Scale Invariance - Scale-invariant Curves and Self-similarity

Scale-invariant Curves and Self-similarity

In mathematics, one can consider the scaling properties of a function or curve under rescalings of the variable . That is, one is interested in the shape of for some scale factor, which can be taken to be a length or size rescaling. The requirement for to be invariant under all rescalings is usually taken to be

for some choice of exponent, and for all dilations . This is equivalent to f being a homogeneous function.

Examples of scale-invariant functions are the monomials, for which one has, in that clearly

An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (r, θ) the spiral can be written as

Allowing for rotations of the curve, it is invariant under all rescalings ; that is is identical to a rotated version of .