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foreign well thank you all for you know coming back repeatedly um so Iamp;#39;m excited today because thereamp;#39;s a bunch of things a bunch of cool facts about apollonian Circle packings that that weamp;#39;re gonna see um so we talked about the apollonian group if you remember it was given in terms of swaps and then the superapalonian group you can add some inversions and so those apollonian Circle packings um arenamp;#39;t in any position in the plane when you describe it that way now I want to give another version of the apollonian group which puts them into the plane so sometimes people call this the geometric apollonian group and the idea is that we can use Mobius Transformations so this I might write like this and it lives inside PSL to well actually turns out it lives inside psl2z to join I um together with if you like complex conjugation so sometimes people call that like generalized Mobius Transformations or something like that if you add complex conjugation and the ide
