Sheet No. 200 · Richardson Sound · soundings in fathoms
The longer you measure, the longer it gets.
In 1951 Lewis Fry Richardson set a pair of dividers on a map of Britain and walked them along the west coast. Each time he shortened the stride, the coast grew. It never stopped growing. This chart runs his survey forever.
The finding stands: span 32 nm reads 151 nm of coast; span 0.5 nm reads 401 nm. D ≈ 1.25.
Divider survey · seven passes · span halves each passfixed seed 1957 · midpoint displacement
01 · The paradox
Ask an atlas how long a coastline is and it answers with confidence. Ask a surveyor with finer instruments and the number rises. Spain once measured its border with Portugal at 987 km; Portugal, using a shorter ruler, found 1,214 km. Neither was wrong.
A coast is crinkled at every scale — bays hide coves, coves hide boulders, boulders hide grains. Halve your stride and you are forced into detours that the longer stride stepped straight across. The length does not converge. There is nothing to converge to.
02 · The method
The instrument here is the walking caliper: a divider set to a fixed span ε, stepped tip over tip along the coast. Count the steps N, and the measured length is N·ε plus the last short remainder.
Each pass on this chart halves the span — 32 nautical miles down to half a mile — and re-walks the same fixed coast. The polygon of chords hugs the shore more tightly every time, and every time it comes back longer.
03 · The dimension
Plot each pass on log axes — span across, length up — and the chaos files into a straight line. Richardson found the slope; Mandelbrot, in 1967, named it. If length obeys L ∝ ε1−D, the slope of that line is 1−D, and D is a fractal dimension.
A tame curve has D = 1. A sheet has D = 2. The west coast of Britain lives between them at about 1.25 — and so, by construction and by honest fit, does this one.
The survey record
Seven passes over one fixed coast. The chart above is drawing these numbers; this table is written by the same walk.
| Pass · span | Steps | Measured length | Change |
|---|
Hover or tab through a pass to light its point on the Richardson plot above. Chart width taken as 128 nautical miles.