It is possible to define a curve which fills every point of a two-dimensional square (or, more generally, an $n$-dimensional hypercube). Such curves are called space-filling curves, or sometimes Peano curves.

More precisely, there is a continuous surjection from the interval $I$ onto the square $I\times I$.

This is related to the (also counter-intuitive?) result of Cantor, that the cardinality of the number of points in the unit interval is the same as the that of the unit square, or indeed any finite-dimensional manifold.