I was recently thinking about space filling curves and the following question popped into my head. Clearly $\mathbb{R} \times \mathbb{R}$ is equinumerous to $\mathbb{R}$ so that there is a bijection between the two sets, naturally this indicates that there is a bijection between the real number line and a plane.
I understand that this indicates that there would then exist a function that takes the length of a given line, and maps it bijectively to points of a plane. Naturally the first idea would be a simple Peano curve, however for any point on the plane the length to that point, following the space filling curve, would be infinite. However the bijection between the two spaces would imply that for every point $(x, y)$ in the plane, there would exist a finite real number $a$, such that $f(a) = (x, y)$.
Taking space filling curves into account, this doesnt seem to work if we parameterise this function under the arc length, as for any point on the plane the length to that point would be infinite. So my initial approach of trying to use a Peano curve under an arc length parameterisation to obtain this bijection would not pan out.
Any space filling curve would have hausdorff dimension 2 so maybe in some sense this would not even be a mapping from 1d onto 2d, however clearly the notion of hausdorff dimension is very different to the classical "number of elements in a tuple/vector" understanding so I don't think it would be relevant in the discussion at all, but I am bringing it up in case it is a problem (I have no formal experience with fractal geometry yet)?
My question is:
Whether there exists a constructed example of a bijective function like this, a univariate vector valued bijection $f(a) = (x, y)$? Or whether we are only aware of it's existence via some spin on a diagonal argument?
Whether anyone could shed some light as to why exactly an arc length parameterisation of a space filling curve approach doesn't work if there exists a bijection between $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$, intuitively, if there were to exist such a bijection, it would necessarily be a space filling curve if it were traced out, so why is it that such an approach seemingly doesn't work?
In the case that we only know of the existence of such a function, without constructed examples, I am really confused as to how it could possibly exist, considering that any map would have to trace out a space filling curve, which means that there cannot exist a bijection parameterized with the arc length and I am pretty sure that this indicates there cannot exist a parameter transform for which it would exist either (intuitively at least, I don't see how that is happening)?
The only possible way I could see working is if the bijection was not a "continouous" curve but rather a "chaotic mapping" where real numbers are assigned arbitrarily to points in space.
Kind regards, Vib