  We prove some rigidity theorems for configurations of closed disks. First,
fix two collections $\mathcal{C}$ and $\tilde{\mathcal{C}}$ of closed disks in
the Riemann sphere $\hat{\mathbb{C}}$, sharing a contact graph which
(mostly-)triangulates $\hat{\mathbb{C}}$, so that for all corresponding pairs
of intersecting disks $D_i, D_j \in \mathcal{C}$ and $\tilde D_i, \tilde D_j
\in \tilde{\mathcal{C}}$ we have that the overlap angle between $D_i$ and $D_j$
agrees with that between $\tilde D_i$ and $\tilde D_j$. We require the extra
condition that the collections are "thin", meaning that no pair of disks of
$\mathcal{C}$ meet in the interior of a third, and similarly for
$\tilde{\mathcal{C}}$. Then $\mathcal{C}$ and $\tilde{\mathcal{C}}$ differ by a
M\"obius or anti-M\"obius transformation. We also prove the analogous
statements for collections of closed disks in the complex plane $\mathbb{C}$,
and in the hyperbolic plane $\mathbb{H}^2$.
  Our method of proof is elementary and self-contained, relying only on plane
topology arguments and manipulations by M\"obius transformations. In
particular, we generalize a fixed-point argument which was previously applied
by Schramm and He to prove the analogs of our theorems in the circle-packing
setting, that is, where the disks in question are pairwise interiorwise
disjoint. It was previously thought that these methods of proof depended too
crucially on the pairwise interiorwise disjointness of the disks for there to
be a hope for generalizing them to the setting of configurations of overlapping
disks.
  We end by stating some open problems and conjectures, including conjectured
generalizations both of our main result and of our main technical theorem.
Specifically, we conjecture that our thinness condition is unnecessary in the
statements of our main theorems.
