the issue is that you then lose the clear boundaries, and it's a little harder to see the fractal.. it still would probably be a pretty cool visualization though.
Fascinating. I wonder how much the stability patterns are dependent on the chosen integration method. That is, I'm wondering how much of what we're seeing is the behavior of the differential equation and how much of it is due to the chosen method and error estimate tolerances, or even floating point accuracy.
I assume the Shadowing Lemma applies to this system, but I could be wrong. The Shadowing Lemma says that every computed orbit stays within some small distance of a true orbit, i.e. small errors don't matter.
I was thinking the same thing. I would be interested to see one particular time step compared across multiple different integretion schemes, e.g. what would time step t=2.0 look like with delta-t of 0.1, 0.01, 0.001, etc.
@@qzamboni The thing here is that according to chaos theory, adding a small error to a step would lead to huge discrepancies down the line so not sure if the Shadowing Lemma would apply here. That said it's the first time I hear of this so I could be wrong.
Copying what someone else said, it's probably an island of stability, where nearby pendulums converge into the same shape rather than falling into chaos
I am thinking about ways to "isolate" the central fractal from the surrounding noise. Performing computation on the noisy regions is, in a sense, "wasted", since a noisy region never seems to further change from being just white noise. In the Mandelbrot set, after each iteration, if the iterated value escapes the unit disk in the complex plane, we can safely discard it, as we know the value of this point will diverge into infinity. Usually corresponding starting point is colored white. We could do a similar thing here, and find a range of values for angles and velocities that we know for certain is in the chaotic region, and discard the pendulums that enter that region, then color the corresponding positions white. I believe this should eventually filter out ALL (or most) of the noise outside of the main fractal, even outside this filter region, since unstable positions will, by pure chance, "trickle into" the filter region, like water circling into a sink drain.
@@quantumsoul3495 I don't think so. The small island of stability at x=101°, y=+115° appears to be a basin of a stable attractor, where the pendulums get more and more similar by all approaching the same single limiting cyclic motion, instead of drifting apart in their behavior. Even with limited numerical precision of the simulation, they will stay similar, as long as the corrective attraction of the attractor is greater than the floating point errors.
i wonder if you could instead use the difference between neighboring pixels, and pixels which are surrounded by other values which are all sufficiently different could be discarded. in this way i think you could also adjust the desired noise amount? idk how to express this idea rigorously or even really coherently lol
@@goji_crafter I think that won't quite work for what I intended, which is filtering out the "useless" white noise (useless in the sense that it has no structure). Because even in the non-white-noise regions, you can still get quite dense changes in behavior. Such as the central oval, which starts as a fairly stable region and gradually fills up with higher frequency noise from the top-left and bottom-right. But notice how it never quite becomes _pure_ white noise, unlike the outside regions. This tells us there's some subtle underlying stable behavior which prevents degradation into white noise. Visually, you can see the boundary between the two different kinds of noise, which still shows the outline of the central oval.
Oh, thank you. Immediately shared with whoever is around. It is interesting if the island of stability is the same with different pendulum angle"resolutions".
I was looking at this graph and I was wondering why it is not symmetrical around all 4 corners and the answer is because of gravity. And a double pendulum with no gravity is not chaotic. I was wondering how does the transition between no gravity and small and large number of gravity will look like
In this case, changing gravity is equivalent to just changing the timescale of the system, or its length. Want to see what this would look like with stronger gravity and the same lengths? Play it on ×2 speed
This is an incredible addition to the collective understanding about Chaos Theory. My god what a visual treat. Your previous one was badass, I’m blown the fuck away by this one. Very well done I really appreciate this kind of thing.
The bulbs of the Mandelbrot converge to periodic orbits. The center region seems to be where the angle is small enough that they swing in unison. I would be curios what the ratio of oscillation period is in the stable region.
Have to wonder how much of what we're watching is real and how much is an artifact of the floating-point representation being used. The nature of the system means that any finite rounding will tend to dominate the simulation over a relatively short period of time, which suggests that if infinite-precision calculations were possible, the picture that comes out would look quite different to what we're observing. What is known about the magnitude of this effect in the context presented here? My math isn't strong enough to have a good intuition beyond the sense that rounding *might* have an outsized impact.
It should be possible to make an "accurate" picture without infinite-precision calculations if you control the error terms well enough. By which i mean that at any given time we can calculate a picture that is as close to the real one as we want (this might fail here because of the imprecisions). it might be very inefficient though.
