When deriving the Rayleigh-Jeans law in physics, one must count the number of solutions $(a,b,c)$ to the Diophantine equation $$a^2+b^2+c^2=R^2.$$ The source I have linked approximates the number of integer solutions as the volume of the sphere with radius $R$ (so that there are approximately $\frac{4\pi}{3}R^3$ solutions).
It is not clear to me why this approximation works. The volume provides us with the number of "unit cubes" one can fit inside the sphere. How is there (almost) a bijection between these unit cubes inside the sphere and lattice points on the sphere?