Integral Apollonian packing and Reduction Theory

Apollonian circle packing arise by repeatedly filing the interstices between mutually tangent circles with further tangent circles. We will discuss two possible apllonian packing on a plane and also the soddy packing for dimension 3. There are several interesting things to ask about the apollonian circle packing. In particular, we will look at the reduction theory and perhaps its connection to the Duke’s theorem on counting the number of primitive packing up to height N.