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.

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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].

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


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References

[1] V. Kaurov. "Measuring Fractal Dimension of Natural Object from Digital Images." Mathematica Stack Exchange. (Oct 16, 2012) mathematica.stackexchange.com/questions/13125/measuring-fractal-dimension-of-natural-objects-from-digital-images.

[2] M. Mitchell et al. "Introduction to Complexity: 3.6 Box-Counting Dimension." Complexity Explorer. (2013) www.complexityexplorer.org/online-courses/3.



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