I find the "mistake" to be much more of a fascinating result than the proper synchronization result. I love how it just goes to the maximally divergent phases.
It kind of makes sense given the model though, rather than each syncing towards each other they're instead syncing away from each other which means the only point they can be stable is if they're equidistant from each other and in fact they move away according to the delta. Interestingly if you set this up so they're overlapping at the start the difference is 0 and they stay balanced, but the instant you add even a tiny difference they move away proportional to the difference between their neighbours which can give some weird results with high K factors(constantly oscillating between two unstable states for example).
As swap of phases produces a result of opposite sign. This is equivalent to setting K to a negative value (because of how sin works). It will force the phases to be spread as much as possible. With 3 metronomes they will separate by 120 deg. But with 4 metronomes, you will get separation by 90 deg, but I suspect there will be a small oscilations around 90 deg. So if for visualization you shift them by 90, 180, 270, or visualize the phase differences they might be never going to zero, depending on the value of K (dampening). It is also sensitive to the integration method. Euler method is known to not be stable for some steps, so you need to be careful. Better method, especially designed for the 'stiff' problems (this is a mathematical term from the theory of ordinary differential equations and ODE solvers), would make it go away, one of the methods to do that is using implicit methods.
One (possibly too) simple yet convincing explanation of what's happening is that the synchronization effect is essentially working to minimize the phase differential. As movax20h put it, subtracting the other way around is the same as using a negative coefficient, so the effect is working in reverse. What's the opposite of minimum phase differential? Maximum phase differential!
@@gorillaau I wouldn't say he forgets to edit it out, rather than deliberately leaving it in. To show that science doesn't always go right and you can get neat results from mistakes, too.
@@rolfs2165 Good point. Also important in maths to be recognise that you have made a mistake and to back track to find the error. Blindly accepting the answer on the calculator or spreadsheet as being correct will lead to trouble down the track.
9:20 I imagine the (former) student watching this video now, seeing that his paper (from 2005) is getting mentioned and even receives some praise from Matt Parker...
When we were learning Excel on a mandatory computer skills course, I did the assigned exercises, and then passed time seeing what it's capable of. Made a formula to extract the largest common substring of two text cells, and a truly horrible formula to draw Langton's ant. Still prefer Perl for readability.
The Kuramoto-Parker coupling model you first did is the equivalent of a negative K, which is an inverse feedback, which in turn ends up pushing the phases apart instead of pulling them together. It's easy to model it for N=2 but not sure there is a generic solution for it for arbitrary N. I can imagine Bryan's (article author) view on this video. "How many references have you got to the article?" "Just two. But in 2019 I suddenly got ten thousand likes!"
Aren't half the metronomes in the at 2:00 locking to a 180 phase shift? Looks like they're all synced, but some of them are backward. That would strongly suggest to me that, whatever the mechanism, there's a very simple physical interpretation of that, somehow.
Reversing the order of subtraction of the phases exactly corresponds to the Kuramoto model with a negative summation. This has the effect of driving the phase of the oscillators away from the mean phase of the system. This can be seen by introduction of the 'mean-field variables' which decouples the oscillators so they satisfy the Adler equation (with some scaling). As for a physical model, nothing comes to mind as this also corresponds to negative coupling strength (K
Daniel Röder They might never have developed the powerful tools they ended up with, because there would be no motivation to simplify their calculations.
@@odenpetersen6028 True, and their brains wouldn't be that trained in doing maths when a computer just does it for you. Let's just imagine they have already done their fair part of discoveries and now you just give them this tool for a while to show even more stuff we haven't discovered yet.
As someone who occasionally does some recreational maths on a computer (mostly in response to one of these youtube videos), simply using a computer is fun but it doesn't get you far. Having a deeper understanding of the maths makes these "mass computation" tools exponentially better.
There's a lot more to maths than algebra. Calculating 32455.23 / 0.9854 (or doing a million "easy" operations) by hand does not make you a better mathematician, it just takes up time that you could be using to think about something more relevant. Computers don't "do maths for you" any more than drills and arc welders "do bridges for you".