@@Galinaceo0 The problem is that with a chaotic system (basically, any system where the result loops back into itself without diminishing returns), the lack of precision always dominates the output after a certain number of iterations. Any region of the image that doesn't converge to some stable value will always look very different after many iterations, even if the error is 1.0x10^-200 or something else very, very small. The question that I can't answer is the amount that this instability would affect the "interesting" parts of the image highlighted in the animation. Would there still be an interestingly-shaped island in a "real" version of this with infinite precision, and would it have a similar shape? No idea but it is interesting to contemplate.
@@cthonianmessiah That can be fixed by recomputing the whole thing from the beginning with more and more precision each time you want to see more into the future. The problem however is, as i said, that this is extremely inefficient.
@@Galinaceo0 Yes, you hit a wall due to nonlinear behavior and no matter how much compute you throw at the simulation, the progress basically stops. This is, for example, why weather forecasts stop at 7-10 days even as available compute resources increase exponentially.
@@cthonianmessiah Well in weather simulation it's actually impossible to measure everything we need to compute it to a given precision. for double pendulums, as long as we got enough memory and time it is always possible to compute.
I wonder what it would look like if we considered starting positions with initial angular velocity as well. Sure, the space would be 4d, but if we only consider a slice of pendulums with the same total energy we would get a 3d shape. Interesting how trajectories might look there. (Since energy is preserved all of them would lie in the slice.)
I believe that if you change the coorsinate system we're gonna see some interesting things too. Like polar coordinates or something like that. There's probably a system (maybe projecting on an object ?) where we recognise pattern since it's deterministic. Maybe the part you zoomed onto is gonna be round or squared ? You did a great job simulating that shit ahah
Interesting how there is a definite boundary between the perturbative regime and the chaotic region where it's just noise. You can see the two oscillatory modes in the perturbative region, each with its own frequency. The yellow/green one is the lowest frequency mode, and the red/blue one is so fast it looks purple. The low frequency mode looks stable at relatively high amplitudes, which really is surprising to me. And then there is that tiny island where it also happens to be stable.
Yes, very close to the center is the perturbative regime where you can study the pendulum quite easily. It turns out there are two modes, one of which oscillates between yellow and green in this color code, and the other oscillates between red and blue. The surprising part is how sharp the boundary is between that quasi-stable regime and the chaotic region. You do get a Newton fractal, because the four colors must always touch.
You inspired me to try it myself with that last video and I got a pretty nice result by computing the average absolute central difference (dx_angle1, dy_angle1, dx_angle2, dy_angle2) / 4. Then the chaotic values are bright and the ordered ones are black. I'm also pretty sure it's fractal in nature so you could do a zoom in which might be cool.
Very cool. I suspect the "stable" part (however one would formalize that) is actually connected at every point in time, maybe even simply connected. Similarly to how the Mandelbrot set is actually simply connected.
What if the angle are relative to each other? Right now they are absolute degrees, but I wonder we get better resolving on the stable regions if the angles are different. I wonder how mass impacts this, also, it wouldn't surprise me if there is a Julia/Mandelbrot set embedded somewhere in here
@@UniHorned Yes. Also, the chart shows that within certain ranges of starting positions where both pendulums start with similar angles, they maintain a periodic pattern. Outside of that, their motions become more chaotic.
There is something about this video, that explains order within chaos, fractals, and other forms in which the Universe is revealing its inner working concretely
This would seem to me to be a two dimensional slice of a 4d object which would include the initial velocities of each pendulum, this slice being where the initial velocities are zero. Would it be possible to do this animation for a slice where the combined potential and kinetic energy of each pendulum is the same by adding whatever clockwise velocity is needed to one of the arms (the clockwise velocity can be negative), with the double pendulum that is pointing straight up at the start having zero added velocity?
would be interesting to see some of the pendulums with starting conditions in that small island of stability. also is the fractal shape accurate, or is it due to bias from the floating point error that accumulates
In case you're curious, the biggest one I found is close to the y-axis, between -30 and 0 degrees. Note, though, that every island has a mirrored version due to the nature of the double pendulum, so there are two of them.
