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