  Polarizability is a key response property of physical and chemical systems,
which has an impact on intermolecular interactions, spectroscopic observables,
and vacuum polarization. The calculation of polarizability for quantum systems
involves an infinite sum over all excited (bound and continuum) states,
concealing the physical interpretation of polarization mechanisms and
complicating the derivation of efficient response models. Approximate
expressions for the dipole polarizability, $\alpha$, rely on different scaling
laws $\alpha \propto$ $R^3$, $R^4$, or $R^7$, for various definitions of the
system radius $R$. Here, we consider a range of single-particle quantum systems
of varying spatial dimensionality and having qualitatively different spectra,
demonstrating that their polarizability follows a universal four-dimensional
scaling law $\alpha = C (4 \mu q^2/\hbar^2)L^4$, where $\mu$ and $q$ are the
(effective) particle mass and charge, $C$ is a dimensionless excitation-energy
ratio, and the characteristic length $L$ is defined via the
$\mathcal{L}^2$-norm of the position operator. %The applicability of this
unified formula is demonstrated by accurately predicting the dipole
polarizability of 36 atoms and 1641 small organic~molecules. This unified
formula is also applicable to many-particle systems, as shown by} accurately
predicting the dipole polarizability of 36 atoms, 1641 small organic
\rrr{molecules, and Bloch electrons in periodic systems.