This doesn’t capture the fact that in reality sometimes the metronomes end up 180 degrees out of phase. It seems in your excel model they always end up perfectly in phase. Why is that?
It might have shown it with different start conditions, if not its wrong. The phase relations will be zero or pi radians, as you say. Equiprobable, I believe.
In the Kuramoto model, the anti-phase configuration is an unstable equilibrium, i.e., you can only stay there over time by starting there exactly. I don't know if this is a deficiency in the model, or if in reality the anti-phase configuration is unstable. EDIT: My guess is that in reality the anti-phase configuration is stable, but the attractor basin around the in-phase configuration is much, much larger.
@@trogdorstrngbd I believe it's a deficiency in the model as the out of phase conditions are very common in physical models and the likelihood of starting one or more oscillators exactly at the counterpoint should be a rare occurrence if Kuramoto is correct. It would only take a small correcting factor to modify the model and make the 180° situation stable as well.
@@khaitomretro Indeed. Watching the video, Matt says two of the five our out of phase, however one of those two was "M0", so actually three of the five metronomes went to the antiphase.
In chapter 1 of the paper is the description of why Christian Huygens considered this phenomenon in the first place. He was sick in bed and noticed his two pendulum clocks on the same wall were always anti-synchronized. When he would restart a clock with a random phase, the clocks would always return to being anti-synchronized over time. So the real world does allow this solution where the model does not.
Place some spheres on the table and a round dish on top of them. Set the metronomers placed 120 degrees apart, all pointing outwards. Like a clock, at 12, 4 and 8 o'clock. So let's see if they behave like your first example!
It would work if the metronome's travel was tangential to the centre of this circle. Like this I think you atain maximum divergent phases. If they were aligned parallel to the circles circumference it would act similarly to the linear experiment as the forces (rotational force pivoting around the COG) acting on each individual metronome are in line with the travel of the metronome mass
Dzhoy Zuckerman If you focus on just the pendulum it looks the same whether it's pointing inward or outward so it should work the same and synchronize. However if you consider turning it around to be swapping negatives with positives and vice versa then things pointing inward should be out of synch with things pointing outward by exactly pi. But they'll still be in sync from the perspective of whether they're turning the lazy Susan clockwise or counterclockwise at any moment
BTW Matt, it looks like the model that you accidentally first used would be called the "Repulsive Kuramoto Model" and it's apparently interesting enough to get published in PRL!: biocircuits.ucsd.edu/lev/papers/repulsive.pdf They define the model in the form you show, but with a minus before the sum of sin(theta)'s. But it has the same effect as reversing the indices like you did, since sin(x) is an odd function so -sin(x)=sin(-x). EDIT: And for those interested in physical systems that this could model, here you go -- "It may serve as a paradigm for many biological networks in which different elements compete against each other. The best known example of such networks are neuronal ensembles with inhibitor coupling [5]. It is well known that in such systems, nonuniform synchronized oscillation patterns may emerge."
I love how you investigated the Kuramoto model using just Excel. Even though you said Excel is way under-powered for this sort of analysis, it highlights nevertheless just how powerful Excel is for prototyping, investigating, and even full models. And I was surprised at just how simple the Kuramoto model is as I was expecting some complicated solution to a PDE. Anyway, I have done this with innumerable kinds of models. And when the clunkiness of a spreadsheet is not adequate, I will quickly jump into VBA and bang out a custom function. One could easily create a custom function in VBA for the Kuramoto model that would enable the input of more than three coupled oscillators and output the results as an array. Furthermore, it would be interesting to see the Kuramoto model extended to more dimensions because in your example with the five metronomes, they weren't perfectly in sync which I hypothesize was due to the metronomes not being lined up perfectly.
Matt, you've outdone yourself yet again! So many videos report the phenomenon, but no-one I'm aware of works out the physics. Your channel is undisputedly one of TH-cam's finest. Do keep up the awesome work! Cheers, mate!
