Langton's Ant on Penrose Tiling producing a Pentaflake-like Fractal 

Alright Penrose experts, is this entirely trivial or despairingly mysterious

Seeing something that seems like a random walk on an aperiodic tiling producing a fractal instead of a random blob around the starting position is mindblowing to me

Audibly gasped when I started recognizing that it was a fractal

My thinking is Langton's ant (always?) eventually falls into an endless loop, you see this in standard square grids all the time. But because the grid is aperiodic, the loop could never actually move forever in a specific direction. I think it's possible to prove that no true ant loop (where a repeated set of moves cycles forever) could occur on an aperiodic grid, because the ant would eventually reach a previously visited cell which is not part of the loop, by necessity. What I can't explain is why the ant has seemed to fall into an even more complex version of a "loop", like it found order in a fractal way. To prove that the fractal pattern goes on forever doesnt seem impossible... If, for example, it was proved that once the "loop" starts the ant could only ever reach a particular and reduced set (compared to the total number of possible configurations) of previously visited cells. Idk.

I've seen a few comments about most ants falling into repeating patterns. Just to say that one of the cool things about Langdon's ant is that it's fundamentally unpredictable as to whether it does or does not fall into a pattern (it suffers from the halting problem, as a result of its Turing completeness). People are wondering if the aperiodicity of the Penrose Tiling breaks this otherwise periodic behaviour. I'll be honest, I don't know if that's necessarily the strongest takeaway, here. A periodic loop of the ant, if it existed, wouldn't explore the total Penrose tiling structure. In fact, it would explore only a subset, with its path having been determined entirely by the connectivity of the subset and surrounding points. As a result, it settling into a periodic pattern would be only to do with a finite subset of the Penrose tiling, and so wouldn't have any relation to the larger aperiodic nature. My guess is that the aperiodicity doesn't change the odds of it forming cycles. And we're in 2D, where the connectivity of our graph doesn't hugely change with position, so I think this _should_ end up largely being a random walk (could be wrong), for a randomly chosen starting configuration. One thing I _do_ find quite interesting is that normally you'd 'program' the ant with some initial distribution of squared. However, presumably you could also program it with some initial set of connectivities on a graph. Because the Penrose tiling is unique at any point, perhaps there's an argument to be made that simply by choosing the starting position on the tiling, you're setting the ant to start evaluating a certain program. That actually _would_ be quite a nice link between the properties of aperiodicity and Langton's ant. If the number of computer programs is countably infinite (i.e., each is a finite string of 1s and 0s, then picking a point on the Penrose tiling actually _might_ encode enough information). Idk, I feel like there's something fun in here from _that_ perspective. The other vibe in all of this is that it's a bit of a meme that aperiodic tilings were initially conceived of to ask questions about Turing completeness. Now we're running (presumably) a Turing machine _on_ one!

What happens on an aperiodic "einstein" tiling? I.e. only one tile is used, but the tiling is still aperiodic, but nonrandom.

I found this "Penrose and Conway independently proved that whenever a curve closes, it has a pentagonal symmetry, and the entire region within the curve has a fivefold symmetry. At the most, a pattern can have two curves of each color that do not close." The curves they are talking about are the side-matching curves Penrose drew on the tiles. I think this may be related to why you can find these pentagons shaped paths.

the fact that it does turn into a snowflake fractal kind of makes sense when you really think about what the penrose tiling *is*, but the way it goes about it kind of boggles my mind

Wow, this is real neat to see! I would love to know more about how you rendered this

Literally the name of the video, still wasn't expecting such beauty.

Imagine if you never simulated the first 20K steps to start seeing the fractal emerge

Does an aperiodic tiling mean that the ant will never fall into a highway like they usually do? Or is this a sorta highway in its own right? very interesting video!

can you try Langton's ant on Einstein tiling?

Is the starting point the centre of the initial generating rhomb, or is it random? Is the pattern sensitive to starting point? If the algorithm runs indefinitely, what proportion of cells are hit? Interesting stuff! Thank you! Bob Mackay

The pattern it ends up making is probably a normal Langton ant highway on a pentagonally curved surface. On a square grid it would have been a straight line.

woa this must have taken ages

love it! very cool

Beautiful!

The world isn't ready for this

I'm a simple guy, I see pentaflakes, I click.

That’s cool. For starters all the tiles are quadrilaterals so the mechanics of making the ant move are the same as a typical square grid, except now you don’t necessarily have 4 edges per vertex like a lattice would and the starting position matters because the tiling is nonperiodic. So you get wildly different patterns just starting in different places or orientations and nothing like how a regular ant would look…

The initial clumpy blob doesn't seem to repeat anywhere else. Any thoughts as to how "it" breaks out of the "blob" and starts doing the "beautiful" fractal pentaflake pattern?

