Integral Apollonian Circle Packings|
Abstract: Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature. In fact, it is even possible for a packing to be oriented in the Euclidean plane so that for each circle $C_i$ in the packing with center $(x_i,y_i)$ and radius of curvature $r_i$, all the quantities $r_i$, $r_i x_i$ and $r_i y_i$ are integers.
In this talk we study these remarkable packings via the Descartes equation, which relates the radii of curvature of four mutually tangent circles: $2 ( x^2 + y^2 + z^2 + w^2) - (x + y + z + w)^2 = 0.$ We classify integral Apollonian packings under Euclidean motions by the root quadruples, which are the smallest quadruples in the packings. We analyze the Apollonian packings by introducing the Apollonian operators and the Apollonian group, and prove that the Apollonian operators generate a conjugate of the proper Lorentz group. Finally we discuss the some congruence conditions and conjectures on the integers represented by an Apollonian packing.