I'd say periodic things express motions on a loop, in the topological sense of looping back to itself. But because topology is only defined up to deformations,

I'd say periodic things express motions on a loop, in the topological sense of looping back to itself. But because topology is only defined up to deformations, thus you cannot really say that loops are circles. However, there is indeed a connection to circles. One way of seeing this by looking at the solutions to this differential equation over the complex "plane" and looking at the exp(i x) and exp(-i x) solutions, and seeing how these functions wrap the real number line around in a circle. The complex plane fundamentally has a Euclidean geometry to it. So much so that if you grew up in a world with a strong non-euclidean geometry, or even grew up as some sort of digital being with no real notion of space at all, so long as you are able perform moderately sophisticated mathematics you are going to wind up discovering the complex numbers, and things like their absolute value and multiplication and the exponential function and how they map values around. Perhaps you never mentally arrange the complex numbers into a "plane", but none the less, all your basic geometric concepts are embedded into the algebraic operations on these complex numbers, and you will wind up effectively doing Euclidean geometry, even if you never identify it as such.