I want to see a simulation of some pendula from that weird island of stability. We should see really stable motion even with small perturbations of initial conditions, but why does this happen, what are the staring angles that do this? I assume it's some form of resonance between the two
Did you use 9.81 for g? The fact that g varies so much (when 0.1% makes a big difference) makes this simulation extra difficult I think. Really cool project, good work!
Wow! Really cool video keep up the good work 😄. I was wondering how were you able to simulate 4 million pendulums? Last time I tried I was only able to simulate 10k pendulums, is your code available on the web? I'm curious to look in to it.
Interesting. What happens if you define the second axis as the second pendulum’s angle relative to the first one, rather than to the absolute reference frame?
Is it possible to define a fractal by the stability of the double pendulum? Will the "island of stability" approach a final shape or it will just keep shrinking?
I'm curious how this fractal compare to the one for time to first flip. The barrier between mess and not mess doesn't appear to be the same as the one between flip and no flip (at least not around theta2=180 and theat1=0)
idea: add a thing that shows the precentage of all pendulums that are each color. example 66.3 % blue 21.2%yellow 41.8%green 11.1% red it would be cool to see how it changes
Nope. The problem is that multiple factors of the system change over time, depending on the state of other factors of the system. As time goes on, the differences compound more and more, pushing the results further and further from its neighbours, until the motion's completely diverged. The exceptions, I assume, are the broad islands of stability where the various changing conditions balance out, and result in a much simpler motion
The most famous fractal - the Mandelbrot Set - is generated from this simple equation: Z' = Z^2 + C where Z and C are complex numbers [each having one 'real number' and one 'imaginary number'] the x-axis is real, y-axis is imaginary Z is initialized at 0,0i and C is set to the x,y coordinates of each pixel. if the magnitude of Z' exceeds say 4.0 that pixel has 'escaped' and will go off to infinity and is [not part of/outside] the set. if Z' repeats or cycles between previous values, it is [part of/inside] the set. count how many iterations a pixel goes through before we know if it is inside or outside the set - that count becomes the pixel's color if it's outside. Pixels inside are typically colored black. the actual color chosen for each value of the iteration count is up to the user. if you aren't familiar with imaginary numbers or complex numbers here's the short explanation. i = the square root of (negative one) i * i = -1 real * real = real real * imaginary = imaginary imaginary * imaginary = negative real Z=(0,0i) C=(2,3i) Z' = (0,0i)*(0,0i) + (2,3i) Z'' = (2,3i)*(2,3i) + (2,3i) ... 2*2 - 3*3 //real part of Z''' is -5 2*3i + 3i*2 //imaginary part is 12i the resulting shape has been called the thumbprint of god. Unless you are following a previously explored set of coordinates given to you, if you zoom in for more than a couple of minutes, you are looking at a part of the set that no one has seen before.
@@mrosskne how can we be sure that nature uses real numbers? We can ever measure with so much precision, quantum physics implies that two non comutating variables cannot be measured both with high precision. Real numbers are just an idealization perhaps. In case everything in nature is quantized then even a pendulum will be quantized.
@@mrosskne no that is not a proof that nature has real number values. Maybe nature has only finite decimals of such size that we were not able to find it yet, that is our measurements are not accurate enough to hit that limit so we think there is more digits.
Very curious what this would look like if you use a gradient instead of 4 separate quadrants
Definitely needed
the issue is that you then lose the clear boundaries, and it's a little harder to see the fractal.. it still would probably be a pretty cool visualization though.
Im pretty sure someone already made that in a different video
It would just look less defined.
There is significance to the divisions
Fascinating. I wonder how much the stability patterns are dependent on the chosen integration method. That is, I'm wondering how much of what we're seeing is the behavior of the differential equation and how much of it is due to the chosen method and error estimate tolerances, or even floating point accuracy.
I assume the Shadowing Lemma applies to this system, but I could be wrong. The Shadowing Lemma says that every computed orbit stays within some small distance of a true orbit, i.e. small errors don't matter.
I was thinking the same thing. I would be interested to see one particular time step compared across multiple different integretion schemes, e.g. what would time step t=2.0 look like with delta-t of 0.1, 0.01, 0.001, etc.
@@qzamboni The thing here is that according to chaos theory, adding a small error to a step would lead to huge discrepancies down the line so not sure if the Shadowing Lemma would apply here. That said it's the first time I hear of this so I could be wrong.
@@qzamboniI'm pretty sure the double pendulum doesn't have any hyperbolic invariant sets
I had the exact same thought
In the bottom right corner: Are we seeing here the shape of chaos or the shape of floating point inaccuracy?
