Yes! The delightful reason why this configuration can repeat within itself is that this Game of Life pattern is a machine with the rules of Game of Life built into it. It's calculating and displaying the game step-by-step. If you watch the machinery you can see the "data" flying around between cells and then the cells turning on and off.
This pattern isn't unique – you could certainly come up with infinitely many more machines in Game of Life that do the same thing. But yes, most Game of Life patterns aren't fractal like this.
It can't be a stronger property, the essence of Turing-completeness is that a Turing-complete automaton can simulate any other Turing-complete automaton, including itself.
Whether an automaton, capable of scale-invariant self-simulation, but not exhibiting Turing-completeness, is even possible, is a very interesting question. I don't know the answer.
If anyone is interested in reading about this idea, I recommend Stephen Wolfram's "A New Kind of Science" [0] and Max Tegmarks's "Our Mathematical Universe" [1].
Seems most likely, but why is there anything at all to begin with?
.. a feeling deep in the pit of my stomach.
Does anyone else sometimes get these "intuitive flashes of insight" - like your brain configures itself in a pattern that "lifts the veil".
I've been thinking of starting meditating to stabilise or cultivate this state if possible because i've feel like i've been losing this slightly psychedelic ability since i was a kid.
Most often i get it in the hypno/hypopompic phases, sudden flashes of eureka that disappears just as fast, like a loud echo of "wow this is existance", "time is unbelievably deep", "demensions and perspective are bizarre in essence", "is this just a fraction of a shared dream" etc.
If you wanna get really lost, consider how the 'aha' feeling could itself be illusory and indicative of nothing at all. Similar to how everything in a dream is convincingly real. Or how ideas sound better when you're high. 'Aha' is not a reliable narrator.
My challenge to people chasing 'aha', is to see how long you can stay in the present moment.
Why shouldn't stuff be allowed to spontaneously come into existence? Why should there be any rules at all to begin with? If there is truly nothing, then there isn't any rule to force the nothing to stay nothing. So the question becomes "how was the universe chosen to be this way, rather than some other way?". The choice must be made, but there is no objective reason to prefer any option over any other. "Nothing exists" is a simple answer, but simplicity is not objectively preferred.
I think the presence of this subjective choice forces a subjective experience into existence with the free will to decide it. Though I haven't a clue how the characteristics of the experience are chosen - that's also a subjective choice. It's subjective choices all the way down.
Requiring infinite amounts of information to explain a finite universe? Occam's Razor comes down pretty hard on the side of "no".
People have tried to explain the universe with cellular automata and so far none of these systems has even been consistent with our current observations of the universe, let alone predicting some new behavior that would allow us to prove or disprove that the theory was true. (If your theory doesn't predict anything new, it's not a new theory at all!)
Requiring infinite recursion of cellular automata would seem to make the whole problem much harder...
Most current models of the universe suggest that it occupies a volume and has a finite number of particles. But these are just models based on observations and constraints.
There is no dispute that the observable universe is as you describe, but there is absolutely no consensus as to whether the universe is finite or infinite.
That's not true in the context of big bang cosmology. It's true that today we would not be able to observe things beyond the observable universe by definition, and so there's not much purpose in reasoning beyond today's observable universe, but the observable universe changes over time and when reasoning about the early universe there would be differences between a finite universe and an infinite universe.
I agree that this is really cool, but this is a very contrived situation where the entire universe is arranged in such a way as to form a sort of fixed point. The vast majority of configurations of the Game of Life or of the real world are not fixed points and would break the clever tricks that are used to make this.
Also, because the simulation is infinite and speeds up and simplifies itself as you zoom out, you don't properly appreciate the fact that every layer is a thousand times slower than the layer that simulates it and contains a thousand times less cells (I don't know if it's a thousand). If you transposed this to the real world, it would take one kilogram of computing substrate an hour to simulate one gram of matter for 3.6 seconds and that's kind of useless.
Well, we don't have the tech to build such a simulator ourselves, so we'd have to be a leaf node of the simulation tree, that has yet to run long enough to discover how to spawn the next levels. But I guess not all leaves are successful at spawning - some just self-destruct.
