  In this paper we introduce the characteristic dimension of a four dimensional
$\mathcal{N}=2$ superconformal field theory, which is an extraordinary simple
invariant determined by the scaling dimensions of its Coulomb branch operators.
We prove that only nine values of the characteristic dimension are allowed,
$-\infty$, 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing
principle to the vast landscape of 4d $\mathcal N=2$ SCFTs. Whenever the
characteristic dimension differs from 1 or 2, only very constrained special
K\"ahler geometries (i.e. isotrivial, diagonal and rigid) are compatible with
the corresponding set of Coulomb branch dimensions and extremely special,
maximally strongly coupled, BPS spectra are allowed for the theories which
realize them. Our discussion applies to superconformal field theories of
arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the
way, we predict the existence of new $\mathcal{N}=3$ theories of rank two with
non-trivial one-form symmetries.
