--- type: Notebook title: "Chapter 1: History of Fractal Mathematics (notebook)" description: >- Companion notebook to Chapter 1: constructs the Cantor set, Koch snowflake, Sierpinski triangle and carpet, Weierstrass function, Julia and Mandelbrot sets with NumPy and Matplotlib, and tabulates the similarity dimensions of the classical fractals. resource: https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/1_history/ tags: [notebook, part-i, classics, geometry, dimension-theory, jupyter] companion_page: https://tyson-swetnam.github.io/fractal-notebooks/foundations/history/ notebook: source: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb raw: https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/foundations/1_history.ipynb nbviewer: https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb colab: https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb kernel: Python 3 (ipykernel) environment: "environments/fractal-foundations-gpu.yml (runs on CPU: numpy, matplotlib, numba)" cells: markdown: 9 code: 9 executed: 8 figures: 6 errors: 1 outputs: stored sources: - id: ipynb resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb title: 1_history.ipynb last_modified: "2025-12-23T00:00:00Z" generated: by: process:scripts/render_notebooks.py at: "2025-12-23T00:00:00Z" status: stable --- # Chapter 1: History of Fractal Mathematics !!! info "Jupyter notebook" This page was rendered from the stored outputs of `1_history.ipynb` as of 2025-12-23; it is not executed at build time. 1 cell raised an error when the notebook was last run (ValueError). [View on GitHub](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb) · [Open in nbviewer](https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb) · [Open in Colab](https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/foundations/1_history.ipynb) · [Raw .ipynb](https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/foundations/1_history.ipynb) Kernel: Python 3 (ipykernel) · Environment: environments/fractal-foundations-gpu.yml (runs on CPU: numpy, matplotlib, numba) · Companion page: [foundations/history/](https://tyson-swetnam.github.io/fractal-notebooks/foundations/history/) This notebook provides interactive mathematical examples corresponding to the historical development of fractal geometry, from the "mathematical monsters" of the late 19th century to modern fractal theory. **Topics covered:** - Georg Cantor and the Cantor Set (1883) - Helge von Koch and the Koch Snowflake (1904) - Waclaw Sierpinski's Triangle and Carpet (1915-1916) - Gaston Julia and Julia Sets (1918) - Karl Weierstrass and Nowhere Differentiable Functions (1872) - Benoit Mandelbrot and the Mandelbrot Set (1975-1980) ```python import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import Polygon from matplotlib.collections import PatchCollection from numba import jit, prange import warnings warnings.filterwarnings('ignore') # Set default figure style plt.rcParams['figure.figsize'] = [10, 8] plt.rcParams['figure.dpi'] = 100 ``` ## 1.1 Georg Cantor and the Cantor Set (1883) Georg Cantor, working on set theory and the continuum, introduced the **Cantor set** in 1883. The construction iteratively removes the middle third of each interval: 1. Begin with $[0, 1]$ 2. Remove the open middle third $(1/3, 2/3)$, leaving $[0, 1/3] \cup [2/3, 1]$ 3. Continue *ad infinitum* **Properties:** - **Measure zero**: Total length removed equals 1 - **Uncountably infinite**: Contains as many points as the real line - **Fractal dimension**: $D = \frac{\log 2}{\log 3} \approx 0.631$ ```python def