| 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. |
|
| Here is another example, illustrating how power law plots
can reveal fractal patterns present when random arrangements obscure
strict geometric hierarchies. |
