This configuration seems to be the [Coxeter's loxodromic sequence of tangent circles](https://en.wikipedia.org/wiki/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)](http://link.springer.com/article/10.1007%2FBF01817563).