An example of the coastline paradox. The coastline of Britain is measured using Fractal Unit = 200 km, then the length of the coastline = 2400 km (approx.)

Unit = 50 km, length = 3400 km (1000 km more)

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.^[1][2] It was first observed by Lewis Fry Richardson.

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.

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.

Over a wide range of measurement scales, down to the atomic, coastlines show a degree of self-similarity, and as the measurement scale is made smaller and smaller, the measured length continues to increase, rather than converging on any one value.

Extreme cases of the coastline paradox include the fjord-heavy coastlines of Norway, Chile and the Pacific Northwest of North America. From the southern tip of Vancouver Island northwards to the southern tip of the Alaska Panhandle, the convolutions of the coastline of the Canadian province of British Columbia make it over 10% of the entire Canadian coastline—25,725 km vs 243,042 km over a linear distance of only 965 km, including the maze of islands of the Arctic archipelago.^[3]

[edit] See also

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

[edit] Notes

1. ^ Weisstein, Eric W., "Coastline Paradox" from MathWorld.
2. ^ Mandelbrot, Benoit (1983). The Fractal Geometry of Nature. W.H. Freeman and Co.. pp. 25–33. 
3. ^ Sebert, L.M., and M. R. Munro. 1972. Dimensions and Areas of Maps of the National Topographic System of Canada. Technical Report 72-1. Ottawa: Department of Energy, Mines and Resources, Surveys and Mapping Branch.

[edit] External links

• The Atlas of Canada - Coastline and Shoreline
• La costa infinita (animation of a coastline with fractal details)

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