Results¶
This work is licensed under a Creative Commons Attribution 4.0 International License.
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):
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.