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.