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. |