Fractal Orthogonal Circles

Geometry Level 5

Let R>1R>1 be the ratio of diameters of successive circles in an infinite self-similar sequence.

Let nn be any nn-th circle in the sequence. Then the following statements are true:

  • Circle nn intersects circles n2,n1,n+1,n+2n-2, n-1, n+1, n+2 at right angles and no other circles.
  • Circle nn is tangent only to circles n4,n+4n-4, n+4, at points where they intersect circles n2,n+2n-2, n+2 respectively

Find 10000×R\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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