The question is does the Shadowing Lemma apply to this system.
Copying what someone else said, it's probably an island of stability, where nearby pendulums converge into the same shape rather than falling into chaos
What even is chaos?
ugh….this youtuber doesn’t check his comments. i guess we’ll never know for sure.
I had the same question. 😀I wonder how this animation would look different with 64 bit floating point math?
I am thinking about ways to "isolate" the central fractal from the surrounding noise.
Performing computation on the noisy regions is, in a sense, "wasted", since a noisy region never seems to further change from being just white noise.
In the Mandelbrot set, after each iteration, if the iterated value escapes the unit disk in the complex plane, we can safely discard it, as we know the value of this point will diverge into infinity. Usually corresponding starting point is colored white.
We could do a similar thing here, and find a range of values for angles and velocities that we know for certain is in the chaotic region, and discard the pendulums that enter that region, then color the corresponding positions white.
I believe this should eventually filter out ALL (or most) of the noise outside of the main fractal, even outside this filter region, since unstable positions will, by pure chance, "trickle into" the filter region, like water circling into a sink drain.
Doesn't it all eventually fall into white noise?
@@quantumsoul3495 I don't think so. The small island of stability at x=101°, y=+115° appears to be a basin of a stable attractor, where the pendulums get more and more similar by all approaching the same single limiting cyclic motion, instead of drifting apart in their behavior.
Even with limited numerical precision of the simulation, they will stay similar, as long as the corrective attraction of the attractor is greater than the floating point errors.
@Adam-zt4cn I disagree
i wonder if you could instead use the difference between neighboring pixels, and pixels which are surrounded by other values which are all sufficiently different could be discarded. in this way i think you could also adjust the desired noise amount?
idk how to express this idea rigorously or even really coherently lol
@@goji_crafter I think that won't quite work for what I intended, which is filtering out the "useless" white noise (useless in the sense that it has no structure).
Because even in the non-white-noise regions, you can still get quite dense changes in behavior. Such as the central oval, which starts as a fairly stable region and gradually fills up with higher frequency noise from the top-left and bottom-right.
But notice how it never quite becomes _pure_ white noise, unlike the outside regions. This tells us there's some subtle underlying stable behavior which prevents degradation into white noise.
Visually, you can see the boundary between the two different kinds of noise, which still shows the outline of the central oval.
It's very cool that some islands of stability appear in some places, like the one you are showing in the bottom right, it's super counterintuitive !
Oh, thank you.
Immediately shared with whoever is around.
It is interesting if the island of stability is the same with different pendulum angle"resolutions".
The 4K version of this video has an average bitrate of 126.6 Mbps! A challenge to Google's VP9 encoder.
"ow! my bitrate!"
I was going to drop a comment about the encoder not knowing what hit it when the sim starts up :)
Thanks for that info, Shota World
Always wanted to see a visualization like this, good work!
All the explanation at the start was excellent and very interesting and then I forgot what everything meant as soon as it started moving 😂
I was looking at this graph and I was wondering why it is not symmetrical around all 4 corners and the answer is because of gravity. And a double pendulum with no gravity is not chaotic. I was wondering how does the transition between no gravity and small and large number of gravity will look like
In this case, changing gravity is equivalent to just changing the timescale of the system, or its length. Want to see what this would look like with stronger gravity and the same lengths? Play it on ×2 speed
@Rotem_S incorrect. It is a non linear differential equation and it wouldn't simply double the speed.
I turned up the resolution of the video on my phone and the TH-cam app crashed
It would be cool to have another plot with a pendulum from a given pixel, so we can see the relative timescale and chaotic nature up close
Chaotic processes leave behind fractal structures :D
So chaos lives in non-integer dimensions, makes sense
@@Bombito_the unknowable becomes more knowable with each passing day
This is an incredible addition to the collective understanding about Chaos Theory. My god what a visual treat. Your previous one was badass, I’m blown the fuck away by this one. Very well done I really appreciate this kind of thing.
The bulbs of the Mandelbrot converge to periodic orbits. The center region seems to be where the angle is small enough that they swing in unison. I would be curios what the ratio of oscillation period is in the stable region.
