Jörg>* Bill Gosper <billgosper@gmail.com> [Sep 01. 2014 08:00]: [...] Holy cr@p, http://gosper.org/semizerp.htm andhttp://gosper.org/semizerp1.htm on steroids! Another 13fold recursion, replete with inhomogeneous scaling: Mandelbrot's recursive Snowflake. Can such be likewise "teardropped"? He also had a 7fold recursion, I think. --rwg Note I computed _all_ curves generated by L-systems with just one non-constant symbol.< Staggering. So somewhere in all that are these three "sevens" gosper.org/IMG_0245.JPG gosper.org/IMG_0246.JPG gosper.org/IMG_0247.JPG (hardcopied in the 70s, pre-laser.) Also, did you notice this way of mis-teardropping the France fractal? gosper.org/jelly7.bmp Lastly, I had forgotten about these two teardroppings of Mandelbrot's seven Snowflake recursion: gosper.org/trozeImage8.gif gosper.org/trozeImage13.gif Maybe someone can do his thirteen, but I can't do it in my head. I just found this: http://www.meden.demon.co.uk/Fractals/frozenteardrop.html . Jörg> As different curves can give the same overall shape, let's count the shapes (by grid and order): The example pictures are by filename (given in http://jjj.de/tmp-extra/ ). Triangular grid, only turns of 120 degs: c3/search-r03-curves.txt: 1 c3/search-r04-curves.txt: 1 c3/search-r07-curves.txt: 3 c3/search-r09-curves.txt: 5 c3/search-r12-curves.txt: 10 c3/search-r13-curves.txt: 15 c3/search-r16-curves.txt: 17 c3/search-r19-curves.txt: 71 c3/search-r21-curves.txt: 212 c3/search-r25-curves.txt: 184 c3/search-r27-curves.txt: 543 c3/search-r28-curves.txt: 842 c3/search-r31-curves.txt: 1848 Example (order 19): all-r19-curves.pdf all-r19-tiles.pdf Example (order 16): all-r16-curves.pdf all-r16-tiles.pdf Cf. https://oeis.org/A234434 Rectangular grid (turns by 90 degs): c4/search-r05-q-curves.txt: 1 c4/search-r09-q-curves.txt: 1 c4/search-r13-q-curves.txt: 4 c4/search-r17-q-curves.txt: 6 c4/search-r25-q-curves.txt: 33 c4/search-r29-q-curves.txt: 39 c4/search-r37-q-curves.txt: 164 c4/search-r41-q-curves.txt: 335 c4/search-r49-q-curves.txt: 603 Example (order 25): all-r25-q-curves-sty1.pdf all-r25-q-tiles-sty1.pdf Any number of curves on the same grid can be "multiplied" (noncommutatively). Figures like http://gosper.org/semizerp.htm can be obtained from both families above by a sort of "Kronecker division", for arbitrary divisor. Triangular grid, containing turns of 60 degs: c6/search-r07-b-curves.txt: 1 c6/search-r13-b-curves.txt: 3 c6/search-r19-b-curves.txt: 7 c6/search-r25-b-curves.txt: 10 c6/search-r31-b-curves.txt: 63 c6/search-r37-b-curves.txt: 157 c6/search-r43-b-curves.txt: 456 c6/search-r49-b-curves.txt: 1830 Example (order 49): all-r49-b-tile-plus-shapes-2.pdf all-r49-b-tile-plus-shape-borders-2.pdf See the files side by side. There are far more curves (with higher orders more and more tend to give the same shape). The above corresponds to roughly a quarter million curves. A rather random collection from the beginning phase of my search (including curves with more than one non-constant symbol) is http://jjj.de/3frac/ Very many of them where created with pencil and paper. Best, jj P.S.: http://dimacs.rutgers.edu/Workshops/OEIS/abstracts.html#arndt < Break a leg. --rwg