How Long Is a Coastline? A Number Between One and Two

Lewis Fry Richardson noticed in the 1950s that published lengths for the same national border differed substantially between the two countries sharing it, and investigating why led to something odder than a clerical error. Measuring a rugged coastline with a long ruler skips the inlets and gives a short answer. Measuring with a shorter ruler follows more of them and gives a longer answer. Shortening the ruler further keeps increasing the total, and for a genuinely rough coast the total does not settle on a limiting value the way the perimeter of a circle does. Length is therefore not a property of such a coast at all, which is the result that made the whole subject necessary and which Benoit Mandelbrot took up two decades later.

Roughness is captured by how the measured amount grows as the measuring scale shrinks:

• •A straight line measured with rulers half as long needs twice as many, giving dimension one
• •A square measured with tiles half as wide needs four times as many, giving dimension two
• •A cube needs eight times as many, giving dimension three
• •A rough curve needs more than twice as many when the ruler halves, so its dimension exceeds one
• •The exponent relating the count to the scale is the dimension
• •For real coastlines that exponent typically falls between about 1.1 and 1.3

A value between one and two describes something that fills space more thoroughly than a line and less than a region, which sounds like a contradiction and is perfectly definite. A curve so wrinkled that it passes near every point in a patch approaches dimension two without ever being a surface. A dust of points scattered so sparsely that it barely holds together has a dimension between zero and one. The classic constructed examples make this concrete, including a curve built by repeatedly replacing each segment with four shorter ones, which has infinite length, encloses a finite area and has a dimension of about 1.26. Several distinct definitions exist, agreeing on well-behaved shapes and diverging on awkward ones, which is a technical matter rather than a disagreement about the idea.

The shapes this applies to share a property worth stating separately, which is that magnifying a piece reveals structure resembling the whole. A coastline viewed from orbit, from an aircraft and from a beach shows bays within bays within bays, and without a scale bar the three images are hard to tell apart. A fern frond is built of leaflets shaped like the frond. A tree divides into branches that divide in the same pattern. Constructed mathematical examples repeat exactly, while natural ones repeat only statistically and only over a limited range of scales, since a coastline stops being coastlike below the size of a grain of sand. That limited range is what separates a useful description of nature from an exact mathematical object.

The concept escaped pure mathematics quickly because roughness is measurable and useful. Materials scientists characterise fracture surfaces by it, since the roughness of a break carries information about how the material failed. Medical imaging uses it to describe the branching of blood vessels, the structure of lung airways and the texture of tissue, where a change in roughness distinguishes healthy from diseased in several contexts. Ecologists apply it to habitat structure and to the outlines of patches. Geologists use it for rock joint surfaces, which determines how fluid flows through them. Signal analysts apply it to time series including heart rate, where the roughness of the trace is diagnostic. In each case it converts a visual impression of complexity into one number that can be compared.

Measuring a rough coast with shorter rulers keeps producing a longer answer that never settles, so length is not a property of such a coast. Roughness is captured instead by how the count grows as the ruler shrinks, giving an exponent between one and two. That number is used to characterise fracture surfaces, blood vessel branching, lung airways and rock joints, converting an impression of complexity into a comparable figure.