cantor_set(ax, x, y, length, depth, max_depth): """Draw the Cantor set recursively, showing all stages.""" # Draw current segment ax.plot([x, x + length], [y, y], color='black', lw=3) if depth < max_depth: new_length = length / 3 new_y = y - 0.15 # Recurse on left and right thirds cantor_set(ax, x, new_y, new_length, depth + 1, max_depth) cantor_set(ax, x + 2 * new_length, new_y, new_length, depth + 1, max_depth) # Visualize Cantor set construction fig, ax = plt.subplots(figsize=(12, 6)) ax.set_title('Cantor Set Construction (6 iterations)', fontsize=14) cantor_set(ax, 0, 1, 1, 0, 6) ax.set_xlim(-0.05, 1.05) ax.set_ylim(-0.1, 1.1) ax.set_aspect('equal') ax.axis('off') # Add annotations for i in range(7): ax.text(-0.08, 1 - i * 0.15, f'n={i}', fontsize=10, va='center') # Calculate dimension D_cantor = np.log(2) / np.log(3) print(f"Cantor Set Fractal Dimension: D = log(2)/log(3) = {D_cantor:.4f}") plt.tight_layout() plt.show() ``` ```text Cantor Set Fractal Dimension: D = log(2)/log(3) = 0.6309 ``` ![Figure 1: output of code cell 3](1_history_files/1_history_3_1.png) ## 1.2 Helge von Koch and the Koch Snowflake (1904) Swedish mathematician Helge von Koch constructed his curve in 1904 to demonstrate a continuous curve that is **nowhere differentiable**. **Construction:** 1. Begin with a line segment 2. Replace the middle third with two sides of an equilateral triangle pointing outward 3. Repeat on every segment **Properties:** - **Infinite perimeter**: Each iteration multiplies length by $4/3$ - **Finite area**: Converges to $\frac{8}{5}$ times the original triangle - **Fractal dimension**: $D = \frac{\log 4}{\log 3} \approx 1.262$ ```python def koch_curve(p1, p2, depth): """Generate Koch curve points between p1 and p2.""" if depth == 0: return [p1, p2] # Calculate intermediate points dx = p2[0] - p1[0] dy = p2[1] - p1[1] # Points dividing the segment into thirds a = p1 b = (p1[0] + dx/3, p1[1] + dy/3) d = (p1[0] + 2*dx/3, p1[1] + 2*dy/3) e = p2 # Peak of the equilateral triangle # Rotate the middle third by 60 degrees cx = b[0] + dx/6 - dy * np.sqrt(3)/6 cy = b[1] + dy/6 + dx * np.sqrt(3)/6 c = (cx, cy) # Recurse on each segment points = [] for seg in [(a, b), (b, c), (c, d), (d, e)]: pts = koch_curve(seg[0], seg[1], depth - 1) points.extend(pts[:-1]) points.append(e) return points def koch_snowflake(depth): """Generate a complete Koch snowflake.""" # Start with equilateral triangle h = np.sqrt(3) / 2 vertices = [(0, 0), (1, 0), (0.5, h)] all_points = [] for i in range(3): p1 = vertices[i] p2 = vertices[(i + 1) % 3] points = koch_curve(p1, p2, depth) all_points.extend(points[:-1]) all_points.append(all_points[0]) # Close the curve return all_points # Visualize Koch curve evolution fig, axes = plt.subplots(2, 3, figsize=(14, 8)) axes = axes.flatten() for i, depth in enumerate(range(6)): points = koch_snowflake(depth) xs = [p[0] for p in points] ys = [p[1] for p in points] axes[i].plot(xs, ys, 'b-', lw=0.5 if depth > 3 else 1) axes[i].fill(xs, ys, alpha=0.3, color='lightblue') axes[i].set_title(f'Iteration {depth}', fontsize=12) axes[i].set_aspect('equal') axes[i].axis('off') fig.suptitle('Koch Snowflake Construction', fontsize=14) # Calculate dimension D_koch = np.log(4) / np.log(3) print(f"Koch Curve Fractal Dimension: D = log(4)/log(3) = {D_koch:.4f}") print(f"Each iteration multiplies perimeter by 4/3 = {4/3:.4f}") plt.tight_layout() plt.show() ``` ```text Koch Curve Fractal Dimension: D = log(4)/log(3) = 1.2619 Each iteration multiplies perimeter by 4/3 = 1.3333 ``` ![Figure 2: output of code cell 5](1_history_files/1_history_5_1.png) ## 1.3 Waclaw Sierpinski: