Please allow for a few day, I am positively under water. A few remarks below (quickly written, so do not trust anything). * Warren D Smith <warren.wds@gmail.com> [Jun 16. 2015 19:59]:
J Arndt:
* Warren D Smith <warren.wds@gmail.com> [Jun 15. 2015 07:58]:
Possible idea: re-express/re-examine that Minsky iteration and/or the Penrose tiling using numbers/coordinates written in "fibonacci number system" or "golden number system." Likely to be helpful.
Fibonacci system: [...]
For those "sparse" expansions, just use the greedy method.
--WDS: uh, what is "the greedy method"? I mean, the problem with number systems like mine is, unless you forbid substrings, the representation will be very nonunique. So are you claiming, just walk right to left, each time you can add a "1" without using a forbidden and without exceeding the target number, then do it? And that this works? And in what circumstances does this work? Always?
Keep subtracting the largest Fib you can, then you will get the NAF: You cannot get ...11... that way, because F(n) + F(n-1) = F(n+1), so you would have subtracted F(n+1) earlier.
JA: 1 http://jjj.de/tmp-rd/rd-p1m1-twindragon-sign-d012.png 2 http://jjj.de/tmp-rd/rd-naf-m1p1-d1.png <--= I really like this one 3 http://jjj.de/tmp-rd/rd-naf-w17.4p0ps2-terdragon-d012.png 4 http://jjj.de/tmp-rd/rd-naf-m1p1-ver2-d012.png
--WDS: so are you saying, you already thought of my whole tiling idea?
Yes, that was 2011 (started IIRC 2010), I spammed math-fun quite a bit, then recalled I hate spam so stopped. There is a bunch of reasons I switched to those curves http://jjj.de/3frac/ All of them give tilings of the plane and they are _much_ easier to render the way I want. Btw. I now have all proofs I need (given 2012 by Dekking), so I should REALLY REALLY REALLY write things up...
The above pictures, 1 seems to be a "conventional" binary with some complex radix, probably 1-i since it seems tile T is tiled by two sqrt(1/2)-scaled, 90-degree-rotated copies.
Yes, the (shape of the ) Twin dragon, cf. https://en.wikipedia.org/wiki/Dragon_curve
Pictures #2-4, I'm not sure what they are. IF they constitute a tiling of T by means of several rescaled copies of T, and the angles of rotation of the pieces are irrational in degrees, then that means you tile the plane aperiodically. But in pictures 3, 4, apparently only rotation angles divisible by 90 are involved, in picture 2 no rotations at all.
So based on these pictures alone, it appears you did not think of my whole idea, or if you did then you did not find examples of the coolest phenomena.
But you're ahead of me in the sense you have some software tools for manufacturing pictures.
Yes, all written from scratch, including const char ascii_pixels[128*16]= { /* OMG */ } for "printing". No pain, no gain. (binary fits into first level cache of the CPU)
I could be wrong, since I have not done any graphical computing of my own, but I suspect some very cool tilings can be got via my forbidden substrings + funny complex radix number systems approach,
YES! But the math appears really nasty when beyond "same old".
including inherently aperiodic ones with only "one" tile type, albeit present in several scaled versions. (And what is the minimum number of scaled versions? I do not know.)
Also, which seems not very related(?), the question of how the hell Gosper, Ziegler-Hunt etc found Penrose-like tiling using Minsky-like iteration, probably has a proof that explains the mystery, and probably that proof will be based on examining what happens to their numbers when you look at them in Fibonacci and Golden number systems. But again, I have not actually done that examination and produced such a proof -- this is just a suggestion that's likely to go somewhere if you try.
The over-arching question would be: to try to classify all possible such tilings and number systems. Which might be doable. Or at least to find all the "small" examples, which ought to be doable via computer searches aided by some human feedback.
Please watch https://www.youtube.com/watch?v=V7HbHK8yc5w (the projector broke the very second I started the talk, that's why I am a bit stressed first) for a reasonable subset of what you are asking for. I should be able to accelerate the search for the tiles (watch the vid.) by another factor 10 thousand or so. But these are only the "regular" (boring!) tiles. Best, jj (kindly forgive typos, signing off now, good night everybody)
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