Coastline paradox

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The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length.

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 limit to the size of the smallest feature that should not be measured around, and hence no single well-defined perimeter to the country. Various approximations exist when specific assumptions are made about minimum feature size.

Of course, this might not be a paradox at all once we find for certain the smallest thing there is. It might not be practical to measure a coastline down to the sub-atomic level (particularly since grains of sand are often washed away), but that length would still exist.

For practical considerations, an appropriate choice of minimum feature size is on the order of the units being used to measure. If a coastline is measured in miles, then small variations much smaller than one mile are easily ignored. To measure the coastline in inches, tiny variations of the size of inches must be considered.

Note that this does not mean that the coastline is infinite length. Mathematically, the length converges to some exact value. Put simply, as smaller perturbations are considered, the contribution of each becomes less significant.

[edit] See also

• How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
• Fractal dimension
• Paradox of the Heap