I know that there are space-filling curves from $[0, 1]$ to the unit square, and [this question](https://math.stackexchange.com/questions/290148/are-there-any-continuous-functions-from-the-real-line-onto-the-complex-plane) addresses curves transforming the real line into the entire plane. But what about transforming the unit interval into the whole plane?

And if such curves exist, can their construction somehow be depicted, perhaps by showing a few examples of a sequence of curve that converges to the space-filling curve (as is often done for curves mapping the unit interval to the unit square, see below)?

![Construction of Hilbert Curve](https://i.stack.imgur.com/L9b5u.png)

Image from <http://en.wikipedia.org/wiki/Space-filling_curve>