As you know, there exists a bijection $\varkappa: [0, 1]^2 \mapsto [0, 1]$. Cantor demonstrated that. You even know how to construct this bijection: if $x, y \in [0, 1]$ and their decimal expansions are $x=0.x_1 x_2 x_3...$ and $y=0.y_1 y_2 y_3...$ then $\varkappa (x, y)=0.x_1y_1x_2y_2x_3y_3...$. It may be slightly more complicated than that, because in order to get a honest-to-god bijection you’ll need to somehow get around the fact that two decimal expansions may correspond to the same decimal number (for example: 0.999…=1), and it may get tricky, but it’s very doable.

You already knew all of this, or nearly all, but have you ever wondered what does the plot of $\varkappa(x, y)$ look like? Well, wonder no more:

What do we see here? First, the surface, or whatever you want to call it, is a fractal. It’s asymmetric, because $\varkappa$ is asymmetric with respect to its arguments. each of its level-sets consists of a single point, and it’s easy to see why. You may also notice that I cheated a bit, and this is not really the plot of the function I described in the beginning – instead this is a slightly modified version that uses binary expansions instead of decimal ones. It makes no difference in theory, but the plot looks better this way.