Re: [math-fun] Doin' the Hilbert walk
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
rwg>If you have Mma, I highly recommend, e.g., mouse-wobbling
Graphics3D[Table[Stick[Treano[(k+1/3)/512],Treano[(k+4/3)/512]],{k,0,510}]]
http://gosper.org/treano512.SWF . (.5M) Thanks to Ed Pegg for suggesting to Export this from a Table rather than an Animate. --rwg FIELDWORKS DISK FLOWER PLENTEOUS PENTULOSE CHLOROSILANE RESIN ALCOHOL
On 11/26/08, rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> wrote:
rwg>If you have Mma, I highly recommend, e.g., mouse-wobbling
Graphics3D[Table[Stick[Treano[(k+1/3)/512],Treano[(k+4/3)/512]],{k,0,510}]]
http://gosper.org/treano512.SWF . (.5M) Thanks to Ed Pegg for suggesting to Export this from a Table rather than an Animate. --rwg
I haven't (as yet) made much of an effort to follow RWG's constructions, I'm afraid --- not speaking Macsyma doesn't help. It looks to me from his diagrams as though the neighbour connections are sometimes through lattice cube vertices, sometimes edges, sometimes faces --- is this correct? WFL
I haven't (as yet) made much of an effort to follow RWG's constructions,
In the case of the Treano function, that's effort well misered.
I'm afraid --- not speaking Macsyma doesn't help. It looks to me from his diagrams as though the neighbour connections are sometimes through lattice cube vertices, sometimes edges, sometimes faces --- is this correct?
WFL
Fred, that (self redefining) exact rational Treano function is Mathematica, not Macsyma! It maps 0 to (0,0,0) and 1 to (1,0,1), and all connections are corner to corner. I (incorrectly) thought my ViewPoint trajectory included a good view of this. Sorry. The problem is, to avoid self-contact, the plot images [1/1536,1-2/1536], not [0,1], so it never actually hits a corner. Once you have the exact function, you are no longer bound by diagrams. --Bill POLYSEMANTIC PLASTIC MONEY AMYLOPECTINS
rwg>that (self redefining) exact rational Treano function is Mathematica,
not Macsyma! It maps 0 to (0,0,0) and 1 to (1,0,1), and all connections are corner to corner. I (incorrectly) thought my ViewPoint trajectory included a good view of this. Sorry. The problem is, to avoid self-contact, the plot images [1/1536,1-2/1536], not [0,1], so it never actually hits a corner.
Amazingly, Treano[1/28] = Treano[27/28] = {1/2,0,1/2} = Treano[1/2]. I.e., it revisits the midpoint when 93% through! More important, Treano[11/14] = {1,1,1}, so Graphics3D[Table[Stick[Treano[11/14/8^3 + (k)/8/8], Treano[11/14/8^3 + (k + 1)/8/8]], {k, 0, 8^2-2}]] produces http://gosper.org/treano64.SWF, connecting the centers of the subcubes. (This bounding box < [0,1]^3.) --rwg INTERPOLATED LATENT PERIOD ACTINOMETRIES REACTION TIMES
rwg>that (self redefining) exact rational Treano function is Mathematica,
not Macsyma! It maps 0 to (0,0,0) and 1 to (1,0,1), and all connections are corner to corner. I (incorrectly) thought my ViewPoint trajectory included a good view of this. Sorry. The problem is, to avoid self-contact, the plot images [1/1536,1-2/1536], not [0,1], so it never actually hits a corner.
Amazingly, Treano[1/28] = Treano[27/28] = {1/2,0,1/2} = Treano[1/2]. I.e., it revisits the midpoint when 93% through! More important, Treano[11/14] = {1,1,1}, so Graphics3D[Table[Stick[Treano[11/14/8^3 + (k)/8/8], Treano[11/14/8^3 + (k + 1)/8/8]], {k, 0, 8^2-2}]] produces http://gosper.org/treano64.SWF, connecting the centers of the subcubes. (This bounding box < [0,1]^3.) --rwg INTERPOLATED LATENT PERIOD ACTINOMETRIES REACTION TIMES
participants (3)
-
Fred lunnon -
rwg＠sdf.lonestar.org -
Steve Witham