A complete description of Langton's Ant. Read more details here: www.flickr.com/photos/aldoaldo... Una descrizione completa della formica di Langton. Ulteriori dettagli qui: www.flickr.com/photos/aldoaldo...
Then you're in trouble, unless you clarify some things. Do they follow the same rule or does each have an individual rule? Do they move simultaneously or alternatively? If they are simultaneous, the problem arises when two of them step into the same square at the same time. That could only work if they have the same order of colours, so if one ant wants to turn it green, so does the other; then the square turns green and each ant turns according to its own rule which may or may not be the same. On the other hand, if they are alternating, you are free to vary rules. The ants still need to have the same set of colours, as every ant needs to know how to react to whichever colour it encounters. But how to change colour can vary, for example ant A will change 0 to 1, 1 to 2, 2 to 3, 3 to 0, ant B: 0->1->3->2->0, ant C: 0->2->0, 1->2, 3->2, ... In this example ant C can be considered as an ant that only knows 2 colours (0 and 2), but with rules extended so that "unknown" colours are treated as colour 0 (white); such extension would mean that ants son't need to have the same set of numbers (as long as they all have 0, which is the initial colour of the whole plane). An alternative extension for an ant that doesn't "know" all colours would be that "unknown" colours are followed by colour 0 and they turn according to the colour that is followed by 0.
Another thing, do they move with the same speed? Do they start at the same time? Where do they start relative to each other, and in which directions? For example, 2 ants (A and B) start in opposite directions, 10 squares apart back to back, alternative moving, ant B, which starts after ant A made 10 moves, is 1.5 times as fast as the first one. That means the moves go: A A A A A A A A A A B A B B A B A B B A B A B B A ... Note that between each two moves of A (once B starts moving) there are alternatively 1 and 2 moves of B. Without loss of generality, the first ant is facing upwards, starting on square (0,0) and has speed 1 and starts immediately (the first coordinate is how far right (or left if negative) the square is, the second how far up (or down if negative)). In the latter example, the ant A is first and the ant B is facing downwards, starting on (0,-11), having speed 1.5 and delay 10. If, on the other hand, you want to delay and A instead, call B first and A is then facing downwards, starting on (0,-11), having speed 2/3 (0.66...) and whatever delay.
@@zachary007 this system works after basic principles: You see red, you turn the square white and turn right. It's a pretty ordened system, but the infinity amount of possibilities leads to an almost unpredictable situation in a larger scale and after total caos, the system tends to be ordened again. It's entropy, if you look superficially
I wrote some programs to draw EACH frame of the video and write it to disk as a bitmap! I don't know if here we can discuss about chaos, since each try (with the same ant's rule) give exactly the same result. The fact is the ant lasts some time to build a pattern that is the root for all the subsequent iterations.
no one knows, that's why it's chaotic. it could have built a highway by the net step, or take trillions of more steps to make a highway. also sorry for being 2 years late lol
None of these are *truly* chaotic. The apparently chaotic state is self-unstabile, and will inevitably eventually transition to a repeating pattern. A repeating pattern is self-stabile, and once entered will never break. Thus, by simple extrapolation, *absolutely all* such simulations will eventually degenerate into a repeating pattern. (in this context "eventually" just means "less than infinity", so a long wait may be required) . For this to ever fail, the base behaviour would need to be exactly 100% chaotic, which it isn't.
There are only two proofs that I am aware of. 1) The ant will leave any boundary eventually. 2) if the rules are symmetric, the pattern will get to a symmetric state repeatedly. Other than that, not much is known about the ant and the question when it will produce a highway is still unanswered.
I have made some software for everybody in Java to simulate this: github.com/lvivtotoro/langtonvis You can make your own cell types, and the direction (Press the "Releases" button above the long brown bar to download it, there is also a tutorial below!)
