The groups, SO(n), admit universal double covers called Spin groups which are generally mysterious, but have very simple descriptions when n=3,4,5,6. These are called the low-dimensional coincidences. We will go through these nice cases and then justify them using a tool from the theory of Lie algebras, Dynkin diagrams. We will also see why these coincidences stop as soon as n=7. If time allows, we will look at the triality of so(8).