On 2015-12-26 11:03, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 19:33]:
[...] 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).
Could you check against my numeration system? See http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.26.pdf (new, better errors!) in section 3.3 pp.21ff, especially Figure 3.3-C. Yes, I'm pretty sure my IMG_0245 is your R7.1 . (btw. negating the base gives a tile that has a region around zero covered, and that I cannot identify with any curve I know). No picture? Anyway, 0.* failing to represent 0 is annoying but nonfatal. E.g., balanced negabinary doesn't even have a 0 digit, but can represent 0 as an infinite string.
And a (xerographically printed) spacefill: http://gosper.org/IMG_0245.JPG .
Just the minute I was staring at this image... I tried to re-create it, but my rendering methods give superficially similar but different images.
It would also be nice to learn how you got the rendering in the image IMG_0246.JPG (looks like morphed from IMG_0247.JPG to me, but I cannot quite see how).
They actually form a trio, starting with http://www.tweedledum.com/rwg/7posies.bmp . They're topologically different--not morphs. Recursively join together clusters of seven to form a tree. Then "ensausage" it. In the limit, inside and outside are replaced by boundary, and all points are hit at least twice.
[...] 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)
(but poorly lighted)
math museum. --rwg
I have quite a few prints where I pencilled (is that a word?)
In the UK. Stateside, it's penciled.
in similar things. But then the triangular grid is sort-of-ish a projection of the simple cubic lattice, looked upon from direction (say) (1, 1, 1).
Somebody thus treated the flowsnake.
Point of grammar: On p22 you say As the set contains a neighborhood of zero no signs are needed neither. Nobody says this because it sounds like a double negative. A chatbot might say it and blow Turing's test. --rwg
[...]
Best regards, jj