Every answer is right

Ask how long Australia's coastline is and you get at least five published answers, from 19320 km to 59681 km. A 2021 paper by Husain, Reddy, Bisht and Sajid asks which is correct. Their answer: all of them.

Five answers, one coast

The paper lists them. The 1978 Year Book of Australia gave 36735 km. The Australian Encyclopedia gave 19658 km, and the Australian handbook 19320 km. Galloway and colleagues walked a divider over 162 NATMAP maps and got 47070 km, islands included. The figure reported now is 59681 km: 35821 km of mainland and 23860 km of islands. The paper's explanation for the spread: each source used a ruler of a different size.

Walk a ruler along it

Lewis Fry Richardson noticed this in the 1920s, measuring Britain, South Africa, Australia and the border between Spain and Portugal. His method is the one Galloway used. Set a divider, a pair of compass points, to a fixed length. Walk it along the coast, point to point, and count the steps. Steps times divider length is the coastline's length.

For a smooth curve, a shorter divider gives a slightly longer answer, and the answers settle towards one true length. A coastline never settles. Every shorter divider fits into bays and headlands the longer one stepped straight across, and each bay has smaller bays in it. So the length keeps growing as the ruler shrinks.

The number that does not change

Richardson plotted length against divider size with both axes on a logarithmic scale, where equal steps are equal factors. The points fell close to a straight line. That is what a power law looks like: shrink the ruler by a fixed factor and the length grows by a fixed factor, at every scale.

The slope of that line is the useful number. It is negative, because length rises as the ruler falls, and one minus the slope is called the divider dimension. A smooth line has dimension 1; the rougher the line, the higher the number. Mandelbrot turned Richardson's slopes into dimensions: 1.25 for the west coast of Britain, 1.15 for Germany's land frontier, 1.14 for Portugal's, 1.13 for Australia's coast, and 1.02 for South Africa's, which is nearly smooth.

The curve in our interactive is not a coast. It is a Koch curve, built by replacing every straight piece with four pieces a third as long, again and again. The paper notes that for an exactly self-similar curve like this one, the different ways of measuring dimension all agree. Our fit gives 1.26, matching log 4 divided by log 3.

Australia, measured again

The authors used a second method, box counting. Lay a grid of square boxes over the map and count how many touch the coast. Shrink the boxes and count again. They used 17 box sizes, from 1000 km down to 0.5 km, and the log-log slope gave a dimension of 1.143, close to Mandelbrot's 1.13.

That fit also says how long the coast is at a chosen ruler. The straight-line stretch of their plot ends at about 1 km, so they used that as the finest honest ruler. At 1 km the mainland comes out near 38910 km, against the 35821 km reported elsewhere. It is a real length, but only once you name the ruler.

In short

A coastline has no single length. Measure it with a shorter ruler and it grows, because there is always finer detail to walk round. What stays fixed is how fast it grows: the slope of length against ruler on a log-log plot. That slope gives the fractal dimension, and a coastline's length only means something once you say which ruler measured it.