Fractals & Self-Similarity
Benoît Mandelbrot · 1975
"Mandelbrot showed that many natural and economic patterns repeat their rough structure at every scale — a coastline looks similarly jagged whether measured in miles or inches, so 'how long is it' has no single answer without choosing a ruler."
Mandelbrot asked in 1967 'How long is the coast of Britain?' and answered: it depends on your ruler. Zoom in and new bays appear inside bays — the same jaggedness, at every scale.
A fractal has self-similarity and fractional dimension: measured more finely, its length grows without bound, and the same statistical roughness appears at multiple magnifications. Mandelbrot's 1975 synthesis connected this geometry to markets, turbulence, and biology — volatility clusters the same way coastline does, so Gaussian smoothing hides the very risk you most need to see. The actionable insight: if a pattern is fractal, averaging across scales loses information rather than gaining precision.
Why does Mandelbrot's coastline question have no single length answer?
Read more about the topic
The explanation above is written with AI assistance. These are the originals — go to them to check it.
- The Fractal Geometry of Nature (1982)Benoît Mandelbrot
The Monty Hall Problem & Conditional Probability
"Switching doors in Monty Hall doubles your chance of winning from 1/3 to 2/3 — because the host's action of opening a losing door is not random, it leaks information about where the prize isn't."