The Hilbert curve([0,1]) double-covers a dense, uncountable subset of [0,1]x[0,i]. E.g., In[55]:= Hilbert /@ {2661/3280, 2677/3280} Out[55]= {3/4 + (5 I)/11, 3/4 + (5 I)/11} It triple-covers a dense, countable set, e.g., In[57]:= Hilbert /@ {1/6, 1/2, 5/6} Out[57]= {1/2 + I/2, 1/2 + I/2, 1/2 + I/2} Less obviously, it quadruple covers a sparser, but presumably dense set, e.g., In[58]:= Hilbert /@ {5/48, 7/48, 41/48, 43/48} Out[58]= {1/2 + I/4, 1/2 + I/4, 1/2 + I/4, 1/2 + I/4} --rwg