So I noticed that it was possible to assign a real number to every lattice point in $R^2$ while looking at a wall somewhere. This got me thinking about a path that could "go through" every lattice point, and then making this path arbitrarily dense so that an arbitrary point in $R^2$ could be represented by a limit for the arbitrarily dense function, call it $T(x)$. This would allow us to represent an arbitrary vector $(a,b)$ as $\lim_{n\to\infty}[T(f(a,b,n))]$ where $f$ is an expression that will be determined by the transformation $T(x)$. This shouldn't be possible though because a 1-dimensional structure (here $T(x)$) can never completely "fill" $R^2$ or (any solid region in $R^2$ for that matter). I realize that this probably makes more sense to me than you so I came up with an example of what I mean. The transformation from $R$ to $R^2$ given by $T(x)=(x-\lfloor x\rfloor, 2^{-\lfloor log_2(x+1)\rfloor}\lfloor x+2-2^{\lfloor log_2(x+1)\rfloor}\rfloor)$ maps the real line for $x\in[0,\infty)$ onto the open unit square on the Cartesian Plane (vertices $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$).
For $x\in[0,1)$ the output is the line y=1, for $x\in[1,2)$ the line $y=1/2$, for $x\in[2,3)$ the line $y=2/2=1$, for $x\in[3,4)$ the line $y=1/4$, for $x\in[4,5)$ the line $y=2/4=1/2$, and so on. If that doesn't make it clear a quick plot on a calculator/computer should clear things up.
Then the "solution" is $lim_{n\to\infty}[T(a+\lfloor 2^{n}(b+1)\rfloor]=(a,b)$ as long as the point $(a,b)$ is inside the unit square. That's what I consider paradoxical: that the left hand side has only one vector component while the right has two that are arbitrary (not really arbitrary since they need to be inside the unit square but it is still a solid region). What's more is that paths like this one can be used to represent an arbitrary vector of arbitrary dimension but the only dimension I can really "wrap my head around" is $R^2$, which is why my example is in $R^2$. So what's going on? Is it because the expression inside $T(x)$ doesn't converge? Is that why? I apologize if maybe this question isn't very well conveyed.