Where are all the holes in circle packings?

What is it about?

Everybody knows that when packing circles (or disks) as tight as possible together there are holes between the circles. Sometimes these holes are large, that if small circles can be moved into these holes, the packing would become tighter. The article presents an exact method to achieve this goal.

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Why is it important?

To pack things as dense as possible is an very old problem since centuries. Many applications in sciences and technology exist. Circle packing is very simple to understand, it needs only a brief mathematical formulation and it inspires enthusiasts all over the world.