Apollonian circle packings: Dynamics and Number theory Hee Oh YaleUniversity ICWM, 2014 Apollonius of Perga (cid:73) Lived from about 262 BC to about 190 BC. (cid:73) Known as “The Great Geometer”. (cid:73) His famous book on Conics introduced the terms parabola, ellipse and hyperbola. Apollonius’ theorem Theorem (Apollonius of Perga) Given 3 mutually tangent circles, there exist exactly two circles tangent to all three. Proof of Apollonius’ theorem We give a modern proof of this ancient theorem using Mobius transformations: For a,b,c,d ∈ C, ad −bc = 1, (cid:18) (cid:19) a b az +b (z) = , z ∈ C∪{∞}. c d cz +d A Mobius transformation maps circles (including lines) to circles, preserving angles between them. In particular, it maps tangent circles to tangent circles. Proof of Apollonius’ theorem We give a modern proof of this ancient theorem using Mobius transformations: For a,b,c,d ∈ C, ad −bc = 1, (cid:18) (cid:19) a b az +b (z) = , z ∈ C∪{∞}. c d cz +d A Mobius transformation maps circles (including lines) to circles, preserving angles between them. In particular, it maps tangent circles to tangent circles. Proof of Apollonius’ theorem We give a modern proof of this ancient theorem using Mobius transformations: For a,b,c,d ∈ C, ad −bc = 1, (cid:18) (cid:19) a b az +b (z) = , z ∈ C∪{∞}. c d cz +d A Mobius transformation maps circles (including lines) to circles, preserving angles between them. In particular, it maps tangent circles to tangent circles. Proof of Apollonius’ theorem 4 mutually tangent circles Four possible configurations Construction of Apollonian circle packings Beginning with 4 mutually tangent circles, we can keep adding newer circles tangent to three of the previous circles, provided by the Apollonius theorem. Continuing this process indefinitely, we arrive at an infinite circle packing called an Apollonian circle packing . We’ll show the first few generations of this process: