Relations of the form f(x) = k⋅xh are called
power law relations. |
Science is filled with power laws. For instance, |
Hooke's law for springs: F(x) = -k⋅x |
Newton's law of gravitation: F(r) = GMm⋅r-2 |
the allometry of animal metabolic rates: metabolic rate =
k⋅(weight)3/4. It turns out this is a bit of a
surprise. |
|
By themselves, power laws do not imply fractal structure. |
For example, the Stellpflug formula relating weight and radius of
pumpkins is |
weight = k⋅radius2.78 |
yet no one would say a pumpkin is a fractal. |
A pumpkin is a
roughly spherical shell enclosing an empty cavity; a pumpkin certainly isn't made
up of smaller pumpkins. |
Rather, this is the scaling relation between the thickness
of the shell and the size of the pumpkin. |
Here are two examples where a power law does give a dimension. |
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Here is another example, illustrating how power law plots
can reveal fractal patterns present when random arrangements obscure
strict geometric hierarchies. |