Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

That fractal that's been up on my wall for years

chriskw.xyz

Post reply on HN
Print view View original

41–46 of 46 posts

Re: That fractal that's been up on my wall for years

#41

Post by chriskw »

leni536 wrote: post #38
leni536 wrote: post #7

Got a bit nerd-sniped by this and came up with an L-system that fills out (I think) "the wallflower": https://onlinetools.com/math/l-system-generator?draw=AB&skip... edit: On second thought, this probably generates the other fractal, but I'm not sure.

Found a space-filling curve for the wallflower: https://onlinetools.com/math/l-system-generator?draw=ABCD&sk... The previous one fills out the Koch island.

That's really cool! I tried to get something to work last week on pen and paper but couldn't get anything to stick. Is there a strategy you used or did you just go by feel?

Edit: just noticed how you encoded a flip (AB CD) between iterations like how the matrix flips the orientation of space. Super neat!

Re: That fractal that's been up on my wall for years

#42

Post by mckeed »

Fun post! I drew the first 5 iterations by hand myself and I'm finding it easiest to think of as a self-similar coloring of a square tesselation.

If you start with the shape of iteration 3, it tessellates as a 5x5 square tile. Make an infinite grid of those tile shapes with one iteration 3 version in the center. Treat that center tile as the center square in the iteration 3 pattern and color the tiles around it according to how the 2nd and 3rd iterations were built of squares. This gives you the 4th and 5th iteration and you can continue to iterate on the coloring outwards to color the grid of tiles in the wallflower pattern.

Re: That fractal that's been up on my wall for years

#43

Post by leni536 »

chriskw wrote: post #41
leni536 wrote: post #38

Earlier quoted context omitted.

Found a space-filling curve for the wallflower: https://onlinetools.com/math/l-system-generator?draw=ABCD&sk... The previous one fills out the Koch island.

That's really cool! I tried to get something to work last week on pen and paper but couldn't get anything to stick. Is there a strategy you used or did you just go by feel? Edit: just noticed how you encoded a flip (AB CD) between iterations like how the matrix flips the orientation of space. Super neat!

> noticed how you encoded a flip (AB CD)

Exactly! There is also a less obvious relationship between A and B too: B is a A "backwards" (A rotated 180°, starting the curve from the opposite end).

The strategy was to put 5 lines on the plus sign on the sides of the 5 cells, with the idea that each line eventually fills out a neighboring cell in subsequent iterations. I found one such path that had a chance of working. Not sure if this makes sense.

Re: That fractal that's been up on my wall for years

#45

Post by mbty »

Really cool and in-depth, thank you!

I believe that there is a typo in the pattern formula (right after "Looking closely you might pick up on the pattern"): it should read

  5**(n/2) instead of 5**n
  5**((n-1)/2) instead of 5**(n-1)
(\overrightarrow{10*4} is [0, 25] but your original formula gives [0, 625])

Also, regarding Knuth's mistake: Youtube comments point out that his fractal is in fact correct; he just mistook the beginning point with the end point. Loosely speaking, the fractal is symmetrical about its middle turn, which is precisely the one Knuth believed to be incorrect. All in all, he still made a fractal-related mistake, so the conclusion holds.

Re: That fractal that's been up on my wall for years

#46

Post by chriskw »

mbty wrote: post #45

Really cool and in-depth, thank you! I believe that there is a typo in the pattern formula (right after "Looking closely you might pick up on the pattern"): it should read 5**(n/2) instead of 5**n 5**((n-1)/2) instead of 5**(n-1) (\overrightarrow{10*4} is [0, 25] but your original formula gives [0, 625]) Also, regarding Knuth's mistake: Youtube comments point out that his fractal is in fact correct; he just mistook t…

Good catch, corrected the formula!
Post reply on HN