  Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0 = \partial
B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent
to $C_0$ red. What proportion of the concentric circle $C_{\epsilon} = \partial
B(z_0, r_0 + \epsilon)$ is red, and what is the behavior of this quantity as
$\epsilon \rightarrow 0$? Using equidistribution of closed horocycles on the
modular surface $\mathbb{H}^2/SL(2, \mathbb{Z})$, we show that the answer is
$\frac{3}{\pi} = 0.9549\dots$ We also describe an observation due to Alex
Kontorovich connecting the rate of this convergence in the Farey-Ford packing
to the Riemann Hypothesis. For the analogous problem for Soddy Sphere packings,
we find that the limiting radial density is $\frac{\sqrt{3}}{2V_T}=0.853\dots$,
where $V_T$ denotes the volume of an ideal hyperbolic tetrahedron with dihedral
angles $\pi/3$.