Triangle and Carpet (1915-1916) Polish mathematician Waclaw Sierpinski introduced two fundamental fractals. ### Sierpinski Triangle - **Construction**: Remove central inverted triangle from each remaining triangle - **$N = 3$** copies at scale $r = 1/2$ - **Dimension**: $D = \frac{\log 3}{\log 2} \approx 1.585$ ### Sierpinski Carpet - **Construction**: Divide square into $3 \times 3$ grid, remove center - **$N = 8$** copies at scale $r = 1/3$ - **Dimension**: $D = \frac{\log 8}{\log 3} \approx 1.893$ ```python def sierpinski_triangle(ax, vertices, depth, patches): """Generate Sierpinski triangle recursively.""" if depth == 0: triangle = Polygon(vertices, closed=True) patches.append(triangle) else: # Calculate midpoints midpoints = [(vertices[i] + vertices[(i + 1) % 3]) / 2 for i in range(3)] # Recurse on three corner triangles 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) # Visualize Sierpinski triangle evolution fig, axes = plt.subplots(2, 3, figsize=(14, 10)) axes = axes.flatten() vertices = np.array([[0, 0], [1, 0], [0.5, np.sqrt(3)/2]]) for i, depth in enumerate(range(6)): patches = [] sierpinski_triangle(axes[i], vertices, depth, patches) collection = PatchCollection(patches, facecolor='darkblue', edgecolor='black', linewidth=0.5) axes[i].add_collection(collection) axes[i].set_title(f'Iteration {depth} ({3**depth} triangles)', fontsize=11) axes[i].set_xlim(-0.1, 1.1) axes[i].set_ylim(-0.1, 1) axes[i].set_aspect('equal') axes[i].axis('off') fig.suptitle('Sierpinski Triangle Construction', fontsize=14) D_sierpinski = np.log(3) / np.log(2) print(f"Sierpinski Triangle Dimension: D = log(3)/log(2) = {D_sierpinski:.4f}") plt.tight_layout() plt.show() ``` ```text Sierpinski Triangle Dimension: D = log(3)/log(2) = 1.5850 ``` ![Figure 3: output of code cell 7](1_history_files/1_history_7_1.png) ```python def sierpinski_carpet(depth, size=243): """Generate Sierpinski carpet as a binary image.""" # Use size that's a power of 3 for clean division carpet = np.ones((size, size), dtype=bool) def remove_center(x, y, s): """Recursively remove center squares.""" if s < 3: return third = s // 3 # Remove center square carpet[y + third:y + 2*third, x + third:x + 2*third] = False # Recurse on 8 surrounding squares for dy in range(3): for dx in range(3): if dx == 1 and dy == 1: continue # Skip center remove_center(x + dx*third, y + dy*third, third) for d in range(depth): step = size // (3 ** (d + 1)) for y in range(0, size, step * 3): for x in range(0, size, step * 3): carpet[y + step:y + 2*step, x + step:x + 2*step] = False return carpet # Visualize Sierpinski carpet evolution fig, axes = plt.subplots(2, 3, figsize=(14, 10)) axes = axes.flatten() for i, depth in enumerate(range(1, 7)): carpet = sierpinski_carpet(depth) axes[i].imshow(carpet, cmap='binary', interpolation='nearest') axes[i].set_title(f'Iteration {depth}', fontsize=11) axes[i].axis('off') fig.suptitle('Sierpinski Carpet Construction', fontsize=14) D_carpet = np.log(8) / np.log(3) print(f"Sierpinski Carpet Dimension: D = log(8)/log(3) = {D_carpet:.4f}") plt.tight_layout() plt.show() ``` !!! failure "Cell raised ValueError" range() arg 3 must not be zero ![Figure 4: output of code cell 8](1_history_files/1_history_8_1.png) ## 1.4 Karl Weierstrass: Continuous but Nowhere Differentiable (1872) Weierstrass constructed a function that is continuous everywhere but differentiable nowhere: $$W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x)$$ where $0 < a < 1$, $b$ is an odd integer, and $ab > 1 + \frac{3\pi}{2}$. These "pathological" functions were essential precursors to fractal