| 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 |
| 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. |
|
| In 2.5 Other Dimensions we shall
interpret this exponent as the mass dimension. |
| Here is another example, illustrating how power law plots
can reveal fractal patterns present when random arrangements obscure
strict geometric hierarchies. |
