holsz@cadence.com (Wlodzimierz Holsztynski) (04/07/90)
In article <13170@thorin.cs.unc.edu> banks@pooh.cs.unc.edu (Father Banks) writes: >Everyone has seen examples of space-filling curves. Wirth's text >gives example code for generating two kinds that lie in the plane. > >Are there simple algorithms to generate space-filling curves >in n dimensions? Even for n=3, hilbert 3curve code becomes >unhappily complicated. > >David Banks ---------------------------- Let f',f",g" : I --> I and g': I --> I^(n-1) be continuous maps such that their diagonal products f = f'/\ f" : I --> I^2 and g = g'/\g" : I --> n are onto (are Peano "space filling" curves). I assume that we already know how to construct f and g for n=2. Let h' = f' o g" and h" = f" o g". Then H = g' /\ h' /\ h" : I --> I^(n+1) is a continuous map of the interval I onto the (n+1)-dim cube. By induction, we construct maps for n=2,3,4,.... In practise, when dealing with a finite approximation, it's better to vary the special coordinate of the construction when passing to the next dimension. Don't choose the last one (as I did above) each time. BTW, all this is well known. There are continuous maps of I onto the hilbert cube Q too (and one may play some algorithmic games in this infinite dimensional case too, if one insists). I = [0;1] can be mapped onto a Hausdorff space X if and only if X is compact, connected and locally connected. I don't remember the author of this beautiful classical theorem :-( but the related one: a complete connected and locally connected metric space is arcwise connected was proved independently by Hahn and Mazurkiewicz (I think). Remark. My symbol /\ above is a weak imitation of the upper case delta. Enjoy Wlodek