Coastlines and the Scale of a Measurement, math that surprises you lesson

Walk a measuring stick along a stretch of coast and count the steps. Repeat with a stick a third as long and the total usually comes out larger, and a shorter stick still gives a larger total again. The coast has not changed between the measurements, so which number is its length? This page measures a fixed synthetic coastline with several ruler lengths, shows how shorter steps follow bays and headlands that longer steps skip, and explains what a reported length means for a rough boundary and where the growth stops for real data.

Key terms

Coastline paradox

Definition Fix a boundary between two endpoints, such as a stretch of coast traced on a map, and measure it with the divider method described in the card on ruler length. A ruler length $r$ is chosen, the boundary is walked in straight steps of length $r$ , and the measured length $L(r)$ is the number of steps times $r$ plus a final leftover. For a smooth curve, $L(r)$ settles down to one value as $r$ shrinks. The coastline paradox is the observation that for coastlines and other rough boundaries, $L(r)$ keeps growing as $r$ shrinks, over the range of ruler lengths that can be used, instead of settling down. The resolution is that the measurements do not contradict each other. A step of length $r$ cuts straight across bays and headlands much smaller than $r$ , a shorter ruler follows more of them, and so $L(r)$ is the length of the boundary as seen at scale $r$ . For a rough boundary a length means something only together with its ruler length, or more generally its measurement scale, and its procedure. In an ideal mathematical fractal such as the Koch curve the growth never stops and the length is infinite. A real coastline is different. Its detail ends at physical limits such as rocks and grains, the shoreline itself moves with the tide, and every data set has a finest resolution, so measurements show the growth only over a limited range of scales, and they do not show that a physical coastline has infinite length or one exact fractal dimension at every scale. Lewis Fry Richardson collected measurements of this kind for coasts and land borders, published in 1961, and Benoit Mandelbrot discussed them in a paper in 1967.

In plain words A long ruler takes shortcuts across small bends and a short ruler follows them, so the short ruler usually reports more length. The coast is the same, and the two numbers describe it at two different levels of detail.

Example Here is a synthetic coastline with exact numbers. Start with a straight segment of length 1. Replace it by four segments of length $\tfrac13$ , with the middle third swapped for the other two sides of an equilateral triangle built on it, as in the Koch curve. Repeat this on every segment, four times in all. The result is a fixed polygon with 256 segments of length $\tfrac{1}{81}$ and total length $(4/3)^4 = 256/81 \approx 3.160$ . With the ruler $r = 3^{-m}$ for $m$ up to 4, every step lands on a corner of the $m$ th round, because the stretch of coast between two neighboring corners of that round stays closer than $r$ to the first corner until it reaches the second. So the walk takes exactly $4^m$ steps.

Ruler length	Steps	Measured length
$1$	$1$	$1$
$1/3$	$4$	$1.333$
$1/9$	$16$	$1.778$
$1/27$	$64$	$2.370$
$1/81$	$256$	$3.160$
$1/243$	$768$	$3.160$

Each division of the ruler by 3 multiplies the measured length by $\tfrac43$ , until the ruler is as short as the smallest segments. Below that the walk only follows straight segments, and the measured length stays at $\tfrac{256}{81}$ , the length of the polygon. This leveling off is a property of the finite model, which has no detail smaller than $\tfrac{1}{81}$ .

Measurement scale

Definition The measurement scale of a length measurement is the size of the smallest detail it can follow. In the divider method it is the ruler length, in a chord measurement it is the spacing between sample points, and for a boundary taken from a map, photograph or survey it is also limited by the resolution of those data. For a smooth curve such as a circle, the measured lengths approach one limit as the scale shrinks, and that limit is the length of the curve. For a rough boundary the measured length depends on the scale throughout the range where new detail keeps appearing. That range has two ends. At the large end, a ruler as long as the whole stretch measures only the straight distance between its endpoints. At the small end, a ruler shorter than the finest detail in the data finds nothing new to follow, and on a real shore, at the size of rocks and grains, the boundary is no longer a sharp line at all. A real estimate therefore needs a documented data set, a stated convention for where the shoreline is, such as a high-water line, and the range of scales that was used.