Have to wonder how much of what we're watching is real and how much is an artifact of the floating-point representation being used. The nature of the system means that any finite rounding will tend to dominate the simulation over a relatively short period of time, which suggests that if infinite-precision calculations were possible, the picture that comes out would look quite different to what we're observing.
What is known about the magnitude of this effect in the context presented here? My math isn't strong enough to have a good intuition beyond the sense that rounding *might* have an outsized impact.
It should be possible to make an "accurate" picture without infinite-precision calculations if you control the error terms well enough. By which i mean that at any given time we can calculate a picture that is as close to the real one as we want (this might fail here because of the imprecisions). it might be very inefficient though.
@@Galinaceo0 The problem is that with a chaotic system (basically, any system where the result loops back into itself without diminishing returns), the lack of precision always dominates the output after a certain number of iterations. Any region of the image that doesn't converge to some stable value will always look very different after many iterations, even if the error is 1.0x10^-200 or something else very, very small.
The question that I can't answer is the amount that this instability would affect the "interesting" parts of the image highlighted in the animation. Would there still be an interestingly-shaped island in a "real" version of this with infinite precision, and would it have a similar shape? No idea but it is interesting to contemplate.
@@cthonianmessiah That can be fixed by recomputing the whole thing from the beginning with more and more precision each time you want to see more into the future. The problem however is, as i said, that this is extremely inefficient.
@@Galinaceo0 Yes, you hit a wall due to nonlinear behavior and no matter how much compute you throw at the simulation, the progress basically stops.
This is, for example, why weather forecasts stop at 7-10 days even as available compute resources increase exponentially.
@@cthonianmessiah Well in weather simulation it's actually impossible to measure everything we need to compute it to a given precision. for double pendulums, as long as we got enough memory and time it is always possible to compute.
very cool! would love to see a version of this where the points are colored by the difference between their angles and their neighbors
I wonder what it would look like if we considered starting positions with initial angular velocity as well. Sure, the space would be 4d, but if we only consider a slice of pendulums with the same total energy we would get a 3d shape. Interesting how trajectories might look there. (Since energy is preserved all of them would lie in the slice.)
I believe that if you change the coorsinate system we're gonna see some interesting things too. Like polar coordinates or something like that.
There's probably a system (maybe projecting on an object ?) where we recognise pattern since it's deterministic.
Maybe the part you zoomed onto is gonna be round or squared ?
You did a great job simulating that shit ahah
I always wondered what the wavy shapes were as you zoom in and out of pictures of a monitor screen
“Still think there’s nothing to chaos theory?”
-Half Life 1
Interesting how there is a definite boundary between the perturbative regime and the chaotic region where it's just noise.
You can see the two oscillatory modes in the perturbative region, each with its own frequency. The yellow/green one is the lowest frequency mode, and the red/blue one is so fast it looks purple.
The low frequency mode looks stable at relatively high amplitudes, which really is surprising to me. And then there is that tiny island where it also happens to be stable.
Loving it. Is there a mathematical something that describes the stable area?
Yes, very close to the center is the perturbative regime where you can study the pendulum quite easily. It turns out there are two modes, one of which oscillates between yellow and green in this color code, and the other oscillates between red and blue.
The surprising part is how sharp the boundary is between that quasi-stable regime and the chaotic region.
You do get a Newton fractal, because the four colors must always touch.
@@Ricocossa1they don't touch all in the start, why do they start touching?
adding damping could be cool. it would start off ordered, then become chaotic and then become ordered again right? ending with a single color
It would always end with both pendulums pointing down due to simulated gravity.
If each of these frames were to be put together into a 3D shape, would one get four three-dimensional fractals?
This video is Taco Bell for compression algorithms.
Thank you so much for sharing this! What should I learn to make an animation like this of my own?
I'd recommend looking through Henon & Heiles seminal paper, "The Applicability of the Third Integral of Motion".
You inspired me to try it myself with that last video and I got a pretty nice result by computing the average absolute central difference (dx_angle1, dy_angle1, dx_angle2, dy_angle2) / 4. Then the chaotic values are bright and the ordered ones are black. I'm also pretty sure it's fractal in nature so you could do a zoom in which might be cool.
Do you have an upload anywhere of the result of that?
It would be interesting to see this mapped on a hilbert curve
Gorgeous! Thank you! Your map deserves Mandelbrot-status
Since there is no friction, the pendulums will move forever. What exaxct state / time are you simulating?
