Coastlines

Mandelbrot's interpretation of the slope

In 1967 Mandelbrot interpreted the slope in Richardson's measurements.
For the moment, assume the number N(d) of segments of length d needed to walk across the coastline is proportional to 1/dD, for some exponent D.
That is, there is a constant M for which
N(d) = M/dD = M⋅d-D
and so

L(d) = N(d)⋅d = M⋅d-D⋅d = M⋅d1-D
Using properties of logarithms, we see this implies
Log(L(d)) = (1-D)⋅Log(d) + Log(M),
the equation of a straight line with slope 1 - D.
If the data points lie along a straight line, then the assumption N(d) = M/dD is justified and the exponent D is the fractal dimension of the coastline.
Richardson's data then can be interpreted as estimating the dimension D of the coastlines:
D = 1.25 for the west coast of Britain
D = 1.15 for the land frontier of Germany,
D = 1.14 for the land frontier of Portugal,
D = 1.13 for the Australian coast, and
D = 1.02 for the South African coast.
In interpreting D as a dimension, Mandelbrot described these geological features as statistically self-similar.
That is, each feature belongs to a (possibly infinite) collection of shapes, each of which is made of scaled copies of members of the collection, and the probabilities of selecting a given shape is independent of the number of pieces used to form the shape.
Sufficiently fine details, sufficiently far apart, likely become asymptotically independent, so the limiting process to compute the dimension likely converges.

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