From: "Fred lunnon" <fred.lunnon@gmail.com>
[Nor unfortunately, has anyone else attempted to discuss exactly what exactly constitutes a Hilbert-like curve in dimension > 2.]
Okay, now I feel qualified to ask what you mean. Probably, "in math language rather than computer code," right? But "exactly?" One sense of an exact constitution of a >2D Hilbert curve requires making some arbitrary choices, which would sound even uglier in math than it does in code. (Since a certain level of arbitrariness and noise is expected in computer code.) Or...do you just want a *class* of curves? That's easier, although I don' speaka good maths.... You would say that the curve passes through the 2^d subcubes in Gray-code order, and the subcubes' curves are miniatures of the whole curve oriented so they connect, i.e., The first subcube's start matches the cube's start--which is a corner. The last subcubes's end matches the cube's end--an adjacent corner. The ends and starts of adjacent subcubes in the Gray code sequence are...right next to each other--you have to put in that connecting step, but so did Hilbert. To me, any arrangement that meets those constraints is a "Hilbert- like" map from one version of the walk to the next. The curve is the limit of the sequence...am I missing something?
From: rwg@sdf.lonestar.org
Looking at stills of these is like trying to parse protein molecules.
I drew some "exploded views" of 3D Hibert walks: http://www.tiac.net/~sw/2008/10/Hilbert/hilbert_pic.pdf (If you need a page to download that from: http://www.tiac.net/~sw/2008/10/Hilbert about 4/5 of the way down.) This just takes the binary of each coordinate and reinterprets it in base 4.125. --Steve