Among the many examples that show the mathematical nature of our Universe, if you ask me to choose anyone, I would choose the mathematical model that describes trajectories of a body in a gravitational field. The thought that some of these curves described by the bodies in heaven are the ones that you get by slicing a cone in different ways can make anyone fall in love with mathematics instantly.
This example lets you add bodies that generate force fields and dynamically see the trajectory of particles in the force field based on the position, velocity, mass, etc.