In 1665, the clockmaker Christiaan Huygens noticed that two pendulum clocks hanging on a wall tend to synchronize the motion of their pendulums. A similar scenario occurs with two metronomes placed on a piano: they interact through vibrations in the wood and will eventually coordinate their motion. These effects are stable against small perturbations. Such stability is not predicted by either Hamiltonian mechanics or by few-body quantum theory. Nonetheless they can be seen as occurring within a simple model introduced by Kolmogorov. Surprisingly, this model leads to a very complex phase diagram. In turn, the complexities of this phase diagram have been observed within experimental observations of fluid flow, solid state devices, and non-linear electrical circuits. It is reflective of the structure of number theory and of the relation between rational and irrational numbers. Of course, the synchronization arises from friction, an effect often neglected in fundamental theories. Should we then regard synchronization, and its deeply mathematical explanation, as an example of an emergent phenomenon? What does emergence mean? Is is just something that surprises us? How are emergent phenomena connected with the fundamentals of our physical theories?