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Coxeter's Loxodromic Sequence of Tangent Circles

An infinite sequence of circles such that every four consecutive circles are mutually tangent, and the circles' radii ..., R_(-n), ..., R_(-1), R_0, R_1, R_2, R_3, R_4, ..., R_n, R_n+1, ..., are in geometric progression with ratio

k=(R_(n+1))/(R_n)=phi+sqrt(phi),

where phi is the golden ratio (Gardner 1979ab). Coxeter (1968) generalized the sequence to spheres.

See also

Arbelos, Bowl of Integers, Golden Ratio, Hexlet, Pappus Chain, Steiner Chain

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References

Update a linkCoxeter, D. "Coxeter on 'Firmament.' " http://www.bangor.ac.uk/SculMath/image/donald.htmCoxeter, H. S. M. "Loxodromic Sequences of Tangent Spheres." Aequationes Math. 1, 112-117, 1968.Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240, 18-28, Jan. 1979a.Gardner, M. "Mathematical Games: How to be a Psychic, Even if You are a Horse or Some Other Animal." Sci. Amer. 240, 18-25, May 1979b.Robinson, J. "Firmament." http://www.popmath.org.uk/sculpture/pages/donald.html.

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Coxeter's Loxodromic Sequence of Tangent Circles

Cite this as:

Weisstein, Eric W. "Coxeter's Loxodromic Sequence of Tangent Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CoxetersLoxodromicSequenceofTangentCircles.html

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