In plain words A length for a rough boundary is always a length at some scale. Changing the scale changes the answer, so saying which scale was used is part of the answer.

Example Compare a circle of radius 1 with the synthetic coastline. Walking the circle with the ruler $2\sin(\pi/N)$ takes exactly $N$ steps to go around, so the measured length is the perimeter of a regular polygon with $N$ sides.

Steps	Ruler length	Measured length
$6$	$1.000$	$6.000$
$12$	$0.518$	$6.212$
$24$	$0.261$	$6.265$
$48$	$0.131$	$6.279$
$96$	$0.065$	$6.282$

Each doubling of the number of steps adds less than the one before, and the values settle on the circumference $2\pi \approx 6.283$ . For the synthetic coastline, each division of the ruler by 3 multiplies the length by the same factor $\tfrac43$ , all the way down to $\tfrac{1}{81}$ , the finest detail of the model, and only then does the length settle.

Statistical self-similarity

Definition A shape is exactly self-similar when it is made of smaller copies of itself, so that suitable pieces, magnified, match the whole exactly, as each of the four pieces of the ideal Koch curve does when magnified 3 times. A random shape is statistically self-similar when a magnified piece is not an exact copy of the whole but follows the same probabilities, so that any feature, such as the direction of the first bump, is as likely in the magnified piece as in the whole. Natural boundaries such as coastlines show at most an approximate statistical resemblance between scales, and only over a limited range of scales. A small bay is not a resized copy of the whole shore. Mandelbrot put the phrase statistical self-similarity in the title of his 1967 paper on the length of coastlines.

In plain words An exactly self-similar shape repeats itself perfectly at smaller scales. A statistically self-similar one only looks alike in character, the way two stretches of rocky shore can look equally rough without being copies of each other.

Example Build the synthetic coastline again, but toss a fair coin for every segment to decide whether its new bump points to the left or to the right of the segment. Whatever the coins show, each round still replaces every segment by four segments one third as long, so after $n$ rounds the length is $(4/3)^n$ , as before. The piece built on the first third of the starting segment, magnified 3 times, is made by the same coin-tossing rule with one round fewer, so it has the same statistics as a whole curve with that many rounds, but its bumps generally point differently from those of the whole. The curve is statistically self-similar but not exactly self-similar. It is a model of the idea rather than of a real shore, and some coin outcomes even make two bumps touch at a point.

Ruler length

Definition In the divider method, the ruler length $r$ is the fixed straight-line distance covered by every step. The method has three rules. Start at one end of the boundary. From the current point, step to the first point further along the boundary whose straight-line distance from the current point is exactly $r$ , and repeat. When no such point is left, stop and add the leftover, the straight-line distance from the last stop to the far end. With $N(r)$ full steps the measured length is

L(r) = N(r) \cdot r + \text{leftover}.

Other conventions drop the leftover or count it as a fraction of a step, so a table of lengths should say which convention was used. Each step is a straight chord, never longer than the stretch of boundary it replaces, so for a curve of finite length $L(r)$ is never more than the true length. A different procedure, the chord method, works for a boundary that crosses each vertical line once. It marks the points of the boundary at equally spaced horizontal positions and adds the straight distances between neighbors. Halving the spacing keeps the old points and adds one between each pair. By the triangle inequality, which says that a detour through a third point is never shorter than the straight segment, the total can then only grow or stay the same. The divider method gives no such guarantee, and the two procedures generally give different numbers.

In plain words The ruler length is the size of the stride. Strides of a fixed size cut across every bend smaller than the stride, and the leftover rule says what to do with the last bit that is too short for a full stride.

