Box-Counting the Dimension of Coastlines

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This Demonstration uses the box-counting method [1] to estimate the fractal dimension of three coastlines. The box-counting dimension of a coastline can be computed using a mesh of evenly spaced squares laid over a map of the coastline.


Let be the number of squares that are required to cover the whole coastline and be the sidelength of one square of the mesh. Then the box-counting dimension is .

can be interpreted as the slope of a straight line in a plot of versus . To approximate this, take five points determined by five values of and compute the slope of the linear fit between those points. Smaller values for and higher map resolutions lead to a more accurate estimate, as can be verified by selecting a finer mesh from the menu. It has been computed that the box-counting dimension of the coastline of Great Britain is 1.25 by using a very fine mesh on a very detailed map. This Demonstration finds values up to 1.18 using mesh sizes and a map resolution compatible with a reasonable computing time.

A detailed description of the box-counting method used in this Demonstration can be found in [2].


Contributed by: Erik Mahieu (January 2014)
Open content licensed under CC BY-NC-SA




[1] V. Kaurov. "Measuring Fractal Dimension of Natural Object from Digital Images." Mathematica Stack Exchange. (Oct 16, 2012)

[2] M. Mitchell et al. "Introduction to Complexity: 3.6 Box-Counting Dimension." Complexity Explorer. (2013)

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