The cottage smells of peat smoke and pipe tobacco. Rain ticks against the window, and beyond it a sea loch turns pewter in the dusk. An old man with kind eyes and wild eyebrows — Lewis Fry Richardson, retired scientist, lifelong Quaker — spreads two atlases on the table and taps them with his pipe stem.

"Portugal says its border with Spain is 1,214 kilometres," he says. "Spain says the very same border is 987. A difference of two hundred and twenty-seven kilometres. Neither country is lying. Neither has made an error. So — which of them is right?"

You start to answer, and stop. It's a line on the ground. It has to have a length… doesn't it?

Richardson has spent years asking whether countries with long shared borders fight more wars, which meant he needed border lengths — and the numbers in the reference books refused to agree. He slides a pair of brass dividers — a two-legged compass with needle points — across the table to you.

"Set them to a fixed step. Walk them along the coast of Britain like a tiny pair of legs, and count the paces. Then shorten the step and walk it again. Go on," he smiles. "Measure the coast. I shall put the kettle on. You may be some time."

You walk the coast at 100 km a pace. Then at 50. Then at 25. Each time, the coast comes out longer — not by a little, but relentlessly, every single time. Real surveys of Britain do exactly this: measured with a 100 km ruler the coastline is about 2,800 km; halve the ruler to 50 km and it grows to about 3,400 km. Six hundred kilometres of coastline appear out of nowhere.

Richardson returns with the tea, looks at your numbers, and nods slowly. "You see? The question 'how long is the coast of Britain?' has no answer. Or rather — the answer is: it depends how small your ruler is. And if the ruler could shrink forever, the length would grow forever."

Spain and Portugal simply used different-sized rulers. The border didn't change. The mathematics of measuring it did.

Mathematicians had actually met this monster before — in captivity. In 1904, the Swedish mathematician Helge von Koch built a curve by taking a triangle and endlessly growing a smaller triangle on the middle of every side. Each fold makes the outline exactly 4/3 as long as before. Fold forever and the perimeter becomes infinite — yet the snowflake still fits on a postage stamp, enclosing only 8/5 the area of the original triangle.

An infinite coastline wrapped around a finite island. Grow one yourself:

Why does the length keep growing? Because a coastline is rough all the way down. From a satellite you see peninsulas. Sail closer: each peninsula has its own bays. Land on a beach: each bay has rocky points, each rock has notches, each notch has crevices, down to pebbles, down to sand grains. Every scale hides new wiggles for a smaller ruler to find. Zoom in and see for yourself — the coast below is self-similar: its close-ups are just as crinkly as the wide view.

Richardson never saw where his dividers would lead. He died in 1953, and his coastline numbers were tucked away as an appendix in an obscure yearbook in 1961 — a curiosity, a footnote. Then a restless mathematician at IBM named Benoît Mandelbrot stumbled on the forgotten paper.

In 1967 he published a short article in Science with a title that sounded like a child's homework problem: "How Long Is the Coast of Britain?" His answer — there is no length, but there is something better — became a turning point in mathematics. The slope of Richardson's plot, Mandelbrot realized, measures a coast's dimension: not the 1 of a smooth line, not the 2 of a filled plane, but something in between. Britain's craggy west coast: about 1.25. South Africa's smooth sweep: barely 1.02. Norway, shredded into fjords: roughly 1.5 — halfway to being a solid sheet of ink.

In 1975 he coined a name for shapes that are rough at every scale, from the Latin fractus, "broken": fractals. And in his great 1982 book he wrote the sentence that opened a thousand eyes: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." The pacifist's puzzle about borders had become a new geometry — the geometry of nature itself.

The United States government has three official answers for the length of its own coastline — 12,383 miles, 29,093 miles, and 88,612 miles — and every one of them is correct. Norway's coast is about 1,573 miles as the crow flies along it, and roughly 18,000 miles if you trace every fjord.

So here is the question to carry with you: what else has no true length? The edge of a cloud? A river? The branching of your own lungs — which pack a tennis court of surface into your chest using exactly this trick? Once you've walked a coastline with Richardson's dividers, you start to see them everywhere: infinities, hiding politely inside finite things.