It may be worth mentioning that for any given dimension = d the d-dimensional Hilbert polygonal curves [0,1] -> [0,1]^d approach a limit as the number of vertices approaches infinity. Which is, of course, a continuous surjection H_d : [0,1] -> [0,1]^d. Although this is old hat to some of us, when it was first discovered (around 1900) it stunned the world of math, forcing people to figure out what the concept of dimension really meant. By making adjustments to the Hilbert polygons, one can easily arrange that the limit function is one-to-one (but of course no longer onto). So the limit function is a homeomorphism of [0,1] onto its image. I suspect the Hausdorff dimension of such a one-one curve can be any real number in [1,d). (Can you confirm this, rwg? Below are some free anagrams for your trouble.) --Dan nawab / bwana bluegill / gullible aquiline / quiniela _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele