Inspired By Islamic Art

Geometry Level 5

Infinitely many tangent circles cover the plane in a repeating pattern as per graphic showing some of them.

Let \(x\) be the fraction of the plane the circles cover.

Evaluate \(\left\lfloor 1000x \right\rfloor\).

Note: There are just 5 different circle radii, corresponding to circles (from smallest to largest) that are tangent to 4, 5, 6, 7, 8 other circles. 14 of these circles can form a pattern that "tiles the plane".


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