The Coastline Paradox

How long is the coast of Britain? It depends on your ruler. Measure in 100 km steps and you get roughly 2,800 km. Switch to 50 km steps and the figure climbs to about 3,400 km. Use a tiny ruler and the total runs past 8,000 km. Smaller rulers catch more bays and inlets, so the measurement never settles.

Mandelbrot published this puzzle in 1967 and gave coastlines a fractal dimension between 1 (a line) and 2 (a surface). The same logic describes mountains, clouds, and blood vessels — shapes too rough for a single whole number.