The classical version of the inverse square force law was introduced to explain the elliptical orbits of planets. Later the quantum version was was used to explain the hydrogen atom. In both cases, the inverse square force law has a surprising extra conserved quantity that plays a crucial role: the Runge-Lenz vector. Noether showed that conserved quantities correspond to symmetries. What symmetries explain the Runge-Lenz vector? If we understand this, we can see that the hydrogen atom is mathematically almost the same as a massless particle moving at the speed of light in the "Einstein universe": an early model of the universe where space is a sphere in 4 dimensions. And this is just the beginning: the story of the inverse square force law keeps taking surprising new turns.
You can see my slides here, check out a more advanced talk on this subject, or read the paper my talk is based on: