios.md) - [Volume and Mass Dimension Predictions](methods.md#volume-and-mass-dimension-predictions)  ·  notebook: [Volume & Mass Dimension](../notebooks/metabolic-scaling/methods/3_volume_mass_dimension.md) - [Differential Box-Counting Methodology](methods.md#differential-box-counting-methodology)  ·  notebooks: [Volume & Mass Dimension](../notebooks/metabolic-scaling/methods/3_volume_mass_dimension.md), [DBC & FracLac](../notebooks/metabolic-scaling/methods/4_dbc_fraclac.md) - [FracLac Analysis Procedures](methods.md#fraclac-analysis-procedures)  ·  notebook: [DBC & FracLac](../notebooks/metabolic-scaling/methods/4_dbc_fraclac.md) ### [Results](results.md) - [Table 1: Synthetic Fractal Dimensions](results.md#table-1-synthetic-fractal-dimensions)  ·  notebook: [Synthetic Dimensions](../notebooks/metabolic-scaling/results/1_synthetic_dimensions.md) - [Table 2: Observed Leaf Mass Dimensions](results.md#table-2-observed-leaf-mass-dimensions)  ·  notebook: [Observed Leaf, Branch & Root Dimensions](../notebooks/metabolic-scaling/results/2_observed_leaf_branch_root.md) - [Table 3: Observed Branch/Root Mass Dimensions](results.md#table-3-observed-branchroot-mass-dimensions)  ·  notebook: [Observed Leaf, Branch & Root Dimensions](../notebooks/metabolic-scaling/results/2_observed_leaf_branch_root.md) - [Table 4: Forest Canopy Height Model Dimensions](results.md#table-4-forest-canopy-height-model-dimensions)  ·  notebook: [Canopy Dimensions & Statistical Validation](../notebooks/metabolic-scaling/results/3_canopy_statistical_validation.md) - [Statistical Validation](results.md#statistical-validation)  ·  notebook: [Canopy Dimensions & Statistical Validation](../notebooks/metabolic-scaling/results/3_canopy_statistical_validation.md) ### [Discussion](discussion.md) - [Closing in on Self-Affinity from the Metabolic Scaling Theory](discussion.md#closing-in-on-self-affinity-from-the-metabolic-scaling-theory) - [Demonstrations of Self-Affine Fractals](discussion.md#demonstrations-of-self-affine-fractals) - [A Fractal Theory of Ecology](discussion.md#a-fractal-theory-of-ecology) - [Fractal Dimensions of Non-Vascular Organisms](discussion.md#fractal-dimensions-of-non-vascular-organisms) - [Extended Applications: Explaining Why Vascular Life Forms Dominate](discussion.md#extended-applications-explaining-why-vascular-life-forms-dominate) - [Toward a Unified Theory](discussion.md#toward-a-unified-theory) ### [Conclusion](conclusion.md) - [Validation of Metabolic Scaling Theory](conclusion.md#validation-of-metabolic-scaling-theory) - [Implications for Ecology](conclusion.md#implications-for-ecology) - [Future Directions](conclusion.md#future-directions) ### [References](references.md) - [Key Papers on Metabolic Scaling Theory](references.md#key-papers-on-metabolic-scaling-theory) - [Fractal Geometry and Mandelbrot](references.md#fractal-geometry-and-mandelbrot) - [Allometry and Historical Background](references.md#allometry-and-historical-background) - [Fractal Analysis Methods](references.md#fractal-analysis-methods) - [Ecological Applications](references.md#ecological-applications) - [Thermodynamics and Entropy](references.md#thermodynamics-and-entropy) ### Supporting Notebooks **Methods** 1. [MST & Allometric Relationships](../notebooks/metabolic-scaling/methods/1_mst_allometry.md) — Kleiber's law $B \propto M^{3/4}$ and the power-law allometric form. 2. [Branching Ratios](../notebooks/metabolic-scaling/methods/2_branching_ratios.md) — $\xi = n^{-1/2}$, $\gamma = n^{-1/3}$, and the space-filling $D=3$ result. 3. [Volume & Mass Dimension](../notebooks/metabolic-scaling/methods/3_volume_mass_dimension.md) — Equations 11–18 and the $d=3/2$ network mass dimension. 4. [DBC & FracLac](../notebooks/metabolic-scaling/methods/4_dbc_fraclac.md) — differential box-counting validated against fBm of known dimension. **Results** 1. [Synthetic Dimensions (Table 1)](../notebooks/metabolic-scaling/results/1_synthetic_dimensions.md) — generated fractal trees, box-counting, DBC, and lacunarity. 2. [Observed Leaf, Branch & Root Dimensions (Tables 2–3)](../notebooks/metabolic-scaling/results/2_observed_leaf_branch_root.md) — reproduction of published mass dimensions with the DBC estimator on synthetic proxies. 3. [Canopy Dimensions & Statistical Validation (Table 4)](../notebooks/metabolic-scaling/results/3_canopy_statistical_validation.md) — forest CHM dimensions and a TOST equivalence test against $d=3/2$. ## Authors [Tyson Lee Swetnam](https://tysonswetnam.com){target=_blank} [![](https://orcid.org/sites/default/files/images/orcid_16x16.png)](http://orcid.org/0000-0002-6639-7181){target=_blank} Institute for Computation and Data-enabled Insight, University of Arizona [Jon D Pelletier](http://jdpellet.github.io/){target=_blank} [![](https://orcid.org/sites/default/files/images/orcid_16x16.png)](http://orcid.org/0000-0002-0702-2646) Department of Geosciences, University of Arizona Brian J. Enquist [![](https://orcid.org/sites/default/files/images/orcid_16x16.png)](http://orcid.org/0000-0002-6124-7096) Department of Ecology and Evolutionary Biology, University of Arizona