I've downloaded saharan's repo and poked around the Life Universe project in Chrome and FF— it seems that the pixels representing the compressed animation are retrieved through a Context2D::drawImage call, and are different in Firefox and other browsers.
It makes sense that ETP would mess with the pixels output from a drawImage call, to fight fingerprinting.
Maybe the ImageLoader can be replaced with a more direct PNG decoder stapled to a fetch(). If I can make that work tonight, I'll create a pull request; this is how I meet the coolest people.
That’s so clever. Nothing stands as particularly revolutionary but the author brilliantly solved all the problems to deliver the final product. They are only a undergraduate student as well, they have a bright future ahead.
Thank you! I still don't understand the relationship between "year 6" and undergraduate, or where does it say "year 6". I'm honestly wondering as I don't know anything about Japan
Nothing to do with undergrad/PhD - I was just commenting on the way the date was written (2024-03-21 in Chinese/big-endian format, 21.03.2024 in European/small-endian format, 03/21/2024 in the muddled US format):
At the bottom of the document, it says "令和 6 年 3 月 21 日“, where the first two characters denote the Reiwa era which started in 2019 (=year 1 of the Reiwa era). 2024 then is year 6 of that era. And that's just what the next two characters denote: "6 年" = year 6. "3 月" is month 3 (the character symbolises a moon), and "21 日" is day 21 (the character symbolises a sun).
This is mind blowing; I'm lucky that I've seen hashlife and life in life before, that these additional layers by themselves don't leave me stunned for days.
I’m assuming this is only achieved by “cheating” in some way? That is, I don’t see how giant pixels can come and go, at a zoomed in level, without breaking the rules of Conway’s game of life?
Still cool, but more curious to understand how this works, and if it’s actually able to pull it off “legitimately”.
If you zoom in sufficiently, you can see that the "giant pixels" are really streams of spaceships, arranged in carefully positioned rows and columns that annihilate each other along the pixel's diagonal. These streams can be turned on or off to form the "display" component of each pixel, which is controlled by other parts of the pattern that do the actual computation.
As you zoom in or out, the simulation speed changes. A pixel seems to undergo a discrete state change when zoomed out, but when you zoom in and slow down, you can see how that change is actually executed by many underlying computational steps.
Other links in this thread go into more detail about how it works. The cleverest part is that the simulation is done in (more or less) "constant time" per frame, no matter how deeply you zoom or how fast you accelerate time.
That is a good summary, but of course, there is "cheating" to do the infinite amounts of work required in a finite time!
The board appears as if there exist at any time infinite levels of cellular automata going all the way down running things. In fact, they only compute one level deep and then short-circuit to using the "hardcoded" rules of Conway's life, and simply pop up the higher or lower levels on demand.
That's a fair point, but I don't consider that "cheating" any more than Hashlife [1] (which it's based on) is cheating. It's an ingenious trick that guarantees that what's displayed is exactly the same pattern as if all of the infinitely-many higher and lower levels had been simulated, in finite time. The rules of the cellular automaton are never broken, only elided.
In a metaphorical way, it reminds me of the "tying the knot" trick in pure functional languages like Haskell, to turn an infinite computation into a finite data structure.
It's no more cheating than the fact that 1 = 1/2 + 1/4 + 1/8 + 1/16 + ...
Despite the fact that you can never actually add up an infinite number of terms one by one, you can however, compute what the limit of the sum will be in a finite amount of time. The same principle applies to this simulation, in which it is computing the limit of the process.
Agreed, I think the key here is that the Game of Life is structured enough that many computational steps can be elided in this way. If it was more chaotic, this couldn't happen. The realization that GoL can be used to run one GoL cell in isolation is the brilliance here I think. Once you can decompose one single cell into GoL, you can also compose GoL into one cell and save computational cycles. Then suddenly it can be GoL all the way down.
I don't know the "tying the knot" trick, but I imagine it to be something like a proof by induction of self-simularity being exploited to share computation at different zoom levels.
I might be missing something, but I don't think you even need Hashlife for this. The pattern is regular enough that you could just flat out store a static (x, y, t, state) map of a single cell and apply it fractally in order to compute the state of any part of the map at any time in constant time. You don't even need to simulate it at all.
