We’ll explore the connections between the rotation Lie groups (SU(n) and SO(n)) in two and three dimensions. In doing so, we’ll prove a remarkable theorem about the way that SU(2) and SO(3) relate using quaternions and then discuss some connections to complex analysis at the end. In doing so, we’ll take a very fun and (in my opinion) mind-blowing journey through spheres of different dimensions. Absolutely no knowledge of Lie theory is expected and I’ll be happy to clarify any details from topology or group theory that come up that you aren’t familiar with.