geometry—they showed that mathematical objects need not be smooth. ```python def weierstrass(x, a=0.5, b=7, n_terms=50): """Compute the Weierstrass function. Parameters: - a: amplitude decay (0 < a < 1) - b: frequency multiplier (odd integer) - n_terms: number of terms in the series """ result = np.zeros_like(x) for n in range(n_terms): result += (a ** n) * np.cos((b ** n) * np.pi * x) return result # Visualize Weierstrass function at different scales fig, axes = plt.subplots(2, 2, figsize=(14, 10)) # Different zoom levels ranges = [(0, 2), (0, 0.5), (0, 0.1), (0, 0.02)] titles = ['Full view [0, 2]', 'Zoom [0, 0.5]', 'Zoom [0, 0.1]', 'Zoom [0, 0.02]'] for ax, (x_min, x_max), title in zip(axes.flatten(), ranges, titles): x = np.linspace(x_min, x_max, 5000) y = weierstrass(x, a=0.5, b=7) ax.plot(x, y, 'b-', lw=0.5) ax.set_title(f'Weierstrass Function: {title}', fontsize=11) ax.set_xlabel('x') ax.set_ylabel('W(x)') ax.grid(True, alpha=0.3) fig.suptitle('Weierstrass Function: Continuous but Nowhere Differentiable\n(a=0.5, b=7)', fontsize=14) plt.tight_layout() plt.show() # The fractal dimension relates to parameters a, b = 0.5, 7 D_weierstrass = 2 + np.log(a) / np.log(b) print(f"Weierstrass function graph dimension: D ≈ 2 + log(a)/log(b) = {D_weierstrass:.4f}") ``` ## 1.5 Gaston Julia and Julia Sets (1918) French mathematician Gaston Julia studied iteration of rational functions in the complex plane. For a quadratic polynomial $f_c(z) = z^2 + c$: - The **filled Julia set** $K_c$ is the set of complex numbers $z$ whose iterates remain bounded - The **Julia set** $J_c$ is the boundary of $K_c$ **Key result:** Julia sets are connected when $c$ is in the Mandelbrot set, and totally disconnected (Cantor-like dust) otherwise. ```python @jit(nopython=True, parallel=True) def compute_julia(c_real, c_imag, x_min, x_max, y_min, y_max, width, height, max_iter): """Compute Julia set using escape-time algorithm.""" result = np.zeros((height, width)) for j in prange(height): for i in range(width): # Map pixel to complex plane x = x_min + (x_max - x_min) * i / width y = y_min + (y_max - y_min) * j / height # Iterate z -> z^2 + c zx, zy = x, y iteration = 0 while zx*zx + zy*zy < 4 and iteration < max_iter: xtemp = zx*zx - zy*zy + c_real zy = 2*zx*zy + c_imag zx = xtemp iteration += 1 result[j, i] = iteration return result # Gallery of Julia sets with different c values fig, axes = plt.subplots(2, 3, figsize=(15, 10)) axes = axes.flatten() # Famous Julia set parameters c_values = [ (-0.7, 0.27015, "Dendrite"), (-0.8, 0.156, "Rabbit"), (-0.4, 0.6, "Spiral"), (0.285, 0.01, "Siegel disk"), (-0.835, -0.2321, "Parabolic"), (-0.7269, 0.1889, "Dragon") ] for ax, (c_re, c_im, name) in zip(axes, c_values): julia = compute_julia(c_re, c_im, -1.5, 1.5, -1.5, 1.5, 600, 600, 256) ax.imshow(julia, cmap='hot', extent=[-1.5, 1.5, -1.5, 1.5]) ax.set_title(f'{name}\nc = {c_re} + {c_im}i', fontsize=10) ax.axis('off') fig.suptitle('Julia Sets: Gallery of Different c Parameters', fontsize=14) plt.tight_layout() plt.show() ``` ![Figure 5: output of code cell 12](1_history_files/1_history_12_0.png) ## 1.6 Benoit Mandelbrot and the Mandelbrot Set (1975-1980) Mandelbrot recognized that "pathological" constructions were not aberrations but the geometric language of natural complexity. He coined "fractal" in 1975. The **Mandelbrot set** $\mathcal{M}$ consists of all parameters $c$ for which the orbit of $z = 0$ under $f_c(z) = z^2 + c$ remains bounded: $$\mathcal{M} = \{c \in \mathbb{C} : |f_c^n(0)| \not\to \infty \}$$ **Properties:** - The boundary has Hausdorff dimension 2 - Contains