Science Puzzle
The Coastline Paradox
Researchers tried to measure the length of the coastline of Britain using different sized measuring sticks. When they used a 200km stick, they got one answer. When they used a 50km stick, they followed more bends and got a longer answer. When they used a 1km stick, the answer was longer still.
What happens to the measured length of a coastline as you use a smaller and smaller measuring stick?
The Answer
It grows without limit. A coastline is fractal: it has detail at every scale. Every bay contains smaller bays, every headland contains smaller headlands, every rock contains smaller bumps, all the way down to individual grains of sand and beyond.
A 200km stick skips over all bays smaller than 200km. A 1km stick catches bays a few kilometres wide. A 1m stick follows every rock. Each halving of the ruler length adds more detail and increases the total, and this process never converges to a fixed value.
This is the Coastline Paradox, identified by Benoit Mandelbrot. It is why published figures for national coastline lengths vary wildly: Norway's coastline is listed as anywhere from 25,000 to 100,000 km depending on the scale used.
The same principle applies to any fractal shape, including river lengths, mountain profiles, and blood vessel networks.
The principle: The Coastline Paradox. Fractal boundaries have detail at every scale, so their measured length grows as the ruler shrinks. There is no single "true" length for a coastline: the answer depends on the scale of measurement.