Very cool. I suspect the "stable" part (however one would formalize that) is actually connected at every point in time, maybe even simply connected. Similarly to how the Mandelbrot set is actually simply connected.
You just gotta zoom in far enough.
What if the angle are relative to each other? Right now they are absolute degrees, but I wonder we get better resolving on the stable regions if the angles are different. I wonder how mass impacts this, also, it wouldn't surprise me if there is a Julia/Mandelbrot set embedded somewhere in here
My youtube aplication crashed after like 20 seconds into the simulation
i understand literally nothing in this video but its interesting
Me to
I found it to be really well explained.
@@NickCombs The beginning, sure
@@UniHorned Yes. Also, the chart shows that within certain ranges of starting positions where both pendulums start with similar angles, they maintain a periodic pattern. Outside of that, their motions become more chaotic.
Only comment on this vid I understood
Beautiful, how finals quadrants are the "mix" of colors from the opposites.
fine visualization 👍👍👍
It’s a fractal isn’t it? Start the zoom videos.
Very cool, great job mate
There is something about this video, that explains order within chaos, fractals, and other forms in which the Universe is revealing its inner working concretely
This would seem to me to be a two dimensional slice of a 4d object which would include the initial velocities of each pendulum, this slice being where the initial velocities are zero. Would it be possible to do this animation for a slice where the combined potential and kinetic energy of each pendulum is the same by adding whatever clockwise velocity is needed to one of the arms (the clockwise velocity can be negative), with the double pendulum that is pointing straight up at the start having zero added velocity?
woaaah nifty evolving fractal you've got there
would be interesting to see some of the pendulums with starting conditions in that small island of stability. also is the fractal shape accurate, or is it due to bias from the floating point error that accumulates
Seems a subset of the three body problem, with some stable paths and other unstable paths.
Fractal Lava that is slowly cooling
Very informative, and very long
Exactly what I need
Is there another "island" in the inverse cornoer as well? Due to the nature of the simulation it would be reasonable to assume so.
cant wait to write a shader for this
Is this the only island of stability?
I think I can see several more, look around the bottom near the -30 on the X axis for example
In case you're curious, the biggest one I found is close to the y-axis, between -30 and 0 degrees. Note, though, that every island has a mirrored version due to the nature of the double pendulum, so there are two of them.
I see two at about 30 degrees x axis, ~145 and ~160 y-axis, I guess that's the mirror of what IsZomg is saying.
I want to see a simulation of some pendula from that weird island of stability. We should see really stable motion even with small perturbations of initial conditions, but why does this happen, what are the staring angles that do this? I assume it's some form of resonance between the two
I wonder about the difference in angle values. Maybe a gradient?
Use inverse kinematics to initialize all the starting positions that would be really cool
Very curious what this would look like if you mask out the instability region (the noise) from the stabil region, and how does it crals inward
its so messed up that this is so different yet not the first time I've seen it
Did you use 9.81 for g? The fact that g varies so much (when 0.1% makes a big difference) makes this simulation extra difficult I think. Really cool project, good work!
Yep, thank you!
Hey! Does the pattern brakes later? What happens, when blue thin waves become less then 1 pendulum?
I would love to see the behaviour of the pendulum in the isolated stable region
I think even more pixels wouldn't be bad.
Wow! Really cool video keep up the good work 😄. I was wondering how were you able to simulate 4 million pendulums? Last time I tried I was only able to simulate 10k pendulums, is your code available on the web? I'm curious to look in to it.
Thank you! Will publish the code alongside the long-form video, which will be coming soon!
Interesting. What happens if you define the second axis as the second pendulum’s angle relative to the first one, rather than to the absolute reference frame?
Is it possible to define a fractal by the stability of the double pendulum? Will the "island of stability" approach a final shape or it will just keep shrinking?
What is happening in that island of stability there? Would be interesting to see what kind of movement it is, and why everyting aroud it is chaotic
Lol nevermind there was already a video about it on this channel
I'm curious how this fractal compare to the one for time to first flip. The barrier between mess and not mess doesn't appear to be the same as the one between flip and no flip (at least not around theta2=180 and theat1=0)
Have you/someone tried, whether it translates to reality? Eg., whether is that region around 100 deg stable in our univesrse?
Could you make this with 65536 colors/quadrants or even more? A friend asks.
youtube video compression’s worst nightmare
idea: add a thing that shows the precentage of all pendulums that are each color. example
66.3 % blue
21.2%yellow
41.8%green
11.1% red
it would be cool to see how it changes
This is with zero friction, correct?
