Summary
We prove two theorems about the circular Apollonian packing. The first states that the circles used in the packing have an infinite total length. The second states that the largest circle is covered by the other circles used in the packing, up to an area of Lebesgue measure zero. Our proofs are elementary and only use basic coordinate geometry.
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Alexandru Tupan
ALEXANDRU TUPAN received his Ph.D. in mathematics from the Johns Hopkins University. He held visiting positions at Concordia University, McGill University, and Michigan State University before joining the faculty at University of Wisconsin, River Falls. He enjoys spending time with Eun Young, their two children Angela and Andrei, and their cat Simba.