--- type: Notebook title: Foundations of Fractal Geometry (notebook) description: >- Overview notebook for the Foundations chapters with minimal examples of the Cantor set, Sierpinski triangle, Menger sponge and fractional Brownian motion, a self-similar dimension reference table, and pointers to the detailed chapter notebooks. resource: https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/fractals/ tags: [notebook, part-i, foundations, self-similarity, geometry, jupyter] companion_page: https://tyson-swetnam.github.io/fractal-notebooks/foundations/ notebook: source: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb raw: https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/foundations/fractals.ipynb nbviewer: https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb colab: https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb kernel: Python 3 environment: "environments/fractal-foundations-gpu.yml (runs on CPU: numpy, matplotlib)" cells: markdown: 7 code: 6 executed: 6 figures: 4 errors: 0 outputs: stored sources: - id: ipynb resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb title: fractals.ipynb last_modified: "2026-10-24T00:00:00Z" generated: by: process:scripts/render_notebooks.py at: "2026-10-24T00:00:00Z" status: stable --- # Foundations of Fractal Geometry !!! info "Jupyter notebook" This page was rendered from the stored outputs of `fractals.ipynb` as of 2026-10-24; it is not executed at build time. [View on GitHub](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb) · [Open in nbviewer](https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb) · [Open in Colab](https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/fractals.ipynb) · [Raw .ipynb](https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/foundations/fractals.ipynb) Kernel: Python 3 · Environment: environments/fractal-foundations-gpu.yml (runs on CPU: numpy, matplotlib) · Companion page: [foundations/](https://tyson-swetnam.github.io/fractal-notebooks/foundations/) This notebook provides an overview of the Foundations section, with quick demonstrations and links to detailed notebooks. ## Notebook Series The Foundations section is organized into four chapters corresponding to the documentation: | Notebook | Topics | Key Concepts | |----------|--------|-------------| | **[1_history.ipynb](1_history.md)** | History of Fractal Mathematics | Cantor, Koch, Sierpinski, Julia, Mandelbrot, Weierstrass | | **[2_mathematics.ipynb](2_mathematics.md)** | Mathematical Foundations | Box-counting, DBC, power laws, Hurst exponent, lacunarity | | **[3_dimensionality.ipynb](3_dimensionality.md)** | Fractal Dimensionality (1D-4D) | Pink noise, fBm, Mandelbrot/Julia, Menger, DLA | --- ## Quick Start: Essential Fractals Below are minimal examples of key fractal types. See the specialized notebooks for detailed explanations and analyses. ```python import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import Polygon from matplotlib.collections import PatchCollection plt.rcParams['figure.figsize'] = [12, 8] plt.rcParams['figure.dpi'] = 100 ``` ## 1D Fractals: Cantor Set The Cantor set (1883) iteratively removes the middle third of each interval. **Dimension:** $D = \frac{\log 2}{\log 3} \approx 0.631$ See [1_history.ipynb](1_history.md) for detailed exploration. ```python def cantor_set(ax, x, y, length, depth, max_depth): """Draw Cantor set recursively.""" ax.plot([x, x + length], [y, y], color='black', lw=3) if depth < max_depth: new_length = length / 3 cantor_set(ax, x, y - 0.12, new_length, depth + 1, max_depth) cantor_set(ax, x + 2 * new_length, y - 0.12, new_length, depth + 1, max_depth) fig, ax = plt.subplots(figsize=(12, 4)) cantor_set(ax, 0, 1, 1, 0, 6) ax.set_xlim(-0.05, 1.05) ax.set_ylim(0, 1.1) ax.set_title(f'Cantor Set (D ≈ {np.log(2)/np.log(3):.3f})', fontsize=14) ax.axis('off') plt.show() ``` ![Figure 1: output of code cell 3](fractals_files/fractals_3_0.png) ## 2D Fractals: Sierpinski Triangle The Sierpinski triangle (1915) removes the central triangle at each iteration. **Dimension:** $D = \frac{\log 3}{\log 2} \approx 1.585$ See [1_history.ipynb](1_history.md) for construction details and [3_dimensionality.ipynb](3_dimensionality.md) for chaos game implementation. ```python def sierpinski_triangle(ax, vertices, depth, patches): """Generate Sierpinski triangle.""" if depth == 0: patches.append(Polygon(vertices, closed=True)) else: midpoints = [(vertices[i] + vertices[(i + 1) % 3]) / 2 for i in range(3)] sierpinski_triangle(ax, np.array([vertices[0], midpoints[0], midpoints[2]]), depth-1, patches) sierpinski_triangle(ax, np.array([vertices[1], midpoints[1], midpoints[0]]), depth-1, patches) sierpinski_triangle(ax, np.array([vertices[2], midpoints[2], midpoints[1]]), depth-1, patches) fig, ax = plt.subplots(figsize=(8, 7)) vertices = np.array([[0, 0], [1, 