How Scale Choice Changes Box-Counting Fractal Dimension Estimates

How does the choice of box-size window $[\varepsilon_{\min}, \varepsilon_{\max}]$ affect the box-counting dimension estimate $D_{\text{est}}$ for finite-iteration approximations $(n = 1..6)$ of the Koch curve and the Sierpiński triangle when the grid is anchored at $(0, 0)$ and scaled to $[0, 1]^2$? Using power-of-two scales $\varepsilon_i = 2^{-i}$ for $i = 1..10$, grids anchored at $(0, 0)$, and normalized geometries in $[0, 1]^2$, we fit log-log regressions across full, coarse, fine, and sliding windows.

We report:

• slope-based estimates $D_{\text{est}} = -m$

• coefficient of determination $R^2$

• residual diagnostics (RMSE, max residual)

• absolute and relative errors against the theoretical benchmarks

  • $D_{\text{Koch}} = \ln(4) / \ln(3) \approx 1.2619$
  • $D_{\text{Sier}} = \ln(3) / \ln(2) \approx 1.5850$

Reproduce Results

1. Create a virtual environment and install dependencies

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

2. Run the end-to-end pipeline

(Produces tables and plots in results/.)

python scripts/run_analysis.py --output results --max-iter 6 --quick-density

Arguments:

• --output: Directory for output files (default: results)
• --max-iter: Maximum iteration depth (default: 6)
• --quick-density: Sample a subset of epsilons for density check (faster)
• --parallel: Run density check in parallel

3. Inspect outputs

Data tables:

• iteration_accuracy.csv: Summary of convergence metrics (D_est, Error, RMSE, Stability) for each iteration.
• *_all_residuals.csv: Detailed residual data for every epsilon and iteration.
• *_window_residuals.csv: Residuals for specific window presets (fine, coarse, full).
• *_counts.csv and *_regressions.csv: Raw box counts and regression results for each fractal/iteration.

Visualizations:

• *_residuals_scatter.png: Scatter plot of residuals vs. epsilon, colored by iteration.
• *_window_residuals_scatter.png: Scatter plot of residuals vs. epsilon, colored by window type.
• *_loglog.png: Log-log plots of box counts vs. epsilon.
• *_convergence_loglog.png: Combined log-log plots for multiple iterations.
• *_error_heatmap.png: Heatmap of relative errors across different window starts.

License

Released under the MIT License. See LICENSE for details.