[math-fun] The base 2-i, digits 0,i^0..3 spacefill
(This Subject is by now completely wrong. Dragons are related to base i-1, digits 0,1.) It seems to me there should be 4 rep4tile "dragons": gosper.org/4flopfour.png , the Heighway and grid 2tiles, because they're also 4tiles, and one other proper 4tile. But my obvious guess makes gosper.org/bogus4tile.png which is not a rep4, maybe not a rep-anything. Jörg, what's the other rep4? Any idea what I blew to make this bogon --rwg -------- Original Message -------- Subject: Re: [math-fun] The base 2-i, digits 0,i^0..3 spacefill Date: 2016-11-21 01:28 From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> * Bill Gosper <billgosper@gmail.com> [Nov 21. 2016 10:12]:
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Even unto the lowly dyadic (Heighway)
dragon which, if you conjugate at every level, becomes simply a
triangular
patch of a square grid. Two back-to-back: gosper.org/dragrid199.png --rwg Is there a gallery of variously flopped dragons anywhere?
Jörg?
See http://jjj.de/fxt/demo/bits/#bit-paper-fold-general for "all" variations (for 2^64 \approx \infty). Crucially: static inline bool bit_paper_fold_general(ulong k, ulong w) // Return element number k of the general paper-folding sequence: // bit number x of the words w determines whether // a left or right fold is made at the step x. // With w==0 the result is ! bit_paper_fold(k). // With w==~0 the result is bit_paper_fold(k). // The result with ~w is the complement of the result with w. { ulong h = k & -k; // == lowest_one(k) h <<= 1; ulong t = h & (k^w); return ( t!=0 ); } For edge-covering curves on the grids (3^6), (4^4), and (3.6.3.6) there is http://jjj.de/3frac/ Dive into the directories p?/ (for ? \in {3,4,6}) for pdfs But these are only curves with simple L-systems. By the way, I am preparing high quality prints (poster size), two of which will be shown in Atlanta in January, see http://gallery.bridgesmathart.org/exhibitions/2017-joint-mathematics-meeting... Will any math-funster be there? I have yet to do all "paper-folding" curves a la Davis/Knuth/Dekking for other orders. It's not hard, but I have been working on other things.
Julian, can they be made with piecewiserecursivefractal?
No, but by another function in his same notebook! --rwg
Unless I am confused, the orders of "paperfold and company" are powers of two. Indeed all of them can be obtained from the paperfolding curve by using the products I described in my pre-print. The paperfolding figures (plural) are instances of Dekking's "folding morphisms" ("Paperfolding morphisms, planefilling curves, and fractal tiles", Theoretical Computer Science, 2012). Not long ago I realized that I should be able to extend my search (at least on the square grid) to find _all_ of them for small orders. For the square grid I could so far only find curves for odd orders because my "simple L-systems" are more restrictive. I plan to do the search as soon as I find time (currently looking for some funding, wish me luck...). Best regards, jj P.S.: another plan is to go for "all Gosper curves", but this is unlikely to happen soon. * Bill Gosper <billgosper@gmail.com> [Dec 04. 2016 09:53]:
(This Subject is by now completely wrong. Dragons are related to base i-1, digits 0,1.)
It seems to me there should be 4 rep4tile "dragons": gosper.org/4flopfour.png , the Heighway and grid 2tiles, because they're also 4tiles, and one other proper 4tile. But my obvious guess makes gosper.org/bogus4tile.png which is not a rep4, maybe not a rep-anything. Jörg, what's the other rep4? Any idea what I blew to make this bogon --rwg
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participants (2)
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Bill Gosper -
Joerg Arndt