Why the Inverse-Square Law Is Dimensional: A Discrete-Medium Illustration from One to Three Dimensions

Students meet the fact that gravity and the electrostatic force both fall off as 1/r² while the strong nuclear force is short-ranged, but the reason is usually asserted rather than shown. This article turns a simple discrete "toy universe" into a teaching device that makes the reason explicit and that students can run for themselves. In one dimension a single exponential force law between localized "disturbances" cannot reproduce an inverse-square law, and the attempt fails for reasons that are themselves the lesson: exponentials are not power laws, the relevant error is relative rather than absolute, and a power law is the unique scale-invariant function, impossible on a lattice that carries a built-in length. Restoring three dimensions makes the inverse-square law appear on its own, from the geometry of flux through a sphere, realized as the lattice Green’s function of a screened Poisson equation; the same single force then behaves like electromagnetism, gravity, or a short-range Yukawa force according to what it couples to and whether its mediator is massive. Finally, the lattice spacing is shown to play two separable roles — a regulator that removes infinities and a putative physics with short-range, anisotropic fingerprints that vanish in the continuum. Accompanying Python scripts reproduce every figure.