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Gravity and electromagnetism are inverse-square laws. This makes geometric sense -- if you build a spherical shell around a lamp then a shell with twice the radius has four times the surface area and hence a quarter of the intensity per unit area.

The strong interaction has a nontrivial distance-strength relation with an exponential term. Is there an intuitive geometrical explanation for this?

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I don't know of a geometrical explanation, but there is an explanation within quantum mechanics. In gauge theories forces are "carried" by vector bosons. For example, the electromagnetic force is carried by the photon, and the nuclear force (related to the strong force) is carried by the $\pi$ meson. The lifetime of the photon is infinite, so the outgoing flux of photons decreases by the inverse square of distance. The $\pi$ meson has a finite lifetime, so the outgoing flux of vector bosons falls off due to both the inverse square geometrical effect, and the declining flux due to the meson's lifetime.

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