Two countries measured the same border and disagreed by 227 kilometres
Measure a coast with a shorter ruler and you get a longer coast. Keep shortening it and the answer never settles. The question is broken, and what replaces it is a dimension that is not a whole number.
Some time before 1951, Lewis Fry Richardson went looking for something dull. He was studying what makes countries go to war, he suspected the length of a shared border might matter, and he needed the numbers.
He found that Spain gave the Spanish–Portuguese border as 1,214 km and Portugal gave it as 987 km. The same border. A gap of 227 km — about 23% — between two governments describing one line on the ground.
Nobody had blundered. They had used different rulers.
Why the shorter ruler always wins
To measure a wiggly line you walk a fixed span along it, end over end, and count. Every inlet narrower than that span gets stepped over. Shorten the span and inlets that were invisible become part of the route. The walk gets longer, and it gets longer every time.
Britain is the standard demonstration. Measured in 100 km steps its coastline comes to roughly 2,800 km. Measured in 50 km steps, roughly 3,400 km. Six hundred kilometres of new coast appeared, and nothing moved.
For an ordinary curve this is a non-event. Measure a circle with finer and finer steps and the answers climb a little and then stop climbing: they converge on the circumference, which is what "the length of a circle" means. The coast never does that. Halve the ruler, gain the same fraction again, and keep going as far as your patience and your maps allow.
Watch it happen where the arithmetic is exact
Real coasts stop cooperating at some point: eventually you are measuring around individual boulders. A constructed shape does not, and the Fractal Explorer has one where every step is exact.
Choose the Koch snowflake and walk the depth slider up from zero. The panel's count row reads 3 segments, then 12, then 48; at depth 5 it reads 3,072, and at depth 7 it reads 49,152. The rule doing that is the ruler argument with the vagueness removed: every segment is replaced by four segments one third as long. Four thirds of the length, at every step, for ever.
Now watch the row beside it. As the count runs from 3 to 49,152, the dimension readout does not move at all: log 4 / log 3 ≈ 1.262. The length is diverging and that number is standing still, which is the whole point of it.
The question that does have an answer
"How long is the coast of Britain?" has no answer, and not because the surveying is hard. The length is a fact about the coast and the ruler, and you cannot report it without reporting both.
What belongs to the coast alone is how fast the length grows as the ruler shrinks. Plot measured length against ruler length, both on logarithmic axes, and the points fall on a line. That is Richardson's plot, and its slope is the property you were after. Fit one yourself in linear regression; the reason a power law straightens out under logarithms is the same reason it works here.
For the west coast of Britain that exponent gives a dimension of about 1.25. For the coast of South Africa, about 1.02 — a coast so smooth it is nearly a line, and it measures nearly the same whatever you use. Britain is not nearly a line. It is not a plane either. It sits between them, permanently, and no finer survey will ever round it off to something tidier.
Richardson published the border figures as an appendix. Sixteen years later Mandelbrot read them, and the answer to the wrong question turned out to be the definition of a new kind of number.
References (2)
- the paper that turned Richardson’s oddity into a definition Mandelbrot (1967). How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science 156(3775), 636–638.
- the original border measurements, collected while studying the causes of war Richardson (1961). The problem of contiguity: an appendix to Statistics of Deadly Quarrels. General Systems Yearbook 6, 139–187.