Self-similarity, fractal dimension, and fractals in nature
Fractals are mathematical objects with a peculiar property: they look similar at every scale. Zoom in on a fractal, and you'll see patterns that resemble the whole - a property called self-similarity.
Unlike smooth geometric shapes like circles and squares, fractals have infinitely complex boundaries. The closer you look, the more detail you see - detail that never stops, no matter how far you zoom.
Zoom into a coastline and you see smaller bays within bays within bays. The total length depends on your ruler size!
A branch looks like a small tree. A fern frond's leaflet looks like a tiny fern. Nature uses fractal branching for efficiency.
Clouds have bumps on bumps on bumps. Mountain ranges contain smaller ranges. Fractal geometry captures their rough complexity.
A line is 1-dimensional. A plane is 2-dimensional. But what about the coastline of Britain? It's more than a line (it winds and curves) but less than a plane (it doesn't fill area). Its dimension is about 1.25.
How long is the coast of Britain? It depends on your ruler:
This is why we need fractal dimension - it measures how the "amount" of a shape scales with zoom level. It can be any number, not just integers!
Fractals and chaos are deeply connected. Chaotic systems often produce fractal structures:
The Lorenz attractor has fractal dimension ~2.06. Chaotic systems are drawn to these fractal shapes in their phase space.
The boundary between different behaviors is often fractal. This is why tiny changes can lead to completely different outcomes.
Iteration - Simple rules applied repeatedly create infinite complexity
Self-similarity - Parts resemble the whole at different scales
Sensitive dependence - Tiny changes lead to vastly different outcomes
Emergence - Complex behavior arises from simple rules