Power Law Relations

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.
Crumpled paper is a fractal.
Clusters of peas are not fractal.
Here is another example, illustrating how power law plots can reveal fractal patterns present when random arrangements obscure strict geometric hierarchies.

Return to 2. B. Box-Counting Dimension.