# [seqfan] Re: OT: paper "Number representations and dragon curves"

Joerg Arndt arndt at jjj.de
Thu Jun 3 10:06:44 CEST 2010

```Thanks for your comments.

Indeed I stepped on seq A106665, see section
1.31.3.2 "The alternate paper-folding sequence"
on pp.90-92 of the fxtbook.

Looks like I should add the following
(brutally fast) routine to A106665:

bool bit_paper_fold_alt(ulong k)
{
ulong h = k & -k;  // == lowest_one(k)
h <<= 1;
ulong t = h & (k ^ 0xaaaaaaaaUL);  // 32-bit version
return  ( t!=0 );
}

and my comments in A014577 and A080846.

I am aware of Gilbert's papers (and many more
publications regarding this topic).
Just wanted to cite the (apparently) earliest paper
and I cannot obtain it since years.

cheers,  jj

* Robert Munafo <mrob27 at gmail.com> [Jun 03. 2010 09:37]:
> Well, then again, maybe I *didn'*t read that Davis & Knuth paper. I just
> checked my Knuth "Art of Computer Programming" vol. 2 "Seminumerical
> Algorithms" (section 4.1) and there I find the discussion of base -1+i that
> I remember, along with the picture of the dragon fractal (this is in the
> 1997 Third Edition of "Seminumerical Algorithms").
>
> And, checking the OEIS, I see that your paper "Number Representations and
> Dragon Curves" is cited only by one sequence (A106665), and by no other
> sequence. Therefore I suspect you are investigating A106665.
>
> So given that assumption, I suppose you're wondering what the "Alternate
> paper-folding" is or means. It's pretty simple. Look at the Wikipedia
> article, http://en.wikipedia.org/wiki/Dragon_curve and note the illustrated
> description under the heading "[Un]Folding the Dragon" and note that the 1's
> and 0's in the A106665 description correspond to the L and R folds in the
> Wikipedia discussion.
>
> On Wed, Jun 2, 2010 at 17:34, Robert Munafo <mrob27 at gmail.com> wrote:
>
> > I'm pretty sure I read that back in my distant past.
> >
> > The content is essentially the same as in this article:
> >
> > William J. Gilbert,
> >   Fractal geometry derived from complex bases
> >   The Mathematical Intelligencer, Volume 4, Number 2 (June 1982), pp. 78-86
> >   (ISSN 0343-6993; DOI 10.1007/BF03023486)
> >
> > PDF available here:
> >
> > http://www.math.uwaterloo.ca/~wgilbert/Research/MathIntel.pdf
> >
> >
> > On Wed, Jun 2, 2010 at 11:31, Joerg Arndt <arndt at jjj.de> wrote:
> >
> >> I cannot access in any way the paper:
> >
> >
> >> Chandler Davis, Donald E.\ Knuth:
> >
> > {Number representations and dragon curves, I and II}
> >
> > Journal for Recreational Mathematics,
> >
> > vol.3, pp.61-81 and pp.133-149, (1970).
> >
> >
> >> If anyone got an electronic copy of this one,
> >
> > kindly email it my way, thanks in advance!
> >
> >
> --
>  Robert Munafo  --  mrob.com

```