I'd like to point out a thing that is underlying much of that "change angle" trickery. Any (self-avoiding, "simple") curve that has 2-fold rotational symmetry is the border of some tile of one of my "simple" curves (as defined in https://arxiv.org/abs/1607.02433 ). Now _some_ of those curves give a proper plane-filler when using some other angle. Example: The L-system with non-constant F --> F+F-F gives A) the border of the tile of the the R5-curve (fig. 1.1-B, p. 2 for curve, fig. 3.1-D p. 16 for tile) when using 90 deg turns and axiom F+F+F+F. B) the terdragon (fig. 1.1-C, p. 2) with turns by 120 deg and axiom F. C) The border of the Gosper island for turns by 60 deg and axiom F+F+F+F+F+F One can do this for _every_ simple plane-filling curve on the triangular grid that has 2-fold rotational symmetry. The example in Bill's gifs is (at turns at 60 degrees, except for three turns by 120 deg at the corners) are the tile for a curve on the tri-hex (Kagomi) grid. The tiling has half of the triangles flipped over, just as the triangle appear in the triangular grid. The above with the exception for the tiles of curves on the triangular grid, where things are somewhat more complicated. The borders can be identified with folding morphisms (a la Dekking). Similar trickery again works in some cases. Best regards, jj P.S.: the border of the tiles of the folding curves appear to be much more involved than both above. Digraph-IFS may be needed to model them; at least L-systems don't seem to be sufficient. See https://jjj.de/tmp-math-fun/lr-curve-search.pdf for those innocent looking tiles. Any comments are welcome! I suspect Tom Karzes knows a few tricks about this... * Mike Stay <metaweta@gmail.com> [Dec 26. 2017 09:12]:
Uploaded to imgur https://imgur.com/a/NHk7t
On Mon, Dec 25, 2017 at 1:44 AM, Joerg Arndt <arndt@jjj.de> wrote:
The host gosper.org seems slow-ish right now (100 kBytes per second). Worsened by the file size of 25 MBytes. Suggest to wget http://gosper.org/drinkle3.gif Then use your favorite image viewer (strangely even vlc does not like this particular file; maybe it does have issues).
The browser (firefox) worked for me.
Best regards, jj
P.S.: I have a piece of software (written by a student of mine) that allows to render things like this with interactive control. Works under Linux, not sure where else.
* Bill Gosper <billgosper@gmail.com> [Dec 25. 2017 09:33]:
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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