infinitely many scaled copies of itself - Connected despite its intricate boundary ```python @jit(nopython=True, parallel=True) def compute_mandelbrot(x_min, x_max, y_min, y_max, width, height, max_iter): """Compute Mandelbrot set using escape-time algorithm.""" result = np.zeros((height, width)) for j in prange(height): for i in range(width): # Map pixel to complex plane (c value) c_real = x_min + (x_max - x_min) * i / width c_imag = y_min + (y_max - y_min) * j / height # Iterate z -> z^2 + c starting from z = 0 zx, zy = 0.0, 0.0 iteration = 0 while zx*zx + zy*zy < 4 and iteration < max_iter: xtemp = zx*zx - zy*zy + c_real zy = 2*zx*zy + c_imag zx = xtemp iteration += 1 result[j, i] = iteration return result # Full Mandelbrot set fig, axes = plt.subplots(2, 2, figsize=(14, 12)) # Different zoom levels showing self-similarity regions = [ ((-2.5, 1.0, -1.25, 1.25), "Full Mandelbrot Set"), ((-0.75, -0.65, 0.1, 0.2), "Seahorse Valley"), ((-0.745, -0.735, 0.12, 0.13), "Deeper Zoom"), ((-0.748, -0.744, 0.098, 0.102), "Mini Mandelbrot") ] for ax, ((x_min, x_max, y_min, y_max), title) in zip(axes.flatten(), regions): mandelbrot = compute_mandelbrot(x_min, x_max, y_min, y_max, 800, 800, 500) ax.imshow(mandelbrot, cmap='twilight_shifted', extent=[x_min, x_max, y_min, y_max]) ax.set_title(title, fontsize=11) ax.set_xlabel('Re(c)') ax.set_ylabel('Im(c)') fig.suptitle('Mandelbrot Set: Self-Similarity at Different Scales', fontsize=14) plt.tight_layout() plt.show() print("The Mandelbrot set boundary has Hausdorff dimension = 2") print("Small copies of the entire set appear at every scale.") ``` ![Figure 6: output of code cell 14](1_history_files/1_history_14_0.png) ```text The Mandelbrot set boundary has Hausdorff dimension = 2 Small copies of the entire set appear at every scale. ``` ## Summary: Fractal Dimensions of Classical Examples | Fractal | Year | $N$ copies | Scale $r$ | Dimension Formula | Value | |---------|------|-----------|-----------|------------------|-------| | Cantor Set | 1883 | 2 | 1/3 | $\log 2 / \log 3$ | 0.631 | | Koch Curve | 1904 | 4 | 1/3 | $\log 4 / \log 3$ | 1.262 | | Sierpinski Triangle | 1915 | 3 | 1/2 | $\log 3 / \log 2$ | 1.585 | | Sierpinski Carpet | 1916 | 8 | 1/3 | $\log 8 / \log 3$ | 1.893 | | Mandelbrot Boundary | 1980 | - | - | - | 2.000 | ```python # Calculate and display all dimensions fractals = [ ("Cantor Set", 2, 3, 1883), ("Koch Curve", 4, 3, 1904), ("Sierpinski Triangle", 3, 2, 1915), ("Sierpinski Carpet", 8, 3, 1916), ] print("Self-Similar Fractal Dimensions: D = log(N) / log(1/r)") print("=" * 60) print(f"{'Fractal':<25} {'Year':>6} {'N':>4} {'1/r':>5} {'Dimension':>12}") print("-" * 60) for name, N, inv_r, year in fractals: D = np.log(N) / np.log(inv_r) print(f"{name:<25} {year:>6} {N:>4} {inv_r:>5} {D:>12.4f}") print("=" * 60) ``` ```text Self-Similar Fractal Dimensions: D = log(N) / log(1/r) ============================================================ Fractal Year N 1/r Dimension ------------------------------------------------------------ Cantor Set 1883 2 3 0.6309 Koch Curve 1904 4 3 1.2619 Sierpinski Triangle 1915 3 2 1.5850 Sierpinski Carpet 1916 8 3 1.8928 ============================================================ ``` ## Further Reading - Mandelbrot, B. B. *The Fractal Geometry of Nature*. W. H. Freeman, 1982. - Falconer, Kenneth. *Fractal Geometry: Mathematical Foundations and Applications*. 3rd ed., Wiley, 2014. - Peitgen, H.-O. and Richter, P. H. *The Beauty of Fractals*. Springer, 1986. **Next:** See `2_mathematics.ipynb` for computational methods (box-counting, Hurst exponent, lacunarity).