Fractal Orthogonal Circles

Geometry Level 5

Let \(R>1\) be the ratio of diameters of successive circles in an infinite self-similar sequence. Let \(n\) be any \(n\)-th circle in the sequence. Then the following statements are true:

  • Circle \(n\) intersects circles \(n-2, n-1, n+1, n+2\) at right angles and no other circles.
  • Circle \(n\) is tangent only to circles \(n-4, n+4\), at points where they intersect circles \(n-2, n+2\) respectively

Find \(\left\lfloor 10000\times R \right\rfloor \).

Bonus: Find exact answer, which is short and pretty.

Note: A self-similar geometrical sequence is congruent to itself, after scaling, translation, and rotation.


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