Coastline Paradox

 Ever wondered how the perimeter of irregular surfaces (like coastlines, borders of a fern leaf) are measured and compared? 

For example, let us consider a coastline, you can see that as we zoom in on the picture more irregularities pop up, there is no stage of zooming in where we'll finally reach a straight line. So, if I had to measure the coastline using a 15 cm scale, it'd obviously ignore the small irregularities which lie beyond the scope of a 15 cm scale. If I measure the coastline using a 5 cm scale, then it would accommodate comparatively more irregularities, as a result, the length of the same coastline will be more in this case. This means as the precision of the scale increases the length of the coastline increases. Comparison of coastlines between different regions becomes difficult due to this dilemma. This is known as the 'Coastline Paradox' or 'Richardson effect' and originally the problem was framed as, "How Long is the Coast of Britain?" 

This paradox was explained using the concept of the 'dimension of a fractal'. In simple language, fractal dimension measures the extent to which its border differs from a straight line or it measures its ruggedness. Fractal dimension does not exactly give the length of the coastline but it surely helps to compare how irregular they are. But often we find lengths of coastlines in different situations but none of these is reliable as they do not specify the scale they've used to measure it. 

So, maths has helped us prove that it is not completely possible to find the perimeter of irregular surfaces but one can for sure measure its roughness.