So basically, a simple set of rules (such as those of physics/chemistry) may for a while seem to yield total chaos, but eventually, out of that chaos, self-repeating patterns are bound to emerge, creating order - as if it were planned by a "Grand Designer"
By some means the law of physics came to existence... Given enough cycles (or time) of random possibilities, eventually a combination would workout, and allow our universe to derive from it. Can't help that ever present feeling that "I" put myself here though. ;)
Although this is one of those odd and old reconnections RU-vid has, this is absolutely sick and I feel should be taken up more to really exploit the limit of this poor little ant!
I guess that happens because it functions like a exercise in math where you can get a number that never ends ( for example : 0,33333333333333333.... never ending )
I don't think so, because these 'never ending numbers' are based on decimal notation, which computers don't care about. I eughter think, that the algorythm of this simulated ant is stuck in aninfinte loop (for example: if x = 1 -> set x to 2; if x = 2 -> set x to 1)
From what I can see this is process whereby a formula which "appears chaotic" simply goes through a lot of iterations before it manifests a boundary that cannot be avoided because of limits built into its formula which enabled the apparent chaos and at the same time necessitated the inevitable ordered forms which bind it and also emanate from it (parallel to it). It "intersects" with itself in such a way that it must inevitably result in some linear pattern.
@mquinson : may be it isn't clear which colour is the first one in the sequence. Actually, the initial colour of the squares is white; and the first change is FROM white to RED: so try shifting the sequence one step.
It really is just one ant. The reason it appears to be expanding on all sides simultaneously is probably because the program is set to run with less updates per second. It's kind of like frames per second, but the FPS of the program is constant. The question is how often the program sends the progress of the ant to your computer to display. If you were to turn the updates per second all the way down [and since that would take a lot of CPU, we'll pretend the ant slows down as well]
Very good question! I did a try to a Langton's KING... that is an ant that moves like a chess king, in the eight directions. Apart from some good highway or fill, I didn't find any remarkable result: so I decided to create this video only based on "classic" Langton's ant. (I am still working on this stuff...)
I already did some experiments with "stright" and "return back" colors. Also did some experiments with triangular and hexagonal cells, as well as with a 3D, cubic space: the ant could go up, down, left, right, ahead and back! In this video my intention was only to show how the simplest rule cuold result in a high complexity: may be in the future I create another video with such variations.
@krawattenfan: the program is an evolution of an old old one, written in early 90s in basic. I modified it to write a bitmap to disk for each frame, a then converted the bitmaps to AVI clips.
I've just read about Chris Langton in the book 'Complexity'; and I just found out that it was him who created this ant simulation (I knew about the concept for several years already).
This really gets me thinking that reality and the formation of life is simpler than what we make it out to be. Very simple rules creating diametric patterns and whatnot
May be it is because I first change the color, then turn right or left. Since the original sheet squares is white, the first move the ant performs is the second in the loop, so there is a shift
This is what got me into computer programming and made me interested in Artificial Intelligence and Artificial Life. The way simple rules interacting in an environment can lead to complex emergent behaviour.
So in the same manner, does the universe create a projection of the fundamental laws into a multidimensional reality? Maybe not by "go left or go right" but with something similar (exist/express, not-exist/not express)?
That is freaking awesome! Haven't seen that before. How about adding a selection method? Mutates the rule slightly and the used decides if they prefer it or prehaps having the computer pattern spot and either select for or against repeating patterns.
yes I ended up leaving it on for 16 hours, and got 851 results, though there are a few repetitions. This was with 2-20 rules. I currently have it doing the same, but with 2-100 rules. It's interesting how some of the same patterns can emerge with much greater complexity in the rule set ( given that those rules are applied ).
I would want to see that happens if some color allows it to continue straight forward. Would also be interesting with a "go backward" (turn 180*) color. And also rules for diagonal movement.
A longer rules set not always result in a more complex result. For instance, the rule R-R-L-R-R-L can just be seen as twice R-R-L: apart from colors, the results of these 6 and 3 steps rules will have exactly the same shape. On the other hand, I copy from wikipedia: "In 2000, Gajardo et al. showed [...] it would be possible to simulate a Turing machine using the Ant's trajectory for computation. This means that the Langton's Ant is capable of universal computation." Not bad, for a smiple Ant!
