Dynamical Systems
Fractal Geometry and Hausdorff Dimension
Mandelbrot asked in 1967: 'How long is the coast of Britain?' His answer - it depends on the ruler, and the true answer is infinite - revolutionized how irregular shapes are measured. Fractal dimension quantifies the complexity that standard integer dimension misses.
- Medical diagnostics: fractal dimension of tumor boundaries correlates with malignancy grade; FDA-cleared tools use this metric for breast cancer diagnosis
- Smartphone antennas: fractal Koch and Sierpinski antennas operate simultaneously on multiple frequency bands, enabling the compact multi-band designs in modern devices
- Image compression: Barnsley's fractal compression (1988) exploits self-similarity for 20:1 compression without the blocking artifacts of JPEG
- Turbulence: Kolmogorov's energy cascade has fractal structure; vortex tubes have dimension ~2.37, explaining energy transfer across scales
Предварительные знания
- Measure theory (basics)
- Phase spaces
- Lyapunov exponents
Hausdorff dimension
Benoit Mandelbrot in 1975 introduced the term 'fractal' and showed that the coastline of Britain has Hausdorff dimension approximately 1.25 - greater than 1 (a line) and less than 2 (a plane). Measuring with a 5 km ruler gives about 3800 km; a 1 km ruler gives about 17000 km - the length grows without bound as resolution increases.
How does Hausdorff dimension differ from topological dimension?
Topological dimension is an integer. Hausdorff dimension is defined through the scaling behaviour of covers and takes non-integer values on fractal sets (Cantor, Koch snowflake). The Cantor set has dim_H = log(2)/log(3) ≈ 0.631.
Self-similarity and iterated function systems
Box-counting method: cover the set with a grid of side epsilon and count non-empty boxes N(eps). For a fractal, N(eps) ~ eps^{-d}, so d = -lim log N(eps)/log(eps). This equals d_H for self-similar fractals and is computable in practice.
Fractal dimension as space-filling power. A line scaled by r needs r copies to cover it: d=1. A square needs r^2: d=2. The Sierpinski triangle needs 3 copies at r=2: d=log3/log2 ~ 1.585. Fractional d measures how actively the set fills the ambient space between integer dimensions.
What is the Hausdorff dimension of the Koch curve?
The Koch curve is made of 4 copies scaled by 1/3: N = 4, r = 1/3. Self-similarity dimension d = log(N)/log(1/r) = log(4)/log(3) ≈ 1.262. The value exceeds 1 (the topological dimension), reflecting the fractal complexity.
Fractals in nature and ML
Hausdorff dimension strictly exceeds topological dimension for true fractals. A curve has topological dimension 1, but its Hausdorff dimension can exceed 1 if it is infinitely wiggly.
Connection to dynamical systems: the Lorenz strange attractor has d_H ~ 2.06 (slightly above a 2D surface). The Kaplan-Yorke dimension D_KY = d_1 + (lambda_1 + ... + lambda_j)/|lambda_{j+1}| links d_H to Lyapunov exponents.
| Fractal | N copies | Scale r | d_H | Space |
|---|---|---|---|---|
| Cantor set | 2 | 1/3 | 0.631 | R^1 |
| Koch curve | 4 | 1/3 | 1.262 | R^2 |
| Sierpinski triangle | 3 | 1/2 | 1.585 | R^2 |
| Sierpinski carpet | 8 | 1/3 | 1.893 | R^2 |
| Lorenz attractor | - | - | ~2.06 | R^3 |
Where does fractal dimension appear in real applications?
The Grassberger-Procaccia correlation dimension estimates strange-attractor dimension. Box-counting is the engine of Barnsley's fractal image compression. In deep learning the fractal dimension of SGD basins correlates with generalisation.
Connections to other areas
Fractal geometry connects dynamical systems, statistical physics, and information theory through the shared language of scaling laws.
- Dynamical systems and chaos — Related topic
- Statistical physics — Related topic
- Measure theory — Related topic
- Computer graphics — Related topic
Итоги
- Hausdorff s-measure H^s(A): infimum over covers by balls of radius < delta of sum r_i^s, as delta -> 0
- Hausdorff dimension d_H: critical exponent where H^s transitions from infinity to zero
- Self-similarity formula: d = ln(N)/ln(1/r) for N copies at scale r, exact for IFS fractals
- Box-counting: d = -lim log N(eps)/log(eps), practical algorithm that equals d_H for self-similar sets
- Classic examples: Cantor d~0.631, Koch d~1.262, Sierpinski d~1.585, Lorenz d~2.06
- Kaplan-Yorke dimension relates d_H of an attractor to its Lyapunov spectrum
Why is the Hausdorff dimension of a coastline greater than 1 but less than 2?
A coastline is topologically one-dimensional (a curve), but its self-similar wiggles cause the measured length to grow as resolution increases. d_H > 1 captures this complexity. Since it does not fill the plane, d_H < 2. The fractional value quantifies how densely the curve fills space between integer dimensions.