Fractal Ireland
How long is Ireland’ s coast? Stefan Hutzler explains why the answers to this question vary so wildly.
You are contemplating taking a holiday.
How about a canoe trip around the island of Ireland, keeping as close to the shore as you can? How long will it take at an average of, say, 20 kilometres a day?
A quick Google search gives the length of the coast line as 1448km( CIA- The World Factbook), 3171km( Ordnance Survey Ireland) and 5631km( http:// www. goireland. com / about-ireland. htm). So the trip might take nearly all of summer( 72 days) or indeed, you might barely be able complete it within a year without canoeing in the winter( 282 days)!
How can it be that these numbers differ so wildly? The answer is that an estimate of the length of the coastline depends on the map that you use, in particular on the fine scale at which it gives up reproducing“ the wriggles of the real coastline. Such an object is called a fractal.” How then are we to describe it? And how does the work of a group of second year Theoretical Physics students at Trinity College Dublin [ 1 ] help to realistically plan a canoe trip?
This kind of problem lay largely unrecognised, except in the minds of deep-thinking mathematicians, until in the 1960s and 70s the mathematician Benoît Mandelbrot brought the topic into the scientific mainstream [ 2, 3 ]. Helped by the huge advances in computer graphics, fractals had an enormous popular impact from the 1980s onwards.
So what is a fractal? Consider a smooth object, such as the S-shape shown in Figure 1( right). An approximation of its perimeter may be determined as follows: overlay the object with square boxes of equal side length; count the
The rugged and indented western coast, infra-red image from the Marine Institute’ s National Coastline Survey.
number of boxes that contain a bit of the perimeter; multiply this number by the side length of each box and record this product. Repeating this procedure for smaller and smaller square boxes( corresponding to an ever finer grid), shows that the product is roughly constant, independent on the box size, as shown in Figure 1( below).
The situation is different, however, when this procedure is applied to a coast-line. Using maps from the almighty maps. google. com site and GIMP image manipulation software
Figure 1: The perimeter of an object may be estimated by superimposing a square grid and counting the number of boxes that contain sections of the perimeter. In the case of a smooth object, such as the S-shape shown on the right, the estimated length of the perimeter, i. e. the number of these boxes multiplied by the width of a box, is essentially independent of the box width( computation by D. Whyte, TCD).
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( http:// www. gimp. org), the four TCD students superimposed various grids of decreasing box length onto a map of Ireland. This is illustrated in Figure 2 for the coast between Dingle and Skibbereen. Using Click Counter software( http:// www. murgee. com /), they then counted the number of boxes which contain bits of the coast.
Again we can estimate the length of the coast by multiplying the number of coastline containing boxes by their width. The result is shown in Table 1 and Figure 3. Unlike in the example of the smooth S-shape, the length of the coast-line is seen to increase as we decrease the box sizes. The value given by the Ordnance Survey Ireland is reproduced for a box