+Carlos Barrena There is a trivial solution, where the Ant always turns right (or left) on every colour: the result is a closed loop, 2x2 cells. Apart from this, I read somewhere all the other possibilities imply an infinite enlargement. Actually, I never found such kind of solutions.
For me, I see the resemblance to the formation of crystalline structures in nature. What happens if you put two or more ants in the same space and they work together to fill the space? What happens at contact? Will a boundary be established?
3-4 hours! I wrote a program with powerbasic, a very powerful language, and only worked in memory (nothing screen). I setteld an array of 16000 x 16000 bytes (pixels), and only when the ant reached the border I wrote a bitmap to disk to see what happened.
I wonder - if we theoretically built an entire system like this, using every one of the 256^3 colors available on our screens - what kind of a chaotic image this would produce. What if we added new rules to these colors? Stuff like going diagonally, going straight, going straight and skip 1 pixel, etc.
Use floating points, possibly divide the color grid into 360 colors, each corresponding to an angle, moves 1 unit in that direction, fills a circle radius 1 unit
By placing multiple Ants and inventing a rule for what happens when two meet, this could turn into some kind of game of life, dependant on initial positions and order of colors/turns.
What would happen if you let loose two or more ants, that responded to the same colors differently? Like some turned right on red, rather than left on red, or one made green increment to blue instead of yellow? What if we applied the ant to three or more dimensions? I'm really intrigued with the idea of a machine that leaves behind its own instructions.
I built one of these recently. It broke for the road at about 12000 iterations as predicted. So then I built a 3D version where if there were no directions along x,y it would have to go z. The result was interesting. If I set the z++ it would go some 12 iterations before moving up z then repeat...resulting in a road almost immediately. BUT if I set it to z-- (same algorithm) the iterations settled into a pattern that resulted in a helix. That was really cool!
I did something like your 3D attempt some 10 years ago, and got similar results. Unfortunately the graphic output was really odd, and never found the time to enhance it
since it is a mathematical ant you can predict its movement using some sort of formula, that's just what I love about math. also the ant is cool, nice programming or whatever
I believe that the highway is an essential part of the importance of mathematical research in this experiment. It’s a puzzle to see how different rules will determine how long it takes an unending pattern to form. Hopefully the data sets can eventually be cross-examined to find a clear algorithm to generate specific runs that are guaranteed to behave certain ways. With easily automated parameters.
This kind of behavior looks like what should have happened at the beginning of life. From 2 or 3 simple rules and apparently chaotic behavior, molecules go through a process or trial and error of different combinations and suddenly order emerges (and maybe the firsts aminoacids)... Interesenting
You need to find a way to use this backwards, so that you could draw a picture and the program finds the right setting to achieve this image (or something nearly the same) 😉
+weylin6 But that doesn't really happen. There are rational numbers, such as 1/3 (0.333...) and irrational numbers, such as pi (3.14159...). We don't see numbers with both of these properties, like 5.1297358513513513513513513513513513513 or something like that.
+SirCutRy Of course they exist! The number you mentioned is rational by the way. if you want to make, let's say, the number 1.23444444..., its just using the same rules we use to obtain 0.3333, for example: x=0.333333... 10x=3.333333... 10x-x=3 9x=3 x=1/3 now, let's use the same rule to 1.234444... 100x=123.4444444... 1000x=1234.4444444... 1000x=100x=1111 x=1111/9000 You can apply these rules to obtain numbers like 5.1237358513513513... 10000000x=51237358.513513513... 10000000000x=51237358513.513513... 9990000000x=51186121155 x=51186121155/9990000000 Just by the fact that you can write this number, it exists. If there's an infinite repetition of algarisms in your number, then it's rational
SirCutRy: No no no! You are doing it wrong. You should call his mother out, correct his spelling and bitch about it, say he is a twelve year old, etc. etc.
I believe it's possible that "seas" or "oceans" of repeating digits exist in pi too. However, this is more of a statistics theory than number theory. If the numbers in pi are truly random, then somewhere in pi must exist long (seemingly infinite) seas of repeating digits. This is more a personal argument than a (proven or tested) mathematical argument.
