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Three circles

Let these circles with radiuses $r1, r2, r3,$ and $r3$ is the smallest radius. Also let the following operator $*$ be defined as $$x * y = \frac{xy}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})^2}$$
then it can be proven that $r3 = r1 * r2, \forall r1, r2 \in [0, \infty)$.
It can also be proven that $x * x * x \ * \ ... \ * \ x = \frac{x}{n^2}$ when $n$ is the number of times $x$ has been composed $\forall n \in \mathbb{N} - \{0\}$.

Let $x \circ y = \pi (x * y)(x * y) + x \circ (x * y) + y \circ (x * y)$.

$x \circ y \ $ will equal an infinite sum of circle areas that are between the two starting circles with radii equal to $x$ respectively $y$.
I want to find out towards what value $x \circ y$ converges.

It can also be reasoned that $x \circ y = \binom{n + m}{n} (\frac{x}{n^2} * \frac{y}{m^2})$ where $(n, m)$ takes all pairs of ordered integers.

Marc Grec
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  • I'm confused how $x \circ y$ is defined in terms of itself. Also confused how the $\binom{n+m}{n}$ appears. – marty cohen May 31 '21 at 14:54
  • $\binom{n + m}{n}$ appears because if you unfold the recursion some terms in the sum will appear more than once and the number of times it appears is $\binom{n + m}{n}$. If you want I could try to reason that. – Marc Grec May 31 '21 at 14:57
  • $x \circ y$ is defined in terms of itself because it's a limit and you have to apply the recursive formula infinitely to get the answer. @martycohen – Marc Grec May 31 '21 at 14:59
  • I wouldn't be astonished that your issue is connected with Ford circles – Jean Marie May 31 '21 at 21:11
  • It would be good to explain the filiation with the Descartes-Soddy relationship that can be found in different places, for example in one of my questions with the explanation that one of the curvatures is $0$. – Jean Marie Jun 01 '21 at 11:28
  • Have you remarked that your rule $x \star y=\frac{xy}{x+y+2\sqrt{xy}}$ is associative: $(x \star y)\star z=x \star (y \star z)$ ? The geometrical interpretation is interesting. – Jean Marie Jun 01 '21 at 11:41
  • I think you should add a visualization of the circles you are working on with your recursive formula (which is akin to the question about Apollonian gaskets I mentionned above) – Jean Marie Jun 01 '21 at 11:42
  • Yes Ford circles are apparently what I was looking for, thank you! @JeanMarie – Marc Grec Jun 01 '21 at 13:32

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