Coxeter's loxodromic sequence of tangent circles

The radii of the circles in the sequence form a geometric progression with ratio

$${\displaystyle k=\varphi +{\sqrt {\varphi }}\approx 2.89005\ ,}$$

where ${\displaystyle \varphi }$ is the golden ratio. This ratio ${\displaystyle k}$ and its reciprocal satisfy the equation

$${\displaystyle (1+x+x^{2}+x^{3})^{2}=2(1+x^{2}+x^{4}+x^{6})\ ,}$$

and so any four consecutive circles in the sequence meet the conditions of Descartes' theorem.[1][2]

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[1]

$${\displaystyle \cos ^{-1}\left({\frac {-1}{\varphi }}\right)\approx 128.173^{\circ }\ .}$$

The angle between consecutive triples of centers is

$${\displaystyle \theta =\cos ^{-1}{\frac {1}{\varphi }}\approx 51.8273^{\circ },}$$

the same as one of the angles of the Kepler triangle, a right triangle whose construction also involves the square root of the golden ratio.[3]

The construction is named after geometer H. S. M. Coxeter, who generalised the two-dimensional case to sequences of spheres and hyperspheres in higher dimensions.[1][4][5] It can be interpreted as a degenerate special case of the Doyle spiral.[2]

• Apollonian gasket

• Weisstein, Eric W., "Coxeter's Loxodromic Sequence of Tangent Circles", MathWorld