Matt, omitting the ω term left out a very interesting feature. In this case, if the platform the metronomes rested on was acted upon by a constant force, that force could be represented as ω. The interesting part is that this constant force would cause the metronomes to go in and out of phase at regular intervals, and if ω/K is rational, then the system is considered phase-locked. Phase-locking is incredibly fascinating and is fundamental to determining the stability of chaotic systems.
I'm so glad the sound of those metronomes irritates you. If it didn't we would have had it through the whole video so *thank you so much for stopping them* every time after only a short while. Some people can really learn from you that once you have made your point you don't have to "hammer" it in. This right there is why I love watching your videos!
The antikuramoto would be useful in sound engineering on any chorus, flanger or phaser effect. You could implement it with a kuramoto for another interesting effect.
Unfortunately it looked like it pushed all the waves to be completely out of phase with each other so the resultant sum of the waves would be 0. Not ideal for an audio effect 😄
@@tomholton235 Choruses work by detuning and delaying and with stereo choruses across multiple channels, so they can obviously work if you use triggers to switch between. Similarly, they can be used in DAW stations' metronomes (as was shown by Parker here) when you need to offset your channels accordingly. Again, using triggers you switch between the algorithms to get the correct beat offsets. This is especially useful if you are trying to implement swing to your music.
Really cool video. This may be obvious to many, but I had to think about it for a second: The equation given for the model is an "equation of motion" (a differential equation that defines d(theta)/dt), but doesn't give you theta(t) outright. So Matt is just doing Euler's method to solve theta(t) and plug that into x(t). Euler's method is the simplest way of numerically solving an ODE when you know theta(0) and d(theta)/dt, by approximating theta(t+dt)=theta(t) + dt*(d(theta)/dt). Of course, there is no factor of dt used here when solving theta(t), but I guess all the scaling here is basically arbitrary.
Enjoying the Christmas lectures so far. Seeing my favourite youtubers Hannah Fry, Matt Parker and Tom Scott on the same TV show made my Xmas, and makes me realise we've reached the point now where on-demand culture now dictates what is on traditional TV, rather than the other way around. All three of these people should have their own shows on TV anyway!
Seeing Tom Scott walk out in his red t-shirt was such a good moment, and Matt bringing out Menace again, not to mention his sock/phone sorter (a Parker sorter if you will, lol) :D When I found out Hannah Fry was presenting them I was more than happy, but seeing the guest hosts? It's made the last couple of days so good. Can't wait for the third one :D
Here is my explanation for it! The out-of-phase model is a pretty neat example of gradient descent instead of gradient ascent. The ordinary Kuramoto model tries to put everything in phase, i.e. maximise the energy in the system, which is why there is a plus sign in front of the term with the difference of the phases. This is like gradient ascent. Physically this means the plank will be moving with maximum amplitude possible. When you flip the phases in the Kuramoto equation, because of the property of the sine function, you get an overall negative sign, which is like gradient descent. Essentially, the sinusoidal functions get "balanced," to steal some electrical engineering lingo, meaning the sum of them is 0. There is no energy loss in the system. Physically this means the plank would not be moving at all, but there is still information being conveyed between the metronomes.
Interesting. Would there be an existing model which more accurately reflects this? EDIT: Haven't found the answer as of yet, but it seems that Kuramoto also found a "chimera state" where some oscillators desync and act chaotically, while others stay in sync. I don't believe there is an explanation or a full model describing and predicting all generalised possibilities.
In July 1980, I took a riverboat from just south of Singapore into the heart of Sumatra. We entered the Siak River just around sunset and all along the riverbank, about every 10th tree was covered in fireflies - and they were all blinking in sync. Quite an astounding sight - and no noise either.