0], [0.5, np.sqrt(3)/2]]) patches = [] sierpinski_triangle(ax, vertices, 6, patches) collection = PatchCollection(patches, facecolor='darkblue', edgecolor='black', linewidth=0.2) ax.add_collection(collection) ax.set_xlim(-0.1, 1.1) ax.set_ylim(-0.1, 1) ax.set_aspect('equal') ax.set_title(f'Sierpinski Triangle (D ≈ {np.log(3)/np.log(2):.3f})', fontsize=14) ax.axis('off') plt.show() ``` ![Figure 2: output of code cell 5](fractals_files/fractals_5_0.png) ## 3D Fractals: Menger Sponge The Menger sponge (1926) is the 3D analog of the Sierpinski carpet. **Dimension:** $D = \frac{\log 20}{\log 3} \approx 2.727$ See [3_dimensionality.ipynb](3_dimensionality.md) for 3D visualization. ```python from mpl_toolkits.mplot3d.art3d import Poly3DCollection def menger_cubes(x, y, z, size, depth): """Generate Menger sponge cube positions.""" if depth == 0: return [(x, y, z, size)] new_size = size / 3 cubes = [] for dx in range(3): for dy in range(3): for dz in range(3): if (dx == 1) + (dy == 1) + (dz == 1) >= 2: continue cubes.extend(menger_cubes(x + dx*new_size, y + dy*new_size, z + dz*new_size, new_size, depth-1)) return cubes def draw_cube(ax, x, y, z, size): v = np.array([[x, y, z], [x+size, y, z], [x+size, y+size, z], [x, y+size, z], [x, y, z+size], [x+size, y, z+size], [x+size, y+size, z+size], [x, y+size, z+size]]) faces = [[v[0],v[1],v[2],v[3]], [v[4],v[5],v[6],v[7]], [v[0],v[1],v[5],v[4]], [v[2],v[3],v[7],v[6]], [v[1],v[2],v[6],v[5]], [v[0],v[3],v[7],v[4]]] ax.add_collection3d(Poly3DCollection(faces, facecolors='steelblue', linewidths=0.1, edgecolors='black', alpha=0.8)) fig = plt.figure(figsize=(10, 8)) ax = fig.add_subplot(111, projection='3d') for x, y, z, size in menger_cubes(0, 0, 0, 1, 2): draw_cube(ax, x, y, z, size) ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.set_zlim(0, 1) ax.set_title(f'Menger Sponge (D ≈ {np.log(20)/np.log(3):.3f})', fontsize=14) plt.show() ``` ![Figure 3: output of code cell 7](fractals_files/fractals_7_0.png) ## Self-Affine Fractals: Fractional Brownian Motion Fractional Brownian motion (fBm) is parameterized by the Hurst exponent $H$: - $H < 0.5$: Anti-persistent (mean-reverting) - $H = 0.5$: Standard Brownian motion - $H > 0.5$: Persistent (trending) **Dimension:** $D = 2 - H$ See [2_mathematics.ipynb](2_mathematics.md) for Hurst exponent estimation and [3_dimensionality.ipynb](3_dimensionality.md) for detailed analysis. ```python def generate_fbm(n, H, seed=42): """Generate fractional Brownian motion.""" np.random.seed(seed) freqs = np.fft.rfftfreq(n) freqs[0] = 1e-10 spectrum = 1.0 / np.abs(freqs) ** ((2*H + 1) / 2) spectrum[0] = 0 phases = np.exp(2j * np.pi * np.random.rand(len(freqs))) return np.cumsum(np.fft.irfft(spectrum * phases, n)) fig, axes = plt.subplots(1, 3, figsize=(15, 4)) for i, H in enumerate([0.2, 0.5, 0.8]): fbm = generate_fbm(2000, H) axes[i].plot(fbm, 'b-', lw=0.7) behavior = 'Anti-persistent' if H < 0.5 else ('Persistent' if H > 0.5 else 'Random Walk') axes[i].set_title(f'H = {H} ({behavior})\nD = {2-H:.2f}', fontsize=11) axes[i].grid(True, alpha=0.3) fig.suptitle('Fractional Brownian Motion', fontsize=14) plt.tight_layout() plt.show() ``` ![Figure 4: output of code cell 9](fractals_files/fractals_9_0.png) ## Fractal Dimension Reference | Fractal | N copies | Scale r | Dimension D | |---------|----------|---------|-------------| | Cantor Set | 2 | 1/3 | 0.631 | | Koch Curve | 4 | 1/3 | 1.262 | | Sierpinski Triangle | 3 | 1/2 | 1.585 | | Sierpinski Carpet | 8 | 1/3 | 1.893 | | Menger Sponge | 20 | 1/3 | 2.727 | **Formula:** $D = \frac{\log N}{\log(1/r)}$ ```python print("Self-Similar Fractal Dimensions") print("=" * 50) fractals = [ ("Cantor Set", 2, 3), ("Koch Curve", 4, 3), ("Sierpinski Triangle", 3, 2), ("Sierpinski Carpet", 8, 3), ("Menger Sponge", 20, 3), ] for name, N, inv_r in fractals: D = np.log(N) / np.log(inv_r) print(f"{name:<22} D = log({N})/log({inv_r}) = {D:.4f}") ``` ```text Self-Similar Fractal Dimensions ================================================== Cantor Set D = log(2)/log(3) = 0.6309 Koch Curve D = log(4)/log(3) = 1.2619 Sierpinski Triangle D = log(3)/log(2) = 1.5850 Sierpinski Carpet D = log(8)/log(3) = 1.8928 Menger Sponge D = log(20)/log(3) = 2.7268 ``` ## Next Steps For deeper exploration: 1. **[1_history.ipynb](1_history.md)** - Historical development: Cantor, Koch, Sierpinski, Julia, Mandelbrot, Weierstrass function 2. **[2_mathematics.ipynb](2_mathematics.md)** - Measurement methods: box-counting, differential box-counting, power law analysis, Hurst exponent, lacunarity 3. **[3_dimensionality.ipynb](3_dimensionality.md)** - Examples across dimensions: 1D (pink noise, fBm), 2D (Mandelbrot/Julia, fBs), 3D (Menger, DLA), 4D (growth dynamics) --- **Related sections:** - `biological-geometry/` - Branching architectures, vascular networks, L-systems - `metabolic-scaling/` - West-Brown-Enquist theory, allometric scaling