Welllll, there seems to at least be some mathematical cheating in that this is representing a non-well-founded set. (Declare each pixel to be the set of subpixels at the next level down defining it, this forms an infinite descending chain).
A digression for anyone interested in Hashlife: I want to recommend Gosper's original paper [1] over the various other explanations of it. It is brief (6 pages), lucid, and was the paper that finally made me understand how hashlife works.
For the experts in the house: would it be possible for this to wrap around, so the highest and lowest level join? A torus of GoLs? Can such a thing, assuming it exists, have a finite number of layers? Just curious, this is amazing.
I guess what they mean is that the "zoom" would be cyclic, e.g. at the same time the game zoomed at 1x would look like the game zoomed at 101x, zoomed at 2x would look like 102x etc...
Totally different thing, but this makes me think of looping procedural animations, which are achieved by sampling noise on a circle (or walls of a cylinder in a 3d noise space).
At a very high level, this is simulating the Game of life at the appropriate resolution level based on an algorithm whose input parameters are the zoom level and your location in the space, similar to a fractal pattern. So I'm not sure what you mean by a "torus" of GoLs.
Have you heard of the thought experiment where a 2D plane is finitely sized instead of infinitely sized, but as you travel within the 2D plane, if you get to one edge, you wrap around to the opposite edge? Even though its size is finite, you can pick a direction and go that way forever.
If you sit in 3D space and look at this finite 2D plane, then it looks like (say) a rectangle. That's displeasing because from inside the plane, it's continuous, but from the outside, it looks discontinuous.
One way to get rid of that annoyance is to map the plane onto a torus. Then it looks continuous from the outside. It's no longer flat, which is un-plane-like, but you can't have everything in life.
Anyway, the zoom level of game of life is an infinite number line. But what if it repeats and can be represented as something that wraps around? You could think of it as a line segment that you wrap around into a circle (to join the two ends). So the same concept as mapping the plane onto a torus, except 1D instead of 2D.
Yes but the question is whether there are repeating non empty patterns. I don't think you've answered this? I don't know but I think that's a fascinating question.
I believe the comment above is talking about the game running as a simulation within another instance of the game. Each instance of the game has a state. Game G1 has state S1, game G2 has state S2, etc. I think they're asking if there's any S1 that can be chosen so that S2 = S1.
You might need 2 or more layers, so whatever number of layers you need (if this is possible, you'd have some sequence infinite sequence (moving through layers) like (S1, S1, S1, ...) or (S1, S2, S1, S2, ...) or (S1, S2, S3, S1, S2, S3, ...).
The length of the repeating pattern (subsequence) is going to be a positive integer, so I don't think real numbers are relevant, if that's the question.
To dispose of this briefly, the cardinality of computable functions is ℵ₀, and Life is a computable function. Although the parent question is underspecified (some good guesses as to what specifically is meant in sibling comments), no variation on that question could increase the ℵ number of the result.
Well... if we wrap around without meddling with time, it will be impossible. On the lowest level cells are switching much more frequently than on the highest level.
But if we allow ourselves to bend time, then probably... It would be like you zoom out and go back in time. My guess it is possible, but I still cannot wrap my head around infinite spatial dimensions of the field. We need infinity for that because without it will hit the problem of different cell count on the lowest level and the highest (which are represent the same sequences of states of GoL). I see no possible problems to buils such a "torus" but to be sure one needs to really prove it.
This reminds me of nested dreams, where you wake up from a dream in your bed, only to slowly realize you’re still dreaming and wake again in the same bed.
It just feels like the span of time of the dream is longer than I have been sleeping. I could sleep for an hour and dream for a day. But it's not really that I go through a day long dream, it's just that it feels like a day has passed.
My theory is that the memory of a dream is actually formed instantaneously in the brain, not over whatever span of time you feel like has passed in the dream.
Have you ever dreamed that you've spent an hour trying to turn off your alarm clock, but in real life you wake up and it's 1-minute past the alarm time? How can you have had this long complicated dream involving an alarm clock that won't turn off, but you've only really slept 1-minute through the alarm? My answer is, as I said, the dreams happen instantly but your brain just thinks time is passing in them because it's a false memory.
