---
type: Paper Section
title: Metabolic Scaling — Results
description: >-
  Results for Part II: FracLac box-counting and differential box-counting mass
  dimensions for synthetic L-system trees (Table 1), x-ray leaves (Table 2), branches
  and roots (Table 3) and LiDAR forest canopy height models (Table 4), all close to
  the MST prediction d = 3/2.
resource: https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/results/
tags: [paper, part-ii, metabolic-scaling, image-analysis, canopy-height-model, statistics]
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
sources:
  - id: fraclac
    resource: https://imagej.net/ij/plugins/fraclac/FLHelp/Introduction.htm
    title: "FracLac for ImageJ: box-counting, differential box-counting and lacunarity (Karperien, user guide)"
  - id: west1999science
    resource: https://doi.org/10.1126/science.284.5420.1677
    title: "The fourth dimension of life: fractal geometry and allometric scaling of organisms (West, Brown and Enquist 1999, Science)"
  - id: west1999nature
    resource: https://doi.org/10.1038/23251
    title: A general model for the structure and allometry of plant vascular systems (West, Brown and Enquist 1999, Nature)
  - id: mandelbrot1982
    resource: https://search.worldcat.org/isbn/0716711869
    title: The Fractal Geometry of Nature (Mandelbrot 1982, W. H. Freeman)
generated:
  by: human:tswetnam
  at: "2024-09-22T00:00:00Z"
status: draft
---

# Results

<a rel="license" href="http://creativecommons.org/licenses/by/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>.

!!! note "Supporting notebooks"
    The tables and statistics in this section are reproduced in executable Jupyter notebooks:
    [Synthetic Dimensions (Table 1)](../notebooks/metabolic-scaling/results/1_synthetic_dimensions.md),
    [Observed Leaf, Branch & Root (Tables 2–3)](../notebooks/metabolic-scaling/results/2_observed_leaf_branch_root.md), and
    [Canopy & Statistical Validation (Table 4)](../notebooks/metabolic-scaling/results/3_canopy_statistical_validation.md).

Images of different computer-generated Lindenmayer systems (i.e., fractal trees) were analyzed using the open-source image analysis software **FracLac**. First, the fractal dimension was calculated using regular box-counting, and then with the differential box-count technique. The box-count analysis estimated the objects to have fractal dimensions between \( 1.81 \pm 0.056 < D_B < 1.90 \pm 0.66 \); however, the differential box-count mass dimension was found to be lower on average, with \( D_M = 1.60 \pm 0.10 \) (Table 1).

In Tables 1-X and Figures X-X, the observed mass dimensions of x-ray leaves, branches, and roots are statistically indistinguishable from the MST and fractional Brownian motion (fBm) predictions of \( \alpha = 3/2 \).

From Equation 18, a differential mass dimension for such an image is expected to equal \( 4/3 \) rather than \( 3/2 \) (see Supplementary Information).

## Table 1: Synthetic Fractal Dimensions

Fractal mass dimension \( d_m \pm \mu \text{SE} \) and coefficient of variation (CV):

\[
\text{CV} = \frac{\sigma}{\mu}
\]

where \( \sigma \) is the standard deviation over the mean number of pixels per box. The associated lacunarity \( \Lambda \) and CV are also reported.

| **Fractal Type**          | **Pixels** | **\( d_m \pm \mu \text{SE} \)** | **\( \mu r^2 \)** | **\( d_m (\frac{\sigma}{\mu}) \)** | **\( \Lambda \)** | **\( \Lambda (\frac{\sigma}{\mu}) \)** |
|---------------------------|------------|---------------------------------|------------------|------------------------------------|-------------------|---------------------------------------|
| **Peano Curve 1 (square)** | 756,030    | 1.846 ± 0.100                   | 0.9966           | 0.0111                             | 0.0133            | 0.2128                                |
| **Peano Curve 2 (rounded)**| 1,440,000  | 1.803 ± 0.109                   | 0.9959           | 0.0161                             | 0.0713            | 0.1348                                |
| **H-fractal**              | 3,932,289  | 1.760 ± 0.124                   | 0.9945           | 0.0102                             | 0.1142            | 0.0746                                |
| **Pythagoras Tree 1**      | 5,000,000  | 1.607 ± 0.106                   | 0.9953           | 0.0030                             | 0.7003            | 0.0871                                |
| **Pythagoras Tree 2**      | 393,216    | 1.655 ± 0.075                   | 0.9976           | 0.0108                             | 0.2267            | 0.0899                                |
| **Barnsley's Fern**        | 180,000    | 1.576 ± 0.073                   | 0.9973           | 0.0116                             | 0.3787            | 0.0680                                |
| **Fibonacci Tree**         | 348,140    | 1.470 ± 0.074                   | 0.9969           | 0.0118                             | 0.8680            | 0.0657                                |