Grooviest beats eva!
a battle between chaos and entropy, good VS evil type aah🗣
can u calculate the magnetic isteresis o a dominion?
wow it looks like the earth switching its magnet field polarity many times in a timelapse
I see that little stability spot and wonder if that is the spot in the multiverse we live in? Lol
It's symmetrical. You only have to simulate one half, and then rotate it 180 degrees around the 0,0 point.
it's not symmetrical, it would be if there were just two colors to represent positive and negative difference of angles but this is not that video
@@mateuszodrzywoek8658The point still holds. You could simulate half, rotate 180, and invert the colors
That's actually what they did
maybe they did that... You wouldn't know😉
Hm…it’s hard to tell in the video, but does the background become static or is the whole background mutating over time?
A imagem parece espelhada em relação à uma reta de 45 graus.
I feel something must be wrong with the human perception of math that a two part system would appear this chaotic.
Nope. The problem is that multiple factors of the system change over time, depending on the state of other factors of the system. As time goes on, the differences compound more and more, pushing the results further and further from its neighbours, until the motion's completely diverged. The exceptions, I assume, are the broad islands of stability where the various changing conditions balance out, and result in a much simpler motion
The most famous fractal - the Mandelbrot Set - is generated from this simple equation:
Z' = Z^2 + C
where Z and C are complex numbers [each having one 'real number' and one 'imaginary number']
the x-axis is real, y-axis is imaginary
Z is initialized at 0,0i and C is set to the x,y coordinates of each pixel. if the magnitude of Z' exceeds say 4.0 that pixel has 'escaped' and will go off to infinity and is [not part of/outside] the set. if Z' repeats or cycles between previous values, it is [part of/inside] the set. count how many iterations a pixel goes through before we know if it is inside or outside the set - that count becomes the pixel's color if it's outside. Pixels inside are typically colored black. the actual color chosen for each value of the iteration count is up to the user.
if you aren't familiar with imaginary numbers or complex numbers here's the short explanation.
i = the square root of (negative one)
i * i = -1
real * real = real
real * imaginary = imaginary
imaginary * imaginary = negative real
Z=(0,0i) C=(2,3i)
Z' = (0,0i)*(0,0i) + (2,3i)
Z'' = (2,3i)*(2,3i) + (2,3i) ...
2*2 - 3*3 //real part of Z''' is -5
2*3i + 3i*2 //imaginary part is 12i
the resulting shape has been called the thumbprint of god. Unless you are following a previously explored set of coordinates given to you, if you zoom in for more than a couple of minutes, you are looking at a part of the set that no one has seen before.
oops, forgot to add C for Z''' should be (-3,15i)
It looks like the side of my mom's car after I ate a package of sweet-tarts on an empty stomach.
I’m in love.
3:56 purple
All glory to the hypnotoad.
TH-cam compression loves this video lol.
Tbh i wonder what this would look like in 3 dimensions.
looks just like a meteorological wind chart 🤔
TH-cam compression final boss
use a polar continuous color scheme
Mandelbrot:
gravity but no friction
YT compression is not liking this
wow
Woah!
close, but no sea gar
Moire effect...
very cool :)
I don't know what this is. Have I completed TH-cam?
it only got started : now you'll want to understand the deep meaning of that mathematical wonder.
😂 I was able to follow until whatever the chaos was.. could anyone explain me what was going on??
a physical pendulum has real valued angles, so they can't be enumerated. this isn't all of them.
Or do they? Shouldn't they be quantized?
@tomaspecl1082 There isn't any should. Pendulums are physical objects, and physical objects are real valued, whether we think they should be or not.
@@mrosskne how can we be sure that nature uses real numbers? We can ever measure with so much precision, quantum physics implies that two non comutating variables cannot be measured both with high precision. Real numbers are just an idealization perhaps. In case everything in nature is quantized then even a pendulum will be quantized.
Almost all numbers are irrational. That, by itself, is proof.
@@mrosskne no that is not a proof that nature has real number values. Maybe nature has only finite decimals of such size that we were not able to find it yet, that is our measurements are not accurate enough to hit that limit so we think there is more digits.
i only clicked this video because ive never seen this "auto dub" thing and it just keeps saying "heat" in my language???????
Holy........ 😲