That's a good point. I've definitely had that alarm experience before, but not in the context of dreams within dreams.
I think the idea of each inner dream happening faster than the outer one is probably just an Inception thing.
Fast dreams might not need to be an "instant" false memory. It seams possible that a dream could actually happen much, much faster than your "normal" sense of time, it's entirely synthetic so there is no sensory bottleneck. But it would take more energy to experience it this way.
I'm basing this theory off of the time I had several long, detailed dreams in the 9 minutes between pressing snooze and the second alarm. I woke up bewildered and worn out, as though I had been up all day. Worst "snooze" ever.
They're not regular. Just whenever I do experience a dream-in-a-dream loop, it's never just one loop and it feels like it's taking forever to actually wake up. During the dream, I do experience the emotion of impatience and annoyance. Sometimes there is also a feeling of urgency and fear because the in dream I am trying to escape some danger and I know that I can escape it by waking up ... but when I do wake up, I'm still in a dream.
I've on a few very rare occasions been awake/conscious in real life, but also still dreaming. That's really weird when that happens. To explain: once I was awoken by my wife getting ready for work, I said some words to her, while in the back of my mind the dream was still happening. I then went back to sleep and returned to the same dream in progress. It wasn't like I was talking to her in my sleep and didn't remember it. I was fully awake and making new memories, while still dreaming.
The best analogy that I can make that you might be able to understand is. I have multiple streams of consciousness. One level I use to think about what I'm going to say when speaking to someone, but at the same time I could be counting in my other stream of consciousness. I assume that's how most people's brains work. I think when the dreaming-awake thing happened, the dream was in one stream and the awareness was in the other.
Looking at a Digital Clocks also work for me. For some reason those always seem to “bug out”, showing or hiding random lines and sometimes even showing an unreadable time.
The worst is when this happens when you need to use the toilet. After walking to the toilet for the nth time, how do you know for sure you won't wet the bed?
Adding one pixel at any level is the equivalent of adding me or infinite numbers of pixels, smaller and smaller levels. However, I think if you add in the pixel in a empty space at its level, it would simply disappear. I don't think it would cause any havoc. But, this is not an actual game of life. This is a PNG of many steps in a game of life. The computation power for actually making it a real game of life would be pretty big.
I think the coolest thing about this is the way time works. One step at each level is computed using a large number of steps at the level below. So as you zoom out you are sinking into an exponential vortex where each step takes days, years, millennia to change.
But ingeniously, the simulator smoothly scales its speed as you zoom, totally erasing the fact of time. I wish there was an indicator somewhere of how much game time was going by per real-life second.
EDIT: ...but in order to do that you'd have to declare one of the levels to be the "bottom" level, the one that runs in real time, and that would ruin the fractalness of it all...
For some reason what really sticks out to me is how when you zoom out, you always come out of a different part of the circuit, so it doesn't feel like the normal repetitive fractal, you're seeing all the parts of the machine from different chronological angles or something.
Interestingly, the author's blog post [1] mentions that this is due to technical limitations and not specifically done by choice, although I do like the effect.
> When we zoom in, we discard the information of the largest meta-metapixel, so it is not always possible to reconstruct the same scene when we zoom out again.
> Since zooming in and out can be continued indefinitely, we also realize that we need infinite information to accurately represent "our current position".
"Another thing I know about screens is, you can go into them as far as you want forever. And you can come out of them too, but you might not end up in the same place that you started. You might get lost. Or you might not."
Imagination is fun. I'm pretty sure I can make your brain break a little bit more. Imagine they're all looped and there is no bottom level fastest super infinite computer.
My read on the author's explanation story is he did actually solve the "consistency with time" and "consistency in position" problems that would make it a "rounds up to real-looking" at some zoom seam.
It's not cheating, but the trick is it's pre-computed.
Basically the structural patterns repeat, and you can pre-compute the patterns (which are also the sub-patterns), then basically tile the configurations and their transitions "across time". A naiive approach takes a few TB to store it all, but there's some clever optimizations that can bring it down. Finally once a "solution" through the tiling space was found, the author encoded it into a 4 mb png and built a display with some clever shaders.
