What I wrote was misunderstood. (By the way, rwg, your last two sentences are quite interesting! I wonder how small, in some appropriate sense, that uncountable set S = {x in [0,1]^n | #(finv(x)) > 1} can be. For one thing, the only thing I was asking for confirmation is that all numbers d, 1 <= d < n can actually be the Hausdorff dimension of some variant version of the (adjusted) Hilbert polygons, in the limit. By writing "adjustments to the Hilbert polygons" I certainly did not mean merely changing the vertex/edge structure of the same point set! I meant redefining the whole thing, while sticking to its original spirit -- just as one can define a variant on the original Cantor set by taking not middle thirds but different fractions: Removing the middle 1/2 at each stage gives the equation L_d = 2(1/4)^d L_d where L_d is the Hausdorffly d-dimensional Hausdorff measure, so d = 1/2. (Compared to the middle-third set, for which L_d = 2(1/3)^d L_d, with d = log_3(2).) That's all I meant, but for the case of Hilbert walks after being adjusted so as to be one-to-one. When a function [0,1] -> [0,1]^n is one-to-one, of course it can't be onto if n > 1. Hey, I'm a professional topologist. The limit function is not some kind of vague limit -- it's just a pointwise limit, since for any x in [0,1], let y_k := f_k(x) be its image under the kth stage function f_k : [0,1]-> [0,1]^n (whose image is a polygon). Then y_k is a Cauchy sequence -- since all variant choices one can make [not that I mentioned this earlier] have the property that, once the jth subinterval (call it [a,b]) of [0,1] at the kth stage is mapped into the jth subcube C of [0,1]^n, then that same subinterval [a,b] must always be mapped into the same subcube C under all stages later than the kth. --Dan _________________________________ And by the way, the word "arc" is used to mean a 1-1 image of [0,1], but the word "curve" is commonly used to mean any continuous map of [0,1] into some topological space. <<

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?)

Unfortunately, no! This is why I detest the misleading term "spacefilling curve". A curve is a set--the graph of a function. A spacefilling curve is therefore a solid area! How do you define the limit of a sequence of polygonal curves? Is it a set or a function? The only coherent notion is spacefilling function-- a continuous map of a closed interval onto a closed 2(or more)D set. All those different patterns in http://www.tweedledum.com/rwg/sampeano.htm are polygonal subsequences of the smae function! You cannot change the function by tweaking the curve and then taking some nebulous limit. A spacefilling function (D>=2) *cannot* be 1-1. Incredibly, every spacefilling function must overfill by visiting infinitely many points at least three times! (And uncountably many points twice.) --rwg

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_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele