The coastline paradox
How long is India's coastline? It depends on the length of your ruler.
Lewis Fry Richardson, 1950s; Benoit Mandelbrot, 1967
Two surveyors, one coast
Two surveyors, Neha and Joseph, are given the same job: measure the coast of Kerala. Neha uses a measuring stick 100 kilometres long, swinging it from headland to headland. She gets her answer quickly.
Joseph uses a stick just 1 kilometre long. With every step he follows the bays and river mouths that Neha's long stick cut straight across. His answer comes out much longer than hers.
Their boss, annoyed, sends a third surveyor with a 1-metre stick, who follows every rock and every little inlet. Her answer is longer still. A fourth surveyor, measuring round every pebble with a 1-centimetre ruler, gets an even bigger number.
The boss throws up his hands. "The coast isn't growing," he says. "So which one of you is right?"
The puzzle
The shorter your ruler, the longer the coastline you measure, with no sign of settling on one true length.
See it for yourself
What's really going on
Smooth shapes like circles settle down: measure with smaller and smaller rulers and you approach one answer. Coastlines aren't smooth. They have wiggles inside wiggles inside wiggles, at every scale. Each smaller ruler catches a new level of detail, so the length keeps growing.
In 1967 Benoit Mandelbrot used this to introduce the idea of fractals: shapes with detail at every scale. A coastline's 'roughness' can be measured by a number called its fractal dimension, somewhere between 1 (a line) and 2 (a filled area). That is why different official sources give quite different lengths for the same coast: they used different rulers.
Think about it
In the animation, each step multiplies the length by 4/3. What happens to the length if you keep going forever?
More space and infinity paradoxes
- Olbers' paradox: why is the night sky dark?If the universe is full of stars, every direction should end on one. So why isn't the night sky blazing?
- The Fermi paradoxThe galaxy is old and huge. So where is everybody?
- Hilbert's hotelA hotel with infinitely many rooms is full. A new guest arrives. No problem.