Pardon my poor vocabulary to describe the following:
But if the patterns are precomputed at whatever timescales, the patterns repeat in a predicable interval, based on the scale of time..
So, I wonder what each layer of pattern looks at from a perspective of a prime or true prime number scale digit...
Like if real-time baseline is 1 - and if you look at the pattern in a still frame at position scale 1. What is the pattern look like at time scale of the primes and true primes?
Whereas the primes and true primes are known discretes, so one could think of them as discrete dimensions of timescale based on the pattern represented at timescale position 1 (real-time)
And what if timescale at position 1 is some reflection of the pattern at the primes and true prime timescales... (assuming a dimension simply means the rate at which times flow at that energy level)
I'm not saying it isn't technically impressive and super cool - but he clearly can't actually be computing infinitely recursive levels of GoL, so in that sense it's an illusion.
The hows-it-done article describes the meta-pixel pattern concept then gives you the "clock-ratio" of the periodicity:
> In addition, the pattern has a period of 35328, so advancing 35328 generations will cause the generation at the meta level to advance by 1.
I would even say this time dilation is necessary because the pattern's self-similarity is across time, and if the two levels operate at different clocks, you need to slow down the next level as it comes up to the self-similar animated view of the prior level.
In other words, the structure at level n requires 35328 iterations of the self-similar structure at level n+1, so if you're bringing n+1 up to the self-similar view of n, you need to slow down n+1 as it's coming up to also hit the time-based self-similarity.
I wonder then if there's something like a time-invariant constant, maybe along the lines of the "computational complexity" of any view remains constant across all levels of zoom.
Note that this is also how our own world works. Typical time scales of processes get faster as the length scales shorten. Somehow it feels inevitable for a dynamic multiscale structure.
armchair physicist moment: isn't this a straightforward outcome of c as a global "speed limit"? effects can't propagate faster than c, so I would expect that interaction "latency" decreases linearly with scale.
Since the grid is infinite, it should be possible for one level to to be completely identical in its history to the next level. Thus you could identify all levels with each other to really be the same level. Then both space and time are circular in their scale. (The now-single level could nevertheless still be non-repeating both in space and time!)
My subconscious reaction when first zooming in was: “f—— you.” They’ve taken something I’ve been familiar with for 20+ years and completely blown my mind. One of the coolest things I’ve seen on here in a long time.
Game of Life must be the first simulation I ever took on as a programmer — and have revisited it over the years.
To this day it continues to come up in my mind when people muse about things like consciousness. Is there a magic dust to us? Or are we the result of simple rules playing out in a grand stage?
As the laws of physics do not seem to stop at the borders of a skull, there is no sovereignty of the Self. But no one minds a thought thinking itself its own thinker, as there is no one at all. Just a thought.
> As the laws of physics do not seem to stop at the borders of a skull
There doesn't need to be a magic dust for consciousness, it could still be a primitive compatible with existing physics. See what Roger Penrose worked on.
I.e. an intelligent computing engine which learns to model both its environment and itself, with some degree of access and control of its internal processes including when it considers itself.
At the point of clear, continuous practical self-awareness, how could it not be “conscious”? It will experience it’s self’s awareness of it’s self-awareness.
Or are the simple rules themselves the "magic dust". These turing complete structures show up everywhere and are in many ways equivalent.
Perhaps this underlying structure is the foundation of our metaphysics and our reality is just one node of it's permutative potentiality. The simple rules of GoL are just an emergent reflection of the fundemental nature of meta-reality. These types of relfections within our neural and semantic structures may be the necessary "magic dust" for consciousness to emerge.
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[ 4.6 ms ] story [ 218 ms ] threadThis pattern isn't unique – you could certainly come up with infinitely many more machines in Game of Life that do the same thing. But yes, most Game of Life patterns aren't fractal like this.
Is the ability to do this tied to the fact that Game of Life is Turing-complete, or is there a weaker/stronger property that allows for this?
Also, does this count as a “quine”?
