What are the "derivations" of the inverse-square law?

Question

Besides the derivation mentioned in this Wiki article, I want to know, if there exists any other derivation of the inverse-square law based on some profound physical/philosophical concepts.

Answer 1

Based on an informal assumption, we could derive the inverse square law for gravitational force and Coulomb force.

Assumption

Suppose everything in the space is scaled up by a factor of $k$, and time stays the same, then we shouldn't expect anything to change.

Derivation

Since we are in a 3D space, any volume would be scaled by $k^3$, and so would mass.

From $F = ma$:

$$F' = (k^3m)(ka) = k^4ma = k^4F$$

Then suppose $F_G = G\frac{m_1m_2}{r^n}$; after scaling up, $F_G' = G\frac{(k^3m_1)(k^3m_1)}{k^nr^n} = k^{6-n}F_G$; also, we know $F_G' = k^4F_G$, so $6-n = 4$, and $n = 2$.

Note: this piece of text is composed by one of my friends, Yushun Cheng.