width of 1.8km. But using an even finer grid would eventually result in a linelength of infinity! Coastlines and indeed also rivers, clouds, ferns, etc. [ 3 ] are examples of self-similar objects which display structure at all length scales.
Mathematically speaking, the determined length of the circumference is proportional to w 1-D, where w is the box width and D is called the fractal dimension D of the object under scrutiny. In the case of our smooth S-shape example of Figure 1 the value of D is one, i. e. the dimension of a line.( Note that this makes L independent of w, as observed in figure 1.) The fractal dimension of the Irish coast-line, however, turns out to be about D = 1.20 ± 0.01, as a numerical analysis of our data shows( see solid line in Figure 3). Since D measures the raggedness of an object, we also expect differences when comparing the relatively smooth east to the“ wild west” coast. And indeed the students found D = 1.1 and D = 1.26, respectively.
How do these numbers compare with values for other countries? The South- African coast-line appears smoothest, with D = 1.05, while Norway with its fjords tops the bill with D = 1.52. Mandelbrot’ s value for the West coast of Britain, determined in 1967, was D = 1.25 [ 2 ].
So what about your canoe trip? In consulting Fig. 3 you might use the length of the canoe as your scale, and get an estimate of more than 12000 kilometres! But that would correspond to unduly meticulous navigation. Let’ s say you will stand off the coast enough that you mostly steer straight for about 100 metres at a time: let us use that. Now we get 5667km... Maybe you should just stay at home and read Mandelbrot’ s magnificent Fractals: Form, Chance and Dimension( 3), opening your eyes to our fractal world?
References
[ 1 ] The TCD students carrying out the project were Killian Walshe, Stephen Burke, John Mark Geary and Liam Kelly. [ 2 ] B. Mandelbrot,“ How long is the coast of Britain? Statistical self-similarity and fractional dimension”, Science, 156, 636- 638( 1967). [ 3 ] B. Mandelbrot, Fractals: Form, Chance and Dimension, W. H. Freeman( 1977).
Stefan Hutzler is Associate Professor at the School of Physics, Trinity College Dublin.
Figure 2: Ireland’ s coast-line between Dingle and Skibbereen, superimposed with square grids of box-width 10.8km( left) and 2.7km( right), respectively. Only boxes containing a section of the coast are shown. It is seen that the finer the grid, the more details of the coast become visible.
Box width in |
Number of |
Estimated |
kilometres |
boxes containing |
total |
|
section of |
length of |
|
coast-line |
coast-line |
36 |
49 |
1764 |
27 |
69 |
1862 |
18 |
112 |
2016 |
14.4 |
147 |
2116 |
10.8 |
203 |
2192 |
7.2 |
363 |
2613 |
3.6 |
777 |
2797 |
2.7 |
1102 |
2975 |
1.8 |
1752 |
3153 |
Table 1: Summary of data obtained for the coast-line of the island of Ireland. The estimated length of the coast increases with decreasing width of the boxes making up the grid.
Figure 3: Unlike the case of our smooth S-shape in Figure 1, the estimated length of the Irish coast increases with the accuracy at which it is mapped. This feature of fractal shapes is written in mathematical terms as L( w) ~ w 1-D, where L( w) is the box width w dependent length and D is the fractal dimension. Determining D numerically( using a leastsquare fit) from our data of Table 1 as D = 1.20 ± 0.01 enables us to interpolate between measured data points to obtain coast lengths for any grid size( box width).
Estimated length of Irish coast in kilometres
Box width in kilometres
SCIENCE SPIN Issue 58 Page 20