CTK Insights

20 Oct

Golden Ratio - Another Sighting

We'll be looking into a chain of circles all tangent to a given line and to the two immediate predecessors in the chain. The chain starts with two circles tangent to each other of radii a and b. For three circles, we have the following well known result, obtained yet in one of the sangaku:

Given three circles tangent to each other and to a straight line, express the radius of the middle circle via the radii of the other two.

Assuming the radii of the three circles are consecutively r, ρ, and R, the relation between the radii is expressed by

\frac{1}{\sqrt{r}} + \frac{1}{\sqrt{R}} = \frac{1}{\sqrt{\rho}}.

Many generations of amateur and professional mathematicians looked at this formula and saw the relation between the radii of three consecutive circles. Very recently, Giovanni Lucca has observed that the relation is actually the one that defines the Fibonacci sequence

F_{n+1} = F_{n-1} + F_{n}.

Denoting the radii of the circles in the chain as R_{n}, n = 1, 2, ...

\frac{1}{\sqrt{R_{n-1}}} + \frac{1}{\sqrt{R_{n}}} = \frac{1}{\sqrt{R_{n+1}}}

And, with G_{n} = \frac{1}{\sqrt{R_{n}}}, we plainly have a recurrence

G_{n+1} = G_{n-1} + G_{n},

with G_{1} = \frac{1}{\sqrt{a}}, G_{2} = \frac{1}{\sqrt{b}}. This is different from the Fibonacci sequence, for which F_{1} = 1, F_{2} = 1. However, the solution for the recurrence is well known:

G_{n+1} = G_{1}F_{n-1} + G_{2}F_{n}.

For both sequence \lim_{n\to\infty}F_{n+1}/F_{n} = \lim_{n\to\infty}G_{n+1}/G_{n} = \phi, the golden ratio.

Let x_{n} be the coordinate on the given line of the center of n^{th} circle. Giovanni Lucca finds that

x_{n+1} = x_{n} + (-1)^{n-1}\frac{2}{G_{n}G_{n+1}} = x_{1} + 2\sum_{k=1}^{n}{\frac{(-1)^{k-1}}{G_{k}G_{k+1}}}.

He shows that the series is convergent, with the limit x_{\infty} determined from

x_{\infty} - x_{1} : x_{2} - x_{\infty} = \phi\sqrt{a} : \sqrt{b}.

References

  1. Giovanni Lucca, Generalized Fibonacci Circle Chains, Forum Geometricorum Volume 10 (2010) 131–133.

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