Plane filling cuvres are a continueous line through a flat (2D) space.
By using a recursive procedure, the curve becomes ever more detailed. It increases in length and at its limit will visit every point in the space.
Hence, the enclosed 2D space has been filled with a curve of infinite length.
We cannot iterate an infinite number of times. Hence we can only draw an approximate curve by iterating just a few times.
Also see L-Systems.
Peano Gosper Curve