Three-dimensional Apollonian spirals

Question

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let $P_{\infty}$ be the point toward which the $S_n$'s tend, let $P_n$ be the center of $S_n$, and let $Q_n$ be the point of tangency between $S_{n}$ and $S_{n+1}$. What can be said about the asymptotic distribution of the unit vectors pointing from $P_{\infty}$ to $P_n$? What about the unit vectors pointing from $P_{\infty}$ to $Q_n$?

This was part of what I had in mind when I posted Spirals in Apollonian circle-packings. My hope is that this process (or something like it) gives a way to generate uniformly-distributed points on the sphere; my fear is that (as in the case of a rotating ellipsoid or rectangular parallelepiped) there's a kind of instability at play, so that asymptotically one gets a uniform distribution on some great circle rather than the whole sphere.

Answer 1

According to the discussion in Coxeter (1968), the tangent points lie asymptotically on a concho-spiral, so the distribution is not uniform on the sphere, but is uniform on a circle.

By the way, the circle is in general not a great circle. In fact, the process can be bidirectional, so there will be two accumulation points. The tangent points are not on a plane but on a "tube in hyperbolic space" / "binodal cyclide" / "inversed circular cone" (help ... what's the standard name for this surface?) .