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