Jack,
As Felix Hausdorff proved during his research of regulated fractals, if you do not factor in the Hausdorff dimension then the chances of calculating the Hilbert curve correctly substantially decreases.
So basically, if the Euclidean line is off by more than 0.01%, you are guaranteed that what you will be witnessing will be just another chaotic dynamical system which is no better than a coin flip.
Read some of the work done by Wac³aw Sierpiñski.