The most simple recipe for D-dim Peano goes like this: Take the (usual) Gray code in base 3 for integers with at most D * f digits (f gives the fine-ness of the approximation). The k-th coordinate (0 <= k < D) of the position in D-dim space at step n (n>=0) is given by taking the equidistant digits k+0*D, k+1*D, k+2*D, ..., k+(f-1)*D. This is pointed out in A. J. Cole: Compaction Techniques for Raster Scan Graphics using Space-filling Curves, The Computer Journal, vol.30, no.1, pp.87-92, (1987). http://comjnl.oxfordjournals.org/content/30/1/87.full.pdf The recipe can be generalized for the Gray code for mixed-radix numbers where all bases are odd and (at least) when the bases for digits apart by D from each other are identical. Dropping the (at least) part may work as well, just read the coordinates as mixed-radix numbers (did not check this). Compared to this the D-dim Hilbert curve is awfully hard (ask WFL!). Best regards, jj * Warren D Smith <warren.wds@gmail.com> [Dec 27. 2015 19:47]:
just to make a rather boring point, the "D-dimensional Peano curve" is based on the fact that a hyperrectangle can be cut into 3 scaled copies of itself... if its edge lengths are
1, A, A^2, ..., A^(D-1) where A^D = 3.
Isn't this meeting Dan Asimov's desires in any dimension?
Then to make a rather more exciting (?) point, Dan said the only real division algebras are in dimensions 1,2,4,8. However, there actually is one in dimension 16 too, depending on the precise technical definition of "division algebra."
That is explained in paper #73 here: http://rangevoting.org/WarrenSmithPages/homepage/works.html
It would be rather nice if anybody figured out a use for it.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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