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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\equiv {R_{n+1}\over R_n} = \phi+\sqrt{\phi}\,,

where $\phi$ is the Golden Ratio (Gardner 1979ab). Coxeter (1968) generalized the sequence to Spheres.

See also Arbelos, Golden Ratio, Hexlet, Pappus Chain, Steiner Chain


Coxeter, D. ''Coxeter on `Firmament.'''

Coxeter, 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.

© 1996-9 Eric W. Weisstein