How Long Is the Coast of Britain?

In 1967 a paper appeared in Science with the title written above. On its first column it gives a number — the west coast of Britain has a dimension D ≈ 1.25, fractional, somewhere between a line and a plane. The result was already in print: Lewis Fry Richardson had measured it six years earlier as an appendix to a study of war, where it sat unread. Mandelbrot did the naming. Below, the same recipe runs live on six real coastlines from the public-domain Natural Earth atlas — pick a ruler, watch the kilometres count themselves.

“The left bank of the Vistula, when measured with increased precision, would furnish lengths ten, hundred or even thousand times as great as the length read off the school map.” Hugo Steinhaus, Colloquium Mathematicum 3 (1954), quoted by Mandelbrot, 1967

I · The disagreement

Two surveyors with two pairs of dividers walk the same shore. They use different step lengths. They come back with different answers, and not by a little — the one with the shorter divider comes back with a coast that is thirty per cent longer, or twice as long, or three times as long, depending on how mismatched their dividers were. Both are right. Both have measured what they meant to measure. The number we usually call length does not exist for them.

This was Richardson's finding. He went looking, between the wars and through the second, for empirical regularities that might predict whether two countries with a common border would fight; the data he collected included national-border lengths reported by both adjacent countries' atlases. Spain said its border with Portugal was 1,214 km; Portugal said 987 km. The two numbers were not made up — each had been measured carefully, with dividers, off a national map. They just used different dividers. (The pair sits in Richardson's posthumous 1961 paper and the verbatim quotation lives in Mandelbrot's later Fractal Geometry of Nature, 1982, p. 33.) Plotting log L against log G (the divider step length), Richardson got a straight line of negative slope. He noticed that the slope depended on which border, that some borders gave nearly flat lines (the smooth ones), and that others — the west coast of Britain, the Norwegian fjords — gave steeper lines. He wrote the law as

L(G) = M · G^1−D, with M a constant of the border and D ≥ 1 a constant of its shape.

Mandelbrot interpreted that D as a dimension — a real number, not an integer. A perfectly straight wall has D = 1; a coastline so crinkled it fills its plane like ink in water would have D = 2; natural shores live somewhere in between. The instrument below does Richardson's recipe in front of you.

II · The dividers

Pick a coast. Drag the ruler. A pair of compasses with that step length walks along the polyline of the chosen shore, counting steps; the orange legs are the chords the dividers traced. The line at the top is the recovered length L(G) = N(G) · G. Notice three things, in order of importance:

1. The length goes up as the ruler goes down. For Britain or Norway, halving the ruler adds a quarter to the kilometres. There is no answer the shore converges to.
2. The circle is the control. Walk its perimeter with any ruler you like — the number stays put within about 5 %, because there is no detail underneath the ruler to find. This is what an actual length is.
3. The South African south coast is the closest natural example to a circle. Its length barely changes. It really is almost a curve in the classical sense — and that is the rare case, not the rule.

III · Richardson's plot — the slope is D − 1

Now do it for every divider length at once. For each coast, log L is plotted against log G; a straight line is fit; its slope is 1 − D, read straight off the data. The flatter the line, the closer the coast is to having an honest length. The steeper, the more fractional its dimension. The legend's D column is what the slope says, recomputed from scratch in your browser.

IV · The table reproduced

In one paragraph on page 637, Mandelbrot quotes Richardson's five dimensions in running prose — west Britain 1.25, German land frontier ~1899 1.15, Spain–Portugal 1.14, Australia 1.13, South Africa 1.02 — and reproduces Richardson's own log-log plot as his Figure 1. Below, those five numbers are recomputed from Natural Earth. The divider column comes from Instrument I, the box-counting column is an independent estimator (project the coast to a local kilometre grid, count the cells the line touches, fit log N vs log G).

V · what this page is honest about

What counts as the coast. "The" coastline does not exist. The line we drew is Natural Earth's mean high water-ish boundary, a single rendering of the ocean–land separation at one resolution. The Ordnance Survey draws a different line; tide tables put it somewhere else every hour. A surveyor walks a line; a sentence does not.

The data sets a floor. The dividers above can shrink to about 10 km before the polyline runs out of detail to find. Real shorelines do not have detail at every scale — they have detail at every scale we have bothered to map. Below that floor the power law has nothing to feed on. Mandelbrot acknowledges this in the original: a coastline is statistically self-similar over a finite range, not all the way down. The dimension is a property of the curve in that range, not of any object in nature.

D depends on the range you fit over. Push the smallest divider down and D climbs; push the largest up and the slope flattens out toward "rectifiable" because dividers comparable to the whole island just measure the bounding box. The instrument fits over a range chosen to span about one log decade, away from both ends — say so, then show the work. Plot the slope locally and you'll see it isn't perfectly constant. Real coasts are approximately self-similar, not exactly.

The projection. Box counting needs a flat plane; we project each coast to local equirectangular kilometres on its own mean latitude. For Australia, which spans 30° of latitude, the high-latitude cells are a little stretched relative to the low. This puts a small bias on the box-counting D (less than 0.05) — visible as the box and divider columns drifting against each other by more than they otherwise would. It is named, not hidden.

What the page is not. The dimension D we recover is not "the" fractal dimension of Britain. There isn't one. There is a D for this base map, this range of G, this projection, and this divider rule. Under another base, another range, another rule, you get another number — close, but not identical. Richardson knew that; he reported several numbers across coastlines and let the spread say what it was going to say. That is the venue's job.

VI · the live check, run in front of you

Everything above re-runs at page load. The checks ask only what the paper claims: that L(G) follows a power law, that the recovered D is in the ballpark Richardson published, that the two methods (divider, box-counting) point at the same number, and that the circle control gives D ≈ 1. If any of them fail in your browser, the verdict at the top of the page will say so out loud.

running…

colophon

Sources, verbatim:
· Mandelbrot, B. B. "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension." Science, New Series, vol. 156, no. 3775 (5 May 1967), pp. 636–638. JSTOR 1721427.
· Richardson, L. F. "The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels." General Systems Yearbook 6 (1961), pp. 139–187. (The original table of D; Fig. 17.)
· Steinhaus, H. Colloquium Mathematicum III, fasc. 1 (1954), p. 1. (The Vistula epigraph, as quoted by Mandelbrot.)
· Natural Earth, ne_10m_coastline.geojson, public domain (Tom Patterson & Nathaniel Vaughn Kelso). naturalearthdata.com. Repository: github.com/nvkelso/natural-earth-vector. The specific lon/lat polylines used here are committed in research/coastline-paradox/coasts.json with the extraction script that produced them.