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