You can build structures within the GoL rules that implement the GoL rules, so…
sounds like GP is aware of that fact :)
Whether an automaton, capable of scale-invariant self-simulation, but not exhibiting Turing-completeness, is even possible, is a very interesting question. I don't know the answer.
Could we be living in a infinitely recursive simulation?
[0] https://www.wolframscience.com/nks/
[1] https://space.mit.edu/home/tegmark/mathematical.html
https://github.com/gwf/CBofN
.. a feeling deep in the pit of my stomach.
Does anyone else sometimes get these "intuitive flashes of insight" - like your brain configures itself in a pattern that "lifts the veil".
I've been thinking of starting meditating to stabilise or cultivate this state if possible because i've feel like i've been losing this slightly psychedelic ability since i was a kid.
Most often i get it in the hypno/hypopompic phases, sudden flashes of eureka that disappears just as fast, like a loud echo of "wow this is existance", "time is unbelievably deep", "demensions and perspective are bizarre in essence", "is this just a fraction of a shared dream" etc.
Does that make any sense?
My challenge to people chasing 'aha', is to see how long you can stay in the present moment.
I think the presence of this subjective choice forces a subjective experience into existence with the free will to decide it. Though I haven't a clue how the characteristics of the experience are chosen - that's also a subjective choice. It's subjective choices all the way down.
People have tried to explain the universe with cellular automata and so far none of these systems has even been consistent with our current observations of the universe, let alone predicting some new behavior that would allow us to prove or disprove that the theory was true. (If your theory doesn't predict anything new, it's not a new theory at all!)
Requiring infinite recursion of cellular automata would seem to make the whole problem much harder...
It's not that your statement is false, but that there is absolutely no consensus on the matter.
Some physicists make strong arguments that at the big bang the total energy of the universe was 0:
https://en.wikipedia.org/wiki/Zero-energy_universe
This simulation is a finite amount of information being used to describe an infinitely large simulation.
A system whose fundamental nature can be modified / controlled by ideas spawned within the system itself would be extremely cool.
I think a decent argument could be made that we may be in just such a system.
https://youtu.be/DoCYY9sa2kU
Also, because the simulation is infinite and speeds up and simplifies itself as you zoom out, you don't properly appreciate the fact that every layer is a thousand times slower than the layer that simulates it and contains a thousand times less cells (I don't know if it's a thousand). If you transposed this to the real world, it would take one kilogram of computing substrate an hour to simulate one gram of matter for 3.6 seconds and that's kind of useless.
(Some extensions conflict with some webpages, too...)
One day we will get it together so every computer program works on every computer, if we don't collapse our ecosystem first.
Though it is working with uBlock origin(default settings)..
It makes sense that ETP would mess with the pixels output from a drawImage call, to fight fingerprinting.
Maybe the ImageLoader can be replaced with a more direct PNG decoder stapled to a fetch(). If I can make that work tonight, I'll create a pull request; this is how I meet the coolest people.
https://en.wikipedia.org/wiki/Reiwa_era
At the bottom of the document, it says "令和 6 年 3 月 21 日“, where the first two characters denote the Reiwa era which started in 2019 (=year 1 of the Reiwa era). 2024 then is year 6 of that era. And that's just what the next two characters denote: "6 年" = year 6. "3 月" is month 3 (the character symbolises a moon), and "21 日" is day 21 (the character symbolises a sun).
Still cool, but more curious to understand how this works, and if it’s actually able to pull it off “legitimately”.
If you zoom in sufficiently, you can see that the "giant pixels" are really streams of spaceships, arranged in carefully positioned rows and columns that annihilate each other along the pixel's diagonal. These streams can be turned on or off to form the "display" component of each pixel, which is controlled by other parts of the pattern that do the actual computation.
As you zoom in or out, the simulation speed changes. A pixel seems to undergo a discrete state change when zoomed out, but when you zoom in and slow down, you can see how that change is actually executed by many underlying computational steps.
Other links in this thread go into more detail about how it works. The cleverest part is that the simulation is done in (more or less) "constant time" per frame, no matter how deeply you zoom or how fast you accelerate time.
