On 2015-12-26 06:16, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 14:40]:
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http://jjj.de/tmp-xmas/thin-3-tiles-sty1.pdf switches from L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R48-1 # dragon # symm-dr to L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R49-1 # dragon # symm-dr about 1/3 of the way in. It starts out identically(?) to a base 2+omega system where the omega^1, omega^3, and omega^5 digits of "France" are replaced with larger ones, producing a feathery tile. --rwg
Dawk, I forgot I had a picture! http://www.tweedledum.com/rwg/tril7.htm . The pictured tile of order 2 is of 7 tiles of order 1, each of which is 7 tiles of order 0 (hexagons), where the grouping at every level is 3 around 3 around 1, vs 6 around 1 for the Island. So the base is still 2+(-1)^(1/3), but the digits are (-1)^(0),(-1)^0+(-1)^(1/3), (-1)^(2,3), (-1)^(2/3)+(-1)^(3/3), (-1)^(4/3), (-1)^(4/3)+(-1)^(5/3) instead of (-1)^(0..5/3).
And a (xerographically printed) spacefill: http://gosper.org/IMG_0245.JPG .
Not sure I understand. One can take _any_ tile for the smallest surrounded sets (the very many tiny triangles, which one can render as hexagons). Choosing one tile for those "atoms" corresponds to the tile of a product curve (two iterates of the curve we are looking at, followed by one iterate of "whatever", as in my Section 5).
Every curve has a tile, we can multiply curves, hence tiles.
The tiles shown in thin-3-tiles-sty1.pdf correspond to two families of curves, each with again two sub-families.
From my file: Family 1: orders 3,4, 12,13, 27,28, 48,49, 75,76, 108,109, ... Family 2: orders 7,9, 19,21, 37,39, 61,63, 91,93, 127,129, ...
And yes, switching back and fourth between orders written next to each other is somewhat lovely (stare at the center of the image to get the difference beyond rotation).
Ah, so the presence or absence in an image of a central hexagon containing a tricolor spiral, which in successive images goes ... YES YES NO NO YES YES NO NO ... is you switching among rules rather than a bizarre consequence of a single rule. I thought the mixing was some kind of editing accident! --rwg
Btw. the "manta" curves show that curves exists that move without any turn as long as theoretically possible (_one_ more straight move and they'd be at the end point!).
Best regards, jj
Btw, that tricolor spiral can make a nasty "physical illusion". (http://gosper.org/esch2.PNG) Just by brightening and darkening the three colors, you can permute which surfaces appear horizontal, and which vertical. I want to see a life-sized contradictory pair of these in a (well-insured and well-carpeted) math museum. --rwg
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