Coastline Paradox: Seven Rulers, Seven Lengths for Britain

Step 1: walk Britain with a 200 km ruler

The coastline paradox is this: measure a coast with a shorter ruler and it comes out longer, so a coastline has no single length until you name the ruler. Husain and colleagues, writing in Scientific Reports in 2021, say the answer depends on "how long is your measuring stick". Watch it happen on one map in four steps.

The map here is Great Britain's main island from Natural Earth version 5.1.2, a public-domain world map drawn for a scale of 1:10 million. Natural Earth's notes say its ocean coastline comes mainly from World Data Bank 2, with modest generalization by line simplification. The island is one closed line of 3,706 points, with no offshore islands. The notes read for this page name no tide level for the line.

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Open a pair of dividers to 200 km and put one point on the outline's westernmost point, at 6.24° W, 56.72° N. Swing the other point clockwise, north first, until it first touches the coast ahead. That touch is step 1; move the first point there and repeat. Each distance is measured on the Earth's surface, on the WGS84 ellipsoid, not across a flat drawing.

Twelve full steps bring the dividers back near the start, with 0.54 of a step left over. So the 200 km walk measures 12.54 × 200 = 2,508 km. The walks on this page are illustrative measurements, computed with a short program that finds, between two data points, the exact spot one ruler length from the last landing point.

Steps 2 and 3: halve the ruler, then halve it again

Halve the opening to 100 km and walk again from the same start. The path cuts fewer corners, and the count rises to 28.59 steps, or 2,859 km. Halve it to 50 km, then 25 km, then 12.5 km. Each time, the path tucks into bays and around headlands it skipped before.

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Every halving adds length. From 200 km down to 12.5 km, each halving adds between 14% and 27%, and the 12.5 km ruler measures about twice what the 200 km ruler did. The last two rows behave differently, and the section on where the rule stops explains why.

The CSV file lists every ruler with its step count, length, start point and data source: coastline-paradox-ruler-walk.csv (CSV; where this file comes from: source, version and license, JSON) Source data: Computed for this page from Natural Earth v5.1.2, 1:10m coastline (ne_10m_coastline), public domain

Step 4: read the slope on a log-log plot

Plot each measured length against its ruler on log-log axes, where equal distances along an axis stand for equal multiples. Husain and colleagues write Richardson's divider rule as L = k × ε^α: the measured length L is a constant k times the ruler length ε raised to a power α. A power law like that turns into a straight line on log-log axes, and its slope α equals 1 − D. They call D the divider dimension.

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Fit a straight line to the five rulers from 200 km to 12.5 km. Its slope is −0.27. That puts D at about 1.27 for this outline over that range. At that slope, each halving of the ruler multiplies the length by 2 to the power 0.27, which is about 1.21, or 21% more.

To see what a dimension like 1.27 means, start with a smooth curve, which has D = 1: once the ruler is short enough, a shorter one adds almost nothing. Husain and colleagues give coastline dimensions as values between 1 and 2, with higher values for more irregular coasts. A D of 1.27 describes how fast this outline gains length between 200 km and 12.5 km, and nothing outside that range.

What Richardson and Mandelbrot reported

The map above was published in Amsterdam in 1743 by Isaak Tirion and engraved by Christiaan Sepp after his own drawing, according to the Rijksmuseum's record. It is here as history only. Nothing on this page was measured on it.

Husain and colleagues describe how Lewis Fry Richardson measured coastlines, Great Britain's among them, by walking dividers of a fixed opening along the coastline and multiplying the count by the opening. They cite his 1961 paper "The problem of contiguity" and summarise his slopes in a table, with Benoit Mandelbrot's reading of each slope as a dimension.

Mandelbrot's 1967 paper in Science, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", says in its abstract that geographical curves are so involved in their detail that their lengths are "often infinite or, rather, undefinable". The abstract adds that many such curves are statistically self-similar, each part a smaller-scale copy of the whole, and describes them with a fractional dimension D above 1.

This page's 1.27 is not a check on the 1.25 in the table. The table's figure is for Britain's west coast as Richardson measured it. This page's covers the whole main island on Natural Earth's generalized line.

So how long is Britain's coastline? There is no single number to quote. Every figure comes with a ruler and a map, and on Natural Earth's 1:10m map alone the seven rulers give anywhere from 2,508 km to 6,579 km.

Husain and colleagues report a similar gap for Australia. By their account, a divider survey on 162 maps by Galloway and colleagues found 47,070 km including islands, while another figure they cite is 59,681 km. They put the difference down to rulers of different sizes.

Where the rule stops on this map

Keep halving and the gains shrink. At 6.25 km the walk measures 6,001 km, 15.4% more than at 12.5 km. At 3.125 km it measures 6,579 km, only 9.6% more. The fitted rule from Step 4 predicted 6,268 km and 7,558 km.

The second prediction is impossible on this file. Add up the straight pieces between neighbouring data points and the whole line is 7,334.5 km long. No walk can measure more than that, because each step is a straight cut across a stretch of the line and is never longer than the stretch it cuts.

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The data points sit 1.4 km apart at the median. Along smooth stretches the map uses long straight edges, five of them over 20 km. In all, 41.5% of the line's length lies in edges longer than 3.125 km. Along such an edge, every ruler shorter than the edge walks the same straight line, so the short rulers find no new bends there.

Is a coastline infinitely long? On a map, no: this one cannot measure more than 7,334.5 km. Richardson saw real coasts keep growing as the ruler shrank, instead of settling toward a limit, as Husain and colleagues summarise his work. They also describe coastlines as similar across scales "at least for some range", so the divider rule describes a measured range, not every scale.

A coast with no bottom: the Koch curve

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Draw a straight segment and cut it into thirds. Replace the middle third with two sides of an equilateral triangle that points outward. Repeat on every segment. Each round turns one segment into four segments a third as long, so each round multiplies the length by 4/3.

Rounds 1 to 4 give 4/3, 16/9, 64/27 and 256/81 of the starting length, about 1.33, 1.78, 2.37 and 3.16. After 10 rounds the curve is about 17.8 times as long as the segment it started from. Every round keeps the corners of the round before, and the limit curve runs through all of them, so it is longer than any round: its length is infinite, and its dimension is log 4 / log 3, about 1.26. Three Koch curves around a triangle make the closed curve called the Koch snowflake, which von Koch first described in 1904, by MathWorld's account.

Each drawing is the rule, caught in the act at a different round. Every piece in one round becomes four pieces, each a third the size, in the next. For another shape built from exact copies of itself, see Sierpinski Triangle: A Dice Game Draws It Dot by Dot.

Keep three things apart. The Koch curve is a mathematical limit with no bottom. This page's walk is a finite measurement on a map that has a smallest bend. A real coast follows the divider rule only across the ruler lengths someone has measured, and the page on checking the construction keeps the same three layers apart.