Coastline paradox

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded by Benoit Mandelbrot.

More concretely, the length of the coastline depends on the method used to measure it. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be measured around, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size.

The basic concept of length originates from Euclidean distance. In the familiar Euclidean geometry, a straight line represents the shortest distance between two points; this line has only one length. The geodesic length on the surface of a sphere, called the great circle length, is measured along the surface curve which exists in the plane containing both end points of the path and the center of the sphere. The length of basic curves is more complicated but can also be calculated. Measuring with rulers, one can approximate the length of a curve by adding the sum of the straight lines which connect the points:

Using a few straight lines to approximate the length of a curve will produce a low estimate. Using shorter and shorter lines will produce sums that approach the curve's true length. A precise value for this length can be established using calculus, a branch of mathematics which enables calculation of infinitely small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length:

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