There's an amazing way to derive Maxwell's equations, and the equations for 3 of the other fundamental forces of nature, directly from Lie group symmetry. You try to write down a theory that is symmetric under -local- symmetry transformations, meaning that the theory should give the same predictions even if you rotate (in some abstract space that you tack into the theory) by an angle that depends on position arbitrarily. At first this seems impossible, because any derivatives with respect to position will depend on the spatial variation of the rotation angle. But if you add an additional field that subtracts off the variation in the rotation angle, you find that this field is a dynamical object that coincides with the electromagnetic field! (Or to be more correct, it's the vector potential, which is directly related to electric and magnetic fields).

So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.