## Table 2: Observed Leaf Mass Dimensions

Observed local mass fractal dimension of five different types of leaves. Results were obtained using the **FracLac** Differential Box Count for a power series with an exponentially increasing box size factor of 0.1.

| **Image**   | **Pixels**  | **\( d_M = \ln(\mu_\varepsilon) / \ln \varepsilon \)** | **\( \mu r^2 \)** | **\( \mu \text{SE} \)** | **\( CV (\frac{\sigma}{\mu}) \)** |
|-------------|-------------|-------------------------------------------------------|--------------------|-------------------------|----------------------------------|
| **Coleus**  | 144,316     | 1.5384                                                | 0.9938             | 0.1008                  | 0.0038                           |
| **Fig**     | 315,495     | 1.4844                                                | 0.9918             | 0.1123                  | 0.0026                           |
| **Nasturtium** | 1,115,114 | 1.5525                                                | 0.9969             | 0.0880                  | 0.0010                           |
| **Ginkgo**  | 1,086,596   | 1.5135                                                | 0.9978             | 0.0705                  | 0.0020                           |
| **Fern**    | 581,196     | 1.5083                                                | 0.9968             | 0.1268                  | 0.0064                           |

## Table 3: Observed Branch/Root Mass Dimensions

Observed local mass fractal dimension of three branching networks. Results were obtained using **FracLac** Differential Box Count for a power series with an exponentially increasing box size factor of 0.1.

| **Image**       | **Pixels**  | **\( d_M = \ln(\mu_\varepsilon) / \ln \varepsilon \)** | **\( \mu r^2 \)** | **\( \mu \text{SE} \)** | **\( CV (\frac{\sigma}{\mu}) \)** |
|-----------------|-------------|-------------------------------------------------------|--------------------|-------------------------|----------------------------------|
| **Single branch**| 255,285     | 1.4946                                                | 0.9953             | 0.0850                  | 0.0024                           |
| **Maple branches**| 473,450    | 1.4775                                                | 0.9944             | 0.0920                  | 0.0057                           |
| **Maple root**   | 617,312     | 1.4549                                                | 0.9970             | 0.0668                  | 0.0016                           |

## Table 4: Forest Canopy Height Model Dimensions

Aerial LiDAR Canopy Height Models (CHM) over various forest types. Results were obtained using **FracLac** Differential Box Count for a power series with an exponentially increasing box size factor of 0.1. The predicted mass fractal dimension is \( \frac{3}{2} \) or 1.5.

| **Image**                   | **Pixels**  | **Mass Dimension \( d_M \)** | **\( \mu r^2 \)** | **\( \mu \text{SE} \)** | **\( CV (\frac{\sigma}{\mu}) \)** |
|-----------------------------|-------------|------------------------------|-------------------|-------------------------|----------------------------------|
| **Lowland Rainforest**       | 361,201     | 1.3313                       | 0.9860            | 0.1969                  | 0.0128                           |
| **Pine/Hardwood (South Carolina)** | 362,403 | 1.5223                       | 0.9898            | 0.1919                  | 0.0135                           |
| **Sierra Madre Oaks (Arizona)** | 362,404   | 1.4973                       | 0.9886            | 0.1988                  | 0.0116                           |
| **Western Ponderosa Pine (New Mexico)** | 361,802 | 1.4566                  | 0.9847            | 0.2252                  | 0.0144                           |
| **Southwest Mixed Conifer (New Mexico)** | 361,802 | 1.5319                 | 0.9869            | 0.2190                  | 0.0177                           |
| **Southwest Spruce-Fir (New Mexico)** | 361,802  | 1.5355                  | 0.9862            | 0.2255                  | 0.0139                           |

## Statistical Validation

The observed mass dimensions across all sample types show remarkable consistency with the MST prediction of \( d_m = \frac{3}{2} \). The mean observed dimension across all leaf samples was \( 1.519 \pm 0.027 \), across branch samples was \( 1.476 \pm 0.020 \), and across forest canopy samples was \( 1.479 \pm 0.080 \).

These results provide strong empirical support for the theoretical predictions of Metabolic Scaling Theory and demonstrate that the differential box-counting method is appropriate for measuring the self-affine fractal dimensions of biological branching networks.