Re: [math-fun] The base 2-i, digits 0,i^0..3 spacefill
On Fri, Nov 11, 2016 at 12:44 PM, Bill Gosper <billgosper@gmail.com> wrote:
(whose?) underlying the pretty gosper.org/dana.PNG has only moderately interesting Fourier gosper.org/rt5-324.png and polygonal gosper.org/rt5.png approximations. --rwg
But we should remember that, for images lacking bilateral symmetry, there are *uncountably many* filled shapes, since you can flip a coin as to whether to mirror image ("flop" --BBM) at each level of recursion. 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? Julian, can they be made with piecewiserecursivefractal?
On Fri, Nov 11, 2016 at 9:47 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Nov 11, 2016 at 12:44 PM, Bill Gosper <billgosper@gmail.com> wrote:
(whose?) underlying the pretty gosper.org/dana.PNG has only moderately interesting Fourier gosper.org/rt5-324.png and polygonal gosper.org/rt5.png a pproximations. --rwg
But we should remember that, for images lacking bilateral symmetry, there are *uncountably many* filled shapes, since you can flip a coin as to whether to mirror image ("flop" --BBM) at each level of recursion.
Thanks to Julian (and earlier Jörg, if I'd been paying attention) I finally realize this was sheer idiocy I've believed for decades. Yes there are uncountably many nonselfsimilar floptiles, but only countably many fractiles. And only two of them are dyadic dragons. Heighway: never flop. Grid triangle: always flop. If you flop every other level you get a frac-4-tile! gosper.org/frac4drag.png (two fill a square, but not this way: gosper.org/fracfourdrag.png ) You only get fractiles (= self similar) by flopping periodically. Two random curves: gosper.org/fourblobs.png gosper.org/hblob.png 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?
Julian, can they be made with piecewiserecursivefractal?
No, but by another function in his same notebook! --rwg
* Bill Gosper <billgosper@gmail.com> [Nov 21. 2016 10:12]:
[...]
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Mon, Nov 21, 2016 at 12:34 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Nov 11, 2016 at 9:47 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Nov 11, 2016 at 12:44 PM, Bill Gosper <billgosper@gmail.com> wrote:
(whose?) underlying the pretty gosper.org/dana.PNG has only moderately interesting Fourier gosper.org/rt5-324.png and polygonal gosper.org/rt5.png a pproximations. --rwg
But we should remember that, for images lacking bilateral symmetry, there are *uncountably many* filled shapes, since you can flip a coin as to whether to mirror image ("flop" --BBM) at each level of recursion.
Thanks to Julian (and earlier Jörg, if I'd been paying attention) I finally realize this was sheer idiocy I've believed for decades. Yes there are uncountably many nonselfsimilar floptiles, but only countably many fractiles. And only two of them are dyadic dragons. Heighway: never flop. Grid triangle: always flop. If you flop every other level you get a frac-4-tile! gosper.org/frac4drag.png (two fill a square, but not this way: gosper.org/fracfourdrag.png ) You only get fractiles (= self similar) by flopping periodically. Two random curves: gosper.org/fourblobs.png gosper.org/hblob.png
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?
Julian, can they be made with piecewiserecursivefractal?
No, but by another function in his same notebook! --rwg
He says it's really just prf dressed up. And he makes the remarkable observation that the frac4 dragon gosper.org/4flopfour.png is the bar graph of x xor 2/3 ! gosper.org/xor667fade.png It appears that x xor r makes a fractile for every rational r. --rwg
* Bill Gosper <billgosper@gmail.com> [Nov 23. 2016 10:21]:
[...] He says it's really just prf dressed up. And he makes the remarkable observation that the frac4 dragon gosper.org/4flopfour.png is the bar graph of x xor 2/3 ! gosper.org/xor667fade.png It appears that x xor r makes a fractile for every rational r. --rwg
Correct: write 1 in binary as 0.11111..., then observe that (x xor r) + ((1-x) xor r) == 1. Best regards, jj
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