The Coastline Paradox and Fractional Dimensions
The coastline paradox arises from the difficulty of measuring shapes with complicated edges such as those of countries like the Britain. As we try and be ever more accurate in our measurement of the British coastline, we get an ever larger answer! We can see this demonstrated below:
This first map represents an approximation of the British coastline with each line representing 200km. With this scale we arrive at an estimation of around 2400km. Yet if we take each line with length 50km we get the following:
This map now has a length of around 3400km. Indeed by choosing ever smaller measuring lengths we can make it much larger still. Coastlines have similar attributes to fractals – which are shapes which exhibit self similarity on ever smaller scales.
We can attempt to classify the dimension of fractals by using decimals. Just as 1 dimension represents a straight line and 2 dimensions represents a surface, we can have a pattern with dimension (say) 1.32. These dimensions make sense in terms of classifying fractal. A fractal with dimension close to 1 will be close to a straight line, one with a dimension close to 2 will be very “crinkly” indeed.
We can use the graph above, which was used by one of the founding fathers of fractal mathematics – Mandelbrot – to help expand his early ideas on the subject. The x axis is a log base 10 scale of the length chosen to measure the coastline in. The y axis is a log base 10 scale of the subsequent coastline length. So for example if we take our first estimate of the British coastline, i.e measurements of 200km, which achieved an estimate of 2400km – then we would plot the coordinate ( log(200), log(2400) ) For our second estimate this achieves the point (log(50), log(3400) ).
We can see that countries with steeper slopes (i.e those whose coastline greatly increases with ever smaller measuring scales) will have a more jagged coastline and so can be regarded as having a higher dimension. Mandelbrot assigned the coastline dimension as related to the gradient of the slope.
Finding the gradient of a log-log graph
However to find the gradient of the lines above is slightly complicated by the fact that we have a log-log plot. There is a formula we should use:
In the formula above, m is the gradient and F1 and F2 are the corresponding y values to x1 and x2. So using our coordinate values ( log(200), log(2400) ) and (log(50), log(3400) ) we would get a slope of:
log(2400/3400)/log(200/50) = -0.251
We then take the absolute value of this and add 1 – which gives a coastline dimension of 1.251 for Britain’s West coast.
We can also read off the approximate values from the graph. If we take the points (1.5, 3.3) and (2.7, 3) then we have a slope of:
log(3/3.3)/log(2.7/1.5) = -0.162 which gives a coastline dimension of 1.162.
Actually, with a more accurate reading of this scale Mandelbrot arrived at a coastline dimension of 1.25 for Britain – agreeing with our previous working out.
The coastline dimensions of other countries
The coastline of the German land frontier was assigned a dimension of 1.15 – i.e it is not as jagged as that of Britain. Meanwhile below we can see the South African coast:
This has a very smooth coastline – and as such the log-log graph looks to have an almost flat gradient. As such it has a dimension of 1.02.
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Mandelbrot and the Koch Snowflake: An exploration of fractal patterns
Julia and Mandelbrot sets: How to use complex numbers to generate pictures of infinity