Introduction to Apollonian circle packings (tangent)

This is not my area of research at all, but I think it’s super cool.  The first time I heard of Apollonian circle packings was at that conference I went to in Marseille last month, during the first lecture of Hee Oh’s minicourse.  So here’s a quick write up  background of the first third of that lecture.

These packings and all this theory come from one dude, Apollonius of Perga, who wrote a bunch of math books back around 200 BC.  Literally this math has been around for 2200 years.  Here’s a paper submitted a month ago which is a generalization of Apollonius’s problem from circles to spheres.  Math is so amazing!  We live in history!

To understand the problem, we’ll have to do a quick geometry brush up.  We say that two shapes or curves or lines in the plane are tangent if they touch at exactly…