9 Circles Packing in a Square, Contraction Mapping, Iterative Algorithm
this paper shows the proof of 9 circles packing in an square.
They show some configuration of 9 points, that achieve packing.
Next, they make 9 square. Each square contains 1 point.
Then, they make iterative algorithm that shrinks the size of square by the restriction of distance between points. Specifically, certain area in the square cannot be used, so they want to delete that.
Then, there is recurrence formula of the side of square. This is contraction, so the 9 squares converges to the 9 points. That completes the proof.
Comment : Very cool. One of the cool point here is they can show the uniqueness of the configuration at the same time.