Example Take the L-shaped path from $(0, 0)$ to $(1, 0)$ and then up to $(1, 1)$ , which has length 2. With $r = \tfrac12$ the stops are $(\tfrac12, 0)$ , $(1, 0)$ , $(1, \tfrac12)$ and $(1, 1)$ , four steps and no leftover, so $L(\tfrac12) = 2$ . With $r = 1.2$ the first point at distance $1.2$ from $(0, 0)$ is on the upright part, at a point $(1, y)$ with

1^2 + y^2 = 1.2^2,

so $y = \sqrt{0.44} \approx 0.663$ . That step cuts the corner. No point further on is $1.2$ away from $(1, 0.663)$ , and the leftover to $(1, 1)$ is about $0.337$ , so $L(1.2) \approx 1.537$ .

Check yourself

Suppose the synthetic coastline is built with five rounds of bumps instead of four. What do the rulers $\tfrac{1}{81}$ , $\tfrac{1}{243}$ and $\tfrac{1}{729}$ report?

Show the answer

The polygon now has $4^5 = 1024$ segments of length $\tfrac{1}{243}$ , and its length is $(4/3)^5 = 1024/243 \approx 4.214$ . The ruler $3^{-m}$ still lands on corners of the $m$ th round for $m$ up to 5. So the ruler $\tfrac{1}{81}$ takes 256 steps and reports $\tfrac{256}{81} \approx 3.160$ , as before, and the ruler $\tfrac{1}{243}$ takes 1024 steps and reports $\tfrac{1024}{243} \approx 4.214$ . The ruler $\tfrac{1}{729}$ walks each straight segment in exactly 3 steps, 3072 steps in all, and reports the same $\tfrac{1024}{243}$ . The extra round of detail moves the leveling off one scale smaller and raises it by a factor of $\tfrac43$ .

On the L-shaped path, measure with $r = 1$ and with $r = 0.9$ . Does a shorter ruler always report a longer length?

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With $r = 1$ the first stop is the corner $(1, 0)$ and the second is $(1, 1)$ , so $L(1) = 2$ . With $r = 0.9$ the first stop is $(0.9, 0)$ . The next point at distance $0.9$ is on the upright part at $(1, y)$ with

0.1^2 + y^2 = 0.9^2,

so $y = \sqrt{0.8} \approx 0.894$ . The leftover to $(1, 1)$ is about $0.106$ , so $L(0.9) \approx 1.8 + 0.106 = 1.906$ .

The shorter ruler gave the shorter length, because the longer ruler happened to land exactly on the corner while the shorter one cut across it. A shorter ruler usually picks up more detail on a rough boundary, but for the divider method this is a tendency, not a rule. The guarantee holds for the chord method, where halving the spacing keeps every old point.

Extend the pattern to the ideal Koch curve, which has bumps at every scale. With the ruler $3^{-m}$ the divider method reports $(4/3)^m$ for every $m$ . What is the smallest $m$ for which the measured length exceeds 10, and what line appears when $\log L$ is plotted against $\log r$ ?

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The lengths are $(4/3)^8 = 65536/6561 \approx 9.99$ and $(4/3)^9 = 262144/19683 \approx 13.32$ , so the smallest $m$ is 9, a ruler of length $\tfrac{1}{19683}$ . Equivalently, $m$ must exceed

\frac{\log 10}{\log (4/3)} \approx 8.004.

For the plot, $r = 3^{-m}$ gives $\log r = -m \log 3$ and $\log L = m \log (4/3)$ , so

\log L = (1 - D) \log r

with $D = \log 4 / \log 3 \approx 1.262$ . The points lie on a straight line of slope $1 - D \approx -0.262$ . The ideal curve follows this line for every ruler $3^{-m}$ , while the synthetic coastline with four rounds follows it only from $r = 1$ down to $r = \tfrac{1}{81}$ and then turns flat. Data from a real coast can at best follow such a line approximately, over a limited range of scales.