This question is based on another question I asked here.
Let $G$ be a torus or a sphere in $\mathbb{R}^3$. I will work with a sphere because I think it is easier. The goal is to color the surface of the sphere using white, black, and grey scales. The goal is to color points that are the farthest apart (the notion of distance is the shortest distance over the surface) with black and white (it does not matter which one white and which one black) and points that are close with similar colors.
Then, the north pole would be black and the south pole would be white (since they are the farthest apart possible). However, one can take any other two opposite points, $a$ and $b$, and would have to color them black and white, respectively, but then $a$ is black and the south pole is white, but they are not the farthest apart. Hence this shows that such coloring is not possible.
My question is: is this equivalent to the statement (with the proof) of the my previous question, namely: that there is no $f:G\rightarrow \mathbb{R}$ such that $d_G(x,y)=d_{\mathbb{R}}(f(x),f(y))$ where $d_{G}(x,y)$ is the length of the shortest path defined over the surface of the torus (or sphere) between points $x$ and $y$, and $d_{\mathbb{R}}(x,y)=|x-y|$.
Or could this reasoning be used a proof of my previous question?