Expressing any given point on plane with one unique number

[Mashiro]

TL;DR

Question on if exists a fractal of a line such that after infinite iterations it could cover any given point on a plane.

Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given point on a plane using only one number.

Intuitionally, I think of lines. I know there exists fractals of a line that could theoretically fill a plane after infinite iterations. Therefore, I believe we can construct a system of expressing a plane based on such fractal:
1. The fractal must pass through all points, therefore arbitrarily define any point as origin, denoted as zero.
2. Based on zero, define a positive direction.
3. To express any given point, simple trace from zero, alone the fractal. The distance from your point to the origin would be the unique number describing the point on the plane.

However, some problems are also raised:
1. Does there exist such rule so that a fractal could pass through every single point on a plane? No matter rational or not.
2. If the theory is correct, due to the fact that set of irrational numbers is a higher level of infinity compared to the set of rational numbers, there is a 100% chance of meeting an "irrational point" by choosing arbitrarily. Is this going to be problematic?

If everything about this theory works out, could we apply the same method to a higher dimension (for instance 3-dimensional), and express any given point in a three dimensional space with two numbers? Or perhaps even one.

I am currently a sophomore and my knowledge about mathematics is basic. I might have made stupid mistakes anywhere above. Please point them out to me if you spot any.

 

[fresh_42]

You can cover a certain square $S\subseteq \mathbb{R}^2$ of the plane with a single line, a Hilbert curve, or a Peano curve. Hence if our curve $\gamma \, : \,[0,1] \longrightarrow S$ hits your point $p\in S$ for the first time (more than once is possible), say $\gamma (t_0)=p,$ then $t_0$ can be thought of as a representation of $p$ by a single number.

The disadvantage is, that different squares lead to different curves lead to different representations.

 

[Mashiro]

You can cover a certain square $S\subseteq \mathbb{R}^2$ of the plane with a single line, a Hilbert curve, or a Peano curve. Hence if our curve $\gamma \, : \,[0,1] \longrightarrow S$ hits your point $p\in S$ for the first time (more than once is possible), say $\gamma (t_0)=p,$ then $t_0$ can be thought of as a representation of $p$ by a single number.

The disadvantage is, that different squares lead to different curves lead to different representations.

Thank you so much! I was learning polar coordinates today in class and could not help thinking about this question. However, I still don't understand why it is possible to hit a same point more than once.

 

[fresh_42]

Thank you so much! I was learning polar coordinates today in class and could not help thinking about this question. However, I still don't understand why it is possible to hit a same point more than once.

A surjective mapping looks easier to create than a bijective one so I wrote this passage a bit as an insurance. Let me look it up.

Cantor wrote in a letter to Dedkind in 1874 that it was impossible and that proof was almost unnecessary.
Three years later in 1877, Cantor wrote Dedekind a letter that he now believed that it is possible and added a sketch of proof.

At least I have been faster than Cantor. It is possible without crossings, but it can last until your point is covered.

 

[Bruzote]

"At least I have been faster than Cantor." - Ha! :-D