A phase transition is a sharp change in the behavior of a mathematical model as one of its parameters varies. This talk describes a striking phase transition that takes place in conic geometry. First, we will explain how to assign a notion of "dimension" to a convex cone. Then we will use this notion of "dimension" to see that two randomly oriented convex cones share a ray with probability close to zero or close to one. This fact has implications for many questions in signal processing. In particular, it yields a complete solution of the "compressed sensing" problem about when we can recover a sparse signal from random measurements. Based on joint works with Dennis Amelunxen, Martin Lotz, Mike McCoy, and Samet Oymak.