The board appears as if there exist at any time infinite levels of cellular automata going all the way down running things. In fact, they only compute one level deep and then short-circuit to using the "hardcoded" rules of Conway's life, and simply pop up the higher or lower levels on demand.
In a metaphorical way, it reminds me of the "tying the knot" trick in pure functional languages like Haskell, to turn an infinite computation into a finite data structure.
[1]: https://en.wikipedia.org/wiki/Hashlife
Despite the fact that you can never actually add up an infinite number of terms one by one, you can however, compute what the limit of the sum will be in a finite amount of time. The same principle applies to this simulation, in which it is computing the limit of the process.
[1] https://www.lri.fr/~filliatr/m1/gol/gosper-84.pdf
https://blog.oimo.io/2023/04/10/life-universe-en/
Totally different thing, but this makes me think of looping procedural animations, which are achieved by sampling noise on a circle (or walls of a cylinder in a 3d noise space).
If you sit in 3D space and look at this finite 2D plane, then it looks like (say) a rectangle. That's displeasing because from inside the plane, it's continuous, but from the outside, it looks discontinuous.
One way to get rid of that annoyance is to map the plane onto a torus. Then it looks continuous from the outside. It's no longer flat, which is un-plane-like, but you can't have everything in life.
Anyway, the zoom level of game of life is an infinite number line. But what if it repeats and can be represented as something that wraps around? You could think of it as a line segment that you wrap around into a circle (to join the two ends). So the same concept as mapping the plane onto a torus, except 1D instead of 2D.
You might need 2 or more layers, so whatever number of layers you need (if this is possible, you'd have some sequence infinite sequence (moving through layers) like (S1, S1, S1, ...) or (S1, S2, S1, S2, ...) or (S1, S2, S3, S1, S2, S3, ...).
The length of the repeating pattern (subsequence) is going to be a positive integer, so I don't think real numbers are relevant, if that's the question.
But if we allow ourselves to bend time, then probably... It would be like you zoom out and go back in time. My guess it is possible, but I still cannot wrap my head around infinite spatial dimensions of the field. We need infinity for that because without it will hit the problem of different cell count on the lowest level and the highest (which are represent the same sequences of states of GoL). I see no possible problems to buils such a "torus" but to be sure one needs to really prove it.
I've had dreams about waking up in succession, but have not experienced a time dilation.
Do you dream you are falling asleep, having an inner dream, and waking up again to the outer dream, to discover less time has passed than expected?
My theory is that the memory of a dream is actually formed instantaneously in the brain, not over whatever span of time you feel like has passed in the dream.
Have you ever dreamed that you've spent an hour trying to turn off your alarm clock, but in real life you wake up and it's 1-minute past the alarm time? How can you have had this long complicated dream involving an alarm clock that won't turn off, but you've only really slept 1-minute through the alarm? My answer is, as I said, the dreams happen instantly but your brain just thinks time is passing in them because it's a false memory.
I think the idea of each inner dream happening faster than the outer one is probably just an Inception thing.
Fast dreams might not need to be an "instant" false memory. It seams possible that a dream could actually happen much, much faster than your "normal" sense of time, it's entirely synthetic so there is no sensory bottleneck. But it would take more energy to experience it this way.
I'm basing this theory off of the time I had several long, detailed dreams in the 9 minutes between pressing snooze and the second alarm. I woke up bewildered and worn out, as though I had been up all day. Worst "snooze" ever.
I've on a few very rare occasions been awake/conscious in real life, but also still dreaming. That's really weird when that happens. To explain: once I was awoken by my wife getting ready for work, I said some words to her, while in the back of my mind the dream was still happening. I then went back to sleep and returned to the same dream in progress. It wasn't like I was talking to her in my sleep and didn't remember it. I was fully awake and making new memories, while still dreaming.
The best analogy that I can make that you might be able to understand is. I have multiple streams of consciousness. One level I use to think about what I'm going to say when speaking to someone, but at the same time I could be counting in my other stream of consciousness. I assume that's how most people's brains work. I think when the dreaming-awake thing happened, the dream was in one stream and the awareness was in the other.
It's a strange feeling to be aware that you're in a dream. I've never really managed to get complete control of the dream.
