system
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human
ELI5: Why does nature like square roots and cube numers?

Late bloomer when it comes to physics here but I wonder why nature is first of all describable in such a precise way with numers and letters and why nature even "obeys" to square roots and cube numbers.
Take for example Newton's law of universal gravitation:
F=GMm/R²

As far as I know, the idea of square roots and cube numbers were invented some time BC, so way earlier than any serious physical discoveries regarding gravitation. It would make sense to me if we invented square roots and cuber numbers AFTER we observed nature and tried to create formulas that gave us precise output, but the fact that it happened the other way around is astonishing to me (not that we came up with the mathematical concept, but that nature "likes" these concepts so much).

It almost feels like a discovery of the "code" of nature that seems almost too simple to be good enough for reality.
gpt
The answer is dimensions. Things that scale in two or three dimensions tend to use square and cube numbers respectively.

To take the law of universal gravitation as an example - when talking about gravity, you have a massive object with a nonzero volume (usually a sphere) that has gravity. The surface of this object is essentially two-dimensional. As you move away from the object (increasing R) gravity "spreads out" in these two dimensions, which results in the force of gravity decreasing with the square of distance.

Light works the same way, light coming from a spherical source (like the sun) spreads out in two dimensions, which means the amount of sunlight you get decreases with the square of distance from the sun.