Explore scale invariance: fractals, power laws, renormalization group flows, and self-similar functions. Dilation x→λx maps power laws f(x)=x^α to themselves.
A function f is scale-invariant if f(λx)=λ^α f(x). Power laws are the unique continuous scale-invariant functions. At critical points (phase transitions, turbulence, fractals), physical systems obey scale invariance → power-law correlations. Renormalization group: fixed points of RG transformation are scale-invariant theories. Weierstrass function W(x)=Σb^{-αn}cos(b^n x) is self-similar with Hausdorff dimension 2-α.