— The intriging geometry of borders
The length of a boundary depends on the scale at which it is measured.
WHEN MATTHEW FLINDERS completed the first circumnavigation of Australia in 1803, he readily established it as the world’s largest island. How large exactly is harder to answer. The 1978 Year Book of Australia, for example, gives the length of the country’s coastline as 36,735 km. The Australian Encyclopaedia, published from 1925 to 1996, quotes the wildly different figure of 19,658 km. What is going on?
Lewis Fry Richardson was an English physicist and meteorologist. He was also a Quaker and pacifist. Appalled by the slaughter of World War I, he decided to analyse conflict mathematically. Following a hunch that the risk of a flareup might depend on the length of two countries’ common borders, he ploughed through some statistics, and noticed that many countries gave highly discrepant estimates. Richardson carried out a careful study to get to the bottom of the confusion. He soon put his finger on the key point that the length of a boundary depends on the scale at which it is measured.
When Flinders sailed around Australia, you might think his journey would be somewhat longer than the length of actual coastline, given the ship was out at sea. In fact the opposite is true. If a ship visited every little bay and inlet, and hugged the coast around every promontory, it would clock up many more kilometres than simply sailing by offshore. If a surveyor walked along every beach and coastal path, measuring the distance around each rocky outcrop, the length would be greater still. The length just seems to go up and up the smaller the scale used to measure it, because the wiggliness doesn’t diminish. Contrast this with a smooth curve, like a sagging rope, where the line gets straighter and straighter on smaller scales, and the total length converges to a definite answer as the segment size of each measurement shrinks to zero.
So does it make any sense to even talk about the length of a coastline? Richardson recognised the boundaries of countries with very wiggly features, like Norway, would in some sense be longer than those, like South Africa, that have relatively smooth coastlines.
How to make this precise? Although all coastlines are longer the smaller the scale on which they are examined, the rate at which that length grows as one zooms in to ever finer scales varies from country to country. Richardson determined that if the ruler length is l, then the total length varies like ld, where D is a number depending on the degree of wiggliness. For a smooth curve, like a sagging rope, D = 1. But for a line that gets ever longer on smaller and smaller scales, D will be greater. A careful analysis shows that D = 1.52 for Norway and 1.05 for South Africa, confirming one’s intuition that South Africa is somehow ‘smoother’ than Norway. Using the same formula, Britain comes out at 1.25 and Australia at 1.13.
Richardson’s proposal went largely ignored until 1975, when the mathematician Benoit Mandelbrot recognised his predecessor had tapped into something mathematically profound. He argued that Richardson’s scaling parameter D could be interpreted as the dimension of the line. In elementary geometry, smooth lines (e.g. sagging ropes) have dimension 1 and areas have dimension 2. An infinitely wiggly line, however, is somehow trying to fill out an area but failing. D, a number between 1 and 2, is a measure of how close the line gets to being an area.
Mandelbrot coined the term ‘fractal dimension’ for D. Thus Australia has a coastline with fractal dimension 1.13 – bigger than a smooth curve but less than Norway’s coastline.
Following Mandelbrot’s work, fractals became all the rage, inspiring works of art as well as advances in scientific areas such as chaos theory. The concept can be extended to any dimension, such as areas that strive to become volumes, or solids full of holes trying to become areas.
Once you start looking, fractals (or at least good approximations) crop up everywhere in nature, wherever there are irregularities over many scales of size – the shapes of fern leaves, the filigree patterns of capillaries or the tributary system of river deltas, the jagged outlines of mountain ranges, and the spiky pathways of lightning, snowflakes and sponges. The concept also has practical value across engineering and medicine, from image data compression to retinal damage in diabetics.