Except, I can't remember if my dream text look that way before I was exposed to AI art, or if my dreams were influenced by my memories of AI art text.
For some reason I want the ability to put a pixel and watch how it destroys the world :)
But ingeniously, the simulator smoothly scales its speed as you zoom, totally erasing the fact of time. I wish there was an indicator somewhere of how much game time was going by per real-life second.
EDIT: ...but in order to do that you'd have to declare one of the levels to be the "bottom" level, the one that runs in real time, and that would ruin the fractalness of it all...
> When we zoom in, we discard the information of the largest meta-metapixel, so it is not always possible to reconstruct the same scene when we zoom out again.
> Since zooming in and out can be continued indefinitely, we also realize that we need infinite information to accurately represent "our current position".
[1] https://blog.oimo.io/2023/04/10/life-universe-en/
https://www.youtube.com/watch?v=Q4OIcwt8vcE
"Another thing I know about screens is, you can go into them as far as you want forever. And you can come out of them too, but you might not end up in the same place that you started. You might get lost. Or you might not."
https://youtu.be/4TI1onWI_IM?si=uBt9Gob0gqnFk8lP
It's not cheating, but the trick is it's pre-computed.
Basically the structural patterns repeat, and you can pre-compute the patterns (which are also the sub-patterns), then basically tile the configurations and their transitions "across time". A naiive approach takes a few TB to store it all, but there's some clever optimizations that can bring it down. Finally once a "solution" through the tiling space was found, the author encoded it into a 4 mb png and built a display with some clever shaders.
But if the patterns are precomputed at whatever timescales, the patterns repeat in a predicable interval, based on the scale of time..
So, I wonder what each layer of pattern looks at from a perspective of a prime or true prime number scale digit...
Like if real-time baseline is 1 - and if you look at the pattern in a still frame at position scale 1. What is the pattern look like at time scale of the primes and true primes?
Whereas the primes and true primes are known discretes, so one could think of them as discrete dimensions of timescale based on the pattern represented at timescale position 1 (real-time)
And what if timescale at position 1 is some reflection of the pattern at the primes and true prime timescales... (assuming a dimension simply means the rate at which times flow at that energy level)
Does that make sense?
> In addition, the pattern has a period of 35328, so advancing 35328 generations will cause the generation at the meta level to advance by 1.
I would even say this time dilation is necessary because the pattern's self-similarity is across time, and if the two levels operate at different clocks, you need to slow down the next level as it comes up to the self-similar animated view of the prior level.
In other words, the structure at level n requires 35328 iterations of the self-similar structure at level n+1, so if you're bringing n+1 up to the self-similar view of n, you need to slow down n+1 as it's coming up to also hit the time-based self-similarity.
I wonder then if there's something like a time-invariant constant, maybe along the lines of the "computational complexity" of any view remains constant across all levels of zoom.
So the higher you go, the less you're impacted by gravity, the faster times goes.
See https://en.wikipedia.org/wiki/Gravitational_time_dilation
computationally, what appears as the experience of “consciousness” is thus Anattā (not-self)
https://en.m.wikipedia.org/wiki/Anattā
Game of Life must be the first simulation I ever took on as a programmer — and have revisited it over the years.
To this day it continues to come up in my mind when people muse about things like consciousness. Is there a magic dust to us? Or are we the result of simple rules playing out in a grand stage?
> As the laws of physics do not seem to stop at the borders of a skull
There doesn't need to be a magic dust for consciousness, it could still be a primitive compatible with existing physics. See what Roger Penrose worked on.
I.e. an intelligent computing engine which learns to model both its environment and itself, with some degree of access and control of its internal processes including when it considers itself.
At the point of clear, continuous practical self-awareness, how could it not be “conscious”? It will experience it’s self’s awareness of it’s self-awareness.
[0] https://www.youtube.com/watch?v=sWblpsLZ-O8
Perhaps this underlying structure is the foundation of our metaphysics and our reality is just one node of it's permutative potentiality. The simple rules of GoL are just an emergent reflection of the fundemental nature of meta-reality. These types of relfections within our neural and semantic structures may be the necessary "magic dust" for consciousness to emerge.