Now that's a highway.

that's one hard working ant

Really cool!

Ant be like (You think i cant make a pattern a repeating pattern out of this... let me show you something)

Most beautiful pentagonal fractal i've ever seen

Ok, now I wanna see a Hilbert curve and a dragon curve family drawn on Penrose tiles.... (by family, I mean try the various variants like Lion, see which ones look interesting)

В любой среде происходят именно те процессы, которые идеально согласуются с природой этой среды. Ничто не может идти наперекор.

This is so cool

the pattern formed because you told it to form. Instead of the ant choosing a direction on it's own you instructed it to move based on how many times it had already stepped on the same tile. What limited randomness we saw was because the tiles weren't all the same shapes. If you gave these instructions to an ant on a square grid you'd see no randomness at all.

My last brain cell exploring my empty brain:

Very cool

Me when I lose my vape

I almost think it got some sort of a handle on the penrose tiling when it started making that fractal

How did you get only 120 views for that Thats awesome

The idea to deploy cellular automata in a Penrose tiling is so interesting! Thumbs up! Has there been much investigation into the idea? There'd be no 'gliders' like in Conway's Game on a grid because the 'terrain' changes!

the BGM is sick!

Wow. Despite using numpy and opencv, I bet making this was still a challenge! Working on an expanding aperiodic tiling patch with good time and space efficiency seems like not exactly a trivial task.

I wonder if it's possible to do this with the Einstein tile (spectre tile or the simpler hat and turtle tiles)

given that the tiles are all quadrilaterals, i definitely expected right and left turns to mean, well, literally leaving via the edge to the left/right of the edge you entered from. can you get as interesting of a pattern that way?

In music we can hear "Fire in the hole" from Geometry Dash if we accelerate it in 2x.

I'm curious about seeing a Langton's ant on a specter tiling, or a hat\turtle tiling now!

A question: after generating the tiles (I guess you used the inflation-deflation method?), how do you check which tiles are adjacent to the current tile? For such a large tiling, the searching may cost a lot of time?

that's cool O_o

I was literally saying "no way" out loud at about 2:12 when I saw what was about to happen

How did you come across this ruleset?

Wait, hold up... How can this possibly... what... WHAT?! Will you next find a pattern in prime numbers, or digits in pi?

I’m curious why the rules are so complex? Surely with each tile being quadrilateral you could just use the standard movement rules for a black and white langton’s ant… Also, this fractal pattern is derived from the layout of an aperiodic tiling, so would that necessarily mean this pattern must also be aperiodic?

Hmm yes, couldnt make a bridge so it decided to make fractals instead

Did you come up with these rules yourself? Or is this already a well established pattern? Very cool 👍

How do you index such a tile in memory ? I would guess you can't index them in a 2d array with XY coords ? Also do you have some kind of space partitioning schem to run it better ?

Could the fractal pattern be a sort of highway? Since penrose tiles form larger blocks that tessellate like the smaller ones, a highway would only be possible by scaling. If that makes any sense.

My god... it's full of stars...

Focus on only the red spots to get second cool fractal

a beautiful flower

So Penrose Tiling + Langton's Ant = Sierpiński's Carpet/Fractal.

My question: I noticed at the beginning that the ant kind of stumbles around for a few thousand turns, before it locks into this regular pattern. Does it always eventually fall into that pattern no matter where you start the ant?

How long did it take to find the L/R rules that produce this pattern?

neat

I wonder if this pattern would reappear if you randomized the walking directions.

What's up with the non-links in the description?

wow!

How do you represent coordinates? There should be a formula. But how does it include different shapes of tiles

inwards koch pantaflakes :O

i love it but it needs more hats

wow it made a fractal

Teehee, beautiful 🥰

That is very cool an unexpected. Can you give a little better description of the algorithm? I know what a penrose tiling is and how langston's ant works, but I'm not sure what your "deflation rule" means in the context of the ant.

Wouldn't this make creating a highway impossible due to the fact that the tiles do not repeat.

i cannot believe i can understand the title. i am a special type of nerd

Now do on spectre tiling.

Can you do hat monotile now?

Now do it with an a periodic mono tile

What's his going rate? Does he do pools?

I'm still blown away by the fact that it drew a flower 🤯

Wouldnt this mean the penrose is a fractal (this is just a question)

Can this turing machine run doom?

I wish I could understand this intuitively. I have the feeling that if I could, i would know God's phone number.

OwO

Pretty worthless.

lol, I do not understand the description :D