Physical model: A number of N coupled oscillators is represented by N coupled differential equations. This system of equations always has N so-called "normal mode" solutions. All other states (phase differences) can be represented as a superposition of the normal modes. In the case of 2 metronomes, the normal modes would be: a) both in phase and b) 180° phase difference. For 3 metronomes, the normal modes you would have a) all 3 in phase, b) 2 metronomes in phase, the third 180° out of phase, c) 3 metronomes with 120° phase difference in respect to each other. The latter is the one settles in when the "order is mixed up" (@17:25)
I absolutely love this video, the way he shows his process. Its so much fun watching him work through each step and even discover something new! He doesn't have all the answers, he's here to find them and ask questions!
I have a theory of why the swap gave you the opposite result. The Kuramoto Model gives yiu the direction of movement of the angle Theta that gets it in sync.The sine wave is an Odd function, which means a change in the argument’s sign shows up as a change of sign in the output.If Kuramoto gave you the direction towards convergence, changing the argument’s sign changed the direction’s sign, and therefore that function gave us the direction to stay Out Of Convergence :3
The inverted version is actually really useful :) we have a game where a bunch of things happen at random times but at a constant frequency after that and using the parker-kurumoto model, they could be spaced out over time so that the load on the server is constant and not bursty
Something you didn't touch on: The Kuramoto model allows for each oscillator to have their own increment/frequency/omega parameters. That's quite useful for real-world models where there is going to be some expected variance in the frequency of oscillators. For example, your metronomes will likely move in/out of phase on their own over long periods of time, despite your attempts to set them identically.
I completely agree with you. I tried making the spreadsheet with each independent omega parameters. It seems to work. 1drv.ms/x/s!AuCyus6wr0yslisdlCyjwfW3P3vf?e=wpuOWF
Having the phases equally spaced would be amazing for building a walker. Maybe somehow measure the position of the occilators and couple it to the position of servos attached to legs. The leg motion could be the coupling force.
The evenly distributed phases cancel each other out, leaving out a null wave if you summed them. It's like the 3 phase system in AC, all of them some degrees apart from each other so they sum to 0.
I saw this in my sub feed. It said it had been posted "14 seconds ago." I found that a bit odd. It is the earliest I've been to a newly posted video (since I clicked on it right away). It currently has no views and 3 likes, which is also odd. It also has five comments... but none have appeared on the screen. Another odd thing. Well, Happy New Year!
Views are only counted if you watch the video far enough (people like before watching) and they need to be more accurate in the end (because the money YT has to pay to the creators depends on it), so they make sure it's done well. That doesn't mean it's always 100% up-to-date, but that they take the time to verify each view to be a genuine view and not e.g. generated by a bot. However, views, being the most common events on YT, are cached a lot more than likes or comments, which are (relatively) rare occurances in comparison.
The first minutes of a TH-cam video being public often have mismatched numbers like that. I think it has something to do with different TH-cam datacenters taking time to talk to each other and get a final version of the various counts.
I would preorder the US release but i couldn't wait and got the UK one. Had to go through the whole thing and cross off all the extra U's but totally worth it.
I pre-ordered the ebook and the waiting is killing me. I'm US but I'm fluent in UK and i don't mind the extra u's and plural "maths", but none of the ebook platforms have the original version available. I'm sure it's a publisher thing. Still, books shouldn't be region locked.
Hi, Matt! I’m not much of a mathematician in any way, but I do find mathematics fascinating, especially as a musician and have always really enjoyed your videos! This particular one has gained my interest because of my familiarity to a similar topic in music. You should check out Steve Reich, a minimalist composer who uses the compositional technique of “phasing” in some of his music. Having played one of these pieces, “Drumming,” I became very familiar with the concept and was excited to see it in mathematics. In music, two (or more) players play the same rhythm, whether in unison or offset, and one player stays steady in tempo while another gradually speeds up, going out of phase, and through that eventually lines up on a different partial of the rhythm to create different interlocking patterns using the same rhythm through this process of “phasing.” This also often achieved using electronics and was how it was discovered, and was originally thought to be almost impossible for a single musician to do on their own. It’s a really awesome thing to listen to and creates an extremely satisfying moment when all the voices eventually sync back up in unison. You can find this in other works of his like “It’s Gonna Rain,” “Violin Phase,” “Piano Phase,” and more. He also takes this gradual process and makes it immediate in a process called “phrase shifting.” This takes out the gradual speeding up and at a certain moment a voice changes where exactly the rhythm starts like in his piece “Clapping Music.” You should definitely check out his music if you found this metronome phase cool!
