Файл:Coxeter circles.pngBlue circle 0 is tangent to circles 1, 2 and 3, as well as to preceding circles −1, −2 and −3.
In geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged so that any four consecutive circles in the sequence are pairwise mutually tangent. This means that each circle in the sequence is tangent to the three circles that precede it and also to the three circles that follow it.
The radii of the circles in the sequence form a geometric progression with ratio
<math display=block>k=\varphi + \sqrt{\varphi} \approx 2.89005 \ ,</math>
where <math>\varphi</math> is the golden ratio. This ratio <math>k</math> and its reciprocal satisfy the equation
<math display=block>(1+x+x^2+x^3)^2=2(1+x^2+x^4+x^6)\ ,</math>
and so any four consecutive circles in the sequence meet the conditions of Descartes' theorem.Шаблон:R
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 isШаблон:R
<math display=block> \cos^{-1} \left( \frac {-1} {\varphi} \right) \approx 128.173 ^ \circ \ .</math>
The angle between consecutive triples of centers is
<math display=block>\theta=\cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ,</math>
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.Шаблон:R
History and related constructions
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.Шаблон:R It can be interpreted as a degenerate special case of the Doyle spiral.Шаблон:R
