TITLE: Spirals in Apollonian circle-packings
QUESTION [6 upvotes]: Given mutually (externally) tangent circles $C_1,C_2,C_3$,
let $C_n$ be the unique circle externally tangent to
$C_{n-1}$, $C_{n-2}$, and $C_{n-3}$ for $n \geq 4$.
Let $P_{\infty}$ be the point toward which the $C_n$'s tend,
let $P_n$ be the center of $C_n$, and
let $Q_n$ be the point of tangency between $P_{n}$ and $P_{n+1}$.
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 $Q_n$?
Although I would like the answer to be that these unit vectors
are uniformly distributed over the unit circle, I see at least
a couple of reasons to doubt this: first, I suspect that for
certain special initial choices of $C_1,C_2,C_3$, the set
of unit vectors that arise is finite, and second, the symmetry
group governing Apollonian packings is the group of conformal
maps, which does not preserve Lebesgue measure on the circle.
Still, I suspect that for generic choices of $C_1,C_2,C_3$,
the unit vectors are asymptotically distributed according to
a measure that is uniformly continuous with respect to
Lebesgue measure on the circle, and I suspect that the
measures that arise in this way admit a nice characterization,
e.g., the set of all measures that are conformally equivalent
to Lebesgue measure.
(This is a more refined version of my MathOverflow post
Isotropy of Apollonian disk-packing . See also my follow-up question Three-dimensional Apollonian spirals .)

REPLY [5 votes]: This configuration seems to be
Coxeter's loxodromic sequence of tangent circles.  According to Wikipedia:

The radii of the circles in the sequence form a geometric progression with ratio
$$k=\varphi + \sqrt{\varphi} \approx 2.89005\ ,$$
where φ is the golden ratio. k and its reciprocal satisfy the equation
$$(1+x+x^2+x^3)^2=2(1+x^2+x^4+x^6)\ .$$
The centres of the circles in the sequence lie on a logarithmic spiral. Viewed from the centre of the spiral, the angle between the centres of successive circles is
$$\cos^{-1} \left( \frac {-1} {\varphi} \right)\ .$$

Higher dimensional generalization was done by Coxeter (1968).