geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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.

Properties

The radii of the circles in the sequence form a

geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...

with ratio

k=\varphi + \sqrt \approx 2.89005 \ ,

where

\varphi

is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. This ratio

k

and its reciprocal satisfy the equation

(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 In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mu ...

. The centres of the circles in the sequence lie on a

logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...

. Viewed from the centre of the spiral, the angle between the centres of successive circles is

\cos^ \left( \frac   \right) \approx 128.173 ^ \circ \ .

The angle between consecutive triples of centers is

\theta=\cos^\frac\approx 51.8273^\circ,

the same as one of the angles of the

Kepler triangle A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...

, a right triangle whose construction also involves the square root of the golden ratio.

History and related constructions

The construction is named after geometer

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, who generalised the two-dimensional case to sequences of spheres and

hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...

s in higher dimensions. It can be interpreted as a degenerate special case of the

Doyle spiral In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposi ...

References

External links

*{{MathWorld, urlname=CoxetersLoxodromicSequenceofTangentCircles , title=Coxeter's Loxodromic Sequence of Tangent Circles, mode=cs2 Circle packing Golden ratio

Properties

History and related constructions

See also

References

External links