On the way from Pappus Chains (previous post) to Clifford’s Circle Theorem I was diverted by the thought that the equations governing Pappus Chains should also govern, possibly with slight modification, Steiner Chains.
Steiner Chains were briefly discussed in Martin Gardner’s 1979 Scientific American article and they are a generalisation of Pappus Chains (or to be more accurate Pappus Chains are a limiting case of Steiner Chains).
A distorted annulus is formed by two circles, one within the other. A Steiner Chain is an infinite sequence of circles tangent (externally) to the circles forming the distorted annulus and to each other, using a particular inscribed circle as the first in the sequence.
In certain circumstances the chain reproduces itself after one ‘circuit’ of the distorted annulus. In these cases the chain will close itself regardless of where the first circle is placed (see diagram below reproduced from Gardner’s article).
This result, referred to as ‘Steiner’s Porism’, is usually explained using inversion geometry (see for example the relevant chapter of ‘Geometry Revisited’ by H S M Coxeter and S L Greitzer).
Pappus Chains are obviously a limiting case of Steiner Chains, where the circles forming the distorted annulus are tangent. The centres of the circles forming Steiner Chains are known to lie on an ellipse (see relevant Mathworld article) so quite possibly the parameters of Steiner Chains, like those of Pappus Chains, are determined by the parameters of the ellipse. In that case, the cases when the chain closes will be determined by a diophantine equation involving those parameters.
Of course it is quite possible that if the chain does not close after one ‘circuit’ of the distorted annulus, it will close after two or more.