Somewhere in the early 1950s, a mathematician named Lewis Fry Richardson was trying to figure out whether countries that share longer borders are more likely to go to war. Reasonable question. He needed border lengths. So he looked them up.
The numbers didn't agree. Spain said the border with Portugal was 987 km. Portugal said 1,214 km. Not close. Not "rounding differently" close. Off by more than 20%.
Richardson checked other borders. Same problem, everywhere. The numbers in encyclopedias, atlases, and government records contradicted each other wildly. He could have blamed sloppy surveying and moved on. Instead he did what mathematicians do when they smell something strange: he pulled the thread.
The ruler problem
Here's what he found. Take a coastline. Measure it with a 200 km ruler, walking along the coast and counting steps. You'll cut across bays, skip inlets, miss peninsulas. You get a number.
Now use a 50 km ruler. You follow the coast more closely. You catch the bays. Your total is longer.
Use a 10 km ruler. Longer still. The ruler snakes into smaller inlets, traces headlands you didn't even notice before.
A 1 km ruler? Longer. A 100 meter ruler? Longer. A 1 meter ruler and suddenly you're tracing around individual rocks.
The measured length of a coastline depends entirely on how small your ruler is. And there's no natural place to stop. The coast just keeps getting longer, forever, as you zoom in.
Those are approximate figures for Britain's coastline. The Ordnance Survey says 31,368 km. The CIA World Factbook says 12,429 km. Wikipedia has yet another number. They're all correct. They just used different rulers.
Try it yourself
Measuring a coast
Drag the slider left. Watch the number climb. There's no trick here, no optical illusion. The coast genuinely has more detail at every scale. A smaller ruler catches more of it.
Richardson was a strange person
This needs saying. Lewis Fry Richardson was a Quaker pacifist who spent his career trying to mathematically model the causes of war. He drove ambulances in World War I, then went home and tried to predict weather using differential equations, by hand, in a factory full of human calculators he designed but never built. His weather forecasts were terrible. His mathematics were decades ahead of everyone else.
He collected arms-race data the way some people collect stamps. He had notebooks full of border lengths, military expenditures, casualty counts. The border length problem was a footnote in his war research. He mentioned it in a paper, noted the pattern, and moved on. He died in 1953. The paper was published posthumously in 1961.
Benoit Mandelbrot read it.
How long is the coast of Britain?
That was literally the title of Mandelbrot's 1967 paper in Science. He took Richardson's footnote and turned it into a new geometry.
Richardson had plotted his measurements on a log-log graph: ruler length on one axis, total measured length on the other. The points fell on a straight line. Every coastline did this. But the slopes were different. Smoother coasts (South Africa) had a gentle slope. Jagged coasts (Norway) had a steep one.
Mandelbrot recognized the slope as a dimension. Not a whole number dimension. A fractional one. A fractal dimension.
A fractal dimension of 1.25 means: for every halving of ruler size, the measured length increases by about 19%. For Norway at 1.52, it's about 43%. The more jagged the coast, the faster the length explodes as you zoom in.
Why Richardson's log-log plots fell on straight lines
Measured length vs. ruler size (schematic)
The straight line on a log-log plot means the same pattern repeats at every scale. That's what "fractal" means. There's no privileged scale. Zoom in on a rocky British headland and the outline looks just as ragged as the whole island. Zoom in on the rocks and the edges of the rocks are ragged too. Self-similarity, all the way down, until you hit atoms.
A straight line like this was weird in 1967. Euclidean geometry had been assuming for two thousand years that curves had definite lengths. A circle has a circumference. An ellipse has a perimeter. Coastlines don't. They have a fractal dimension instead, and a length that depends on your patience and your ruler.
Spain versus Portugal
Back to Richardson's original problem. Spain's border with Portugal: 987 km or 1,214 km? The answer is both, obviously. Portugal is a smaller country with more detailed maps of its own border. Their surveys used a finer scale. More detail, longer measurement.
There's something almost philosophical about this. Two countries sharing the exact same physical border, measuring it with equal competence, and getting numbers that differ by 23%. Neither one is wrong. The border doesn't have a length. It has a relationship between scale and measurement, and your answer depends on where you stop.
Richardson's war research never went anywhere, by the way. The border-length-causes-wars hypothesis didn't pan out. Turns out the borders don't even have consistent lengths. Hard to build a regression model on that.
But the footnote about the rulers turned into fractal geometry, which turned into the Mandelbrot set, which turned into a whole branch of mathematics that describes ferns, lungs, river networks, stock markets, and galaxies. All because a pacifist couldn't get two countries to agree on the length of a line.