en>Jahoe: Removed mergeto template. See talk page

2010-12-21T20:21:04Z

Removed mergeto template. See talk page

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[[Image:Coxeter circles.png|thumb|right|350px|Blue 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>k=\varphi + \sqrt{\varphi} \approx 2.89005 \ ,</math>

where φ is the [[golden ratio]]. ''k'' and its reciprocal satisfy the equation

:<math>(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]].

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

:<math> \cos^{-1} \left( \frac {-1} {\varphi} \right) \approx 128.173 ^ \circ \ .</math>

The construction is named after geometer [[H.S.M. Coxeter|Donald Coxeter]], who generalised the two-dimensional case to sequences of spheres and [[hypersphere]]s in higher dimensions.

==See also==
*[[Apollonian gasket]]

==External links==
*{{MathWorld|urlname=CoxetersLoxodromicSequenceofTangentCircles |title=Coxeter's Loxodromic Sequence of Tangent Circles}}
*[http://www.math.ca/Events/winter98/w98-abs/node6.html H. S. M. Coxeter - The Descartes circle theorem and Fibonacci numbers]

[[Category:Euclidean plane geometry]]

Sphere mapping - Revision history

en>Jahoe: Removed mergeto template. See talk page