In 1643, Ren\'{e} Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called \emph{thin} subgroup $\Gamma$ of $\textrm{GL}_4(\mathbb Z)$. This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs associated to $\Gamma$ -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.