Box-Counting the Dimension of Coastlines

coastline

Japan

Norway

Great Britain

mesh size

coarse

medium

fine

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

N

s

be the number of squares that are required to cover the whole coastline and

s

be the sidelength of one square of the mesh. Then the box-counting dimension

D

is

lim

s0

log(

N

s

)

log(1/s)

.

D

can be interpreted as the slope of a straight line in a plot of

log(

N

s

)

versus

log(1/s)

. To approximate this, take five points determined by five values of

s

and compute the slope of the linear fit between those points. Smaller values for

s

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].