You could have just made K negative. Also would be interesting to see how much you could perturb the angular frequencies before the synchronisation stops working
Since sin(-x)=-sin(x), you get dphi/dt being the negative of the normal dphi/dt. So what previously was an unstable equilibrium turns into a stable equilibrium and vice versa. And equally spaced phases being an unstable equilibrium for the Kuramoto model makes sense.
Would this also work in 3 Dimensons? Like pendulums swinging in different directions while hanging on a board that's also hanging? Or would this just lead to a Chaotic pendulum ?
The inverse phase one makes total sense, if you think about it. In the correct model the "energy" of the system is lowest if the system has the lowest total phase shift (all oscillators are in sync). If you inverse the coupling (sin(x) is an odd function, so sin(-x) = -sin(x)), the energy of the system gets lower, if the phase shift gets larger. Therefore the system will tend towards maximizing the total phase shift. Sidenote: Total phase shift does not mean sum of all phase shifts directly, but the sum of the sin-s of all phase shifts.
What happens if you, instead of putting the oscillators on a "linear coupling board", put them on a "circular coupling board", like on the tips of a huge fidget spinner/ball bearing?
As the metronome ( placed correctly) will jerk the spinner clockwise and anticlockwise, instead of left-right, it will be the same as in linear coupling, only in a circle.
I'm not (necessarily) a math kind of person but I am a sort of "intuitive engineer/physics observer". As such, I think I understand that the movable platform is the (ϴj - ϴi) part of the equation. If put the right way around, the board is finding the difference between the two movements (result of the combined but differently timed momentum of the pendulums) and over time canceling that difference with it's own opposing movements. Kind of like using bent knees to cancel your own bouncing on a trampoline. If placed the wrong way around (not sure but I think that's only a mathematical option) the movement of the board still finds the difference but applies it in an additive way - like working a paddle ball where, when done properly, the motion of the paddle is directly opposite to the motion of the ball except it amplifies the speed and impact of the ball. The elastic band is finding the difference of movement and adding it to the motion of the ball and paddle maintaining a 180° out of phase movement. So ... right way around, the platform moderates or filters the movement(s) - wrong way around the platform controls the movement(s). That's my spit-ball evaluation.
That is a nice generalized solution. As an engineer, I like a more specific solution that is based on physical properties. This would be more fun to build as a dynamic analysis in an FEM model.
I decided to learn more (beyond just GCSE) about maths and computing at 33, I'm currently taking a proper run at learning excel fluently as a stepping stone to learn more elsewhere - thanks for giving us a fun brief/scenario to explore the tools!
The perfectly out-of-sync sine waves at 13:15 (aka. "The Parker Sync") reflect metronomes that never will go in sync with each other. Another physical analogy is the blades of a propeller or the glass tubes of a centrifuge. Here's why: Start out with two (or more) metronomes with the same frequency and a random phase shift, just like at the start of the video. If you add up the sine waves of their positions, you will get another sine wave with the same frequency, and the average of their phase shifts as its own phase shift. This is actually the movement of the table, and is also the phase shift all the metronomes will end up with once they're in sync. However, if you start with phase shifts that are perfectly out-of-sync ("The Parker Sync"), once you add up all the sine waves they will cancel out and give you 0. The table will not move and the metronomes will not synchronise. Coincidentally, these are the phase shifts (or angles) you would use for the blades of a propeller or the glass tubes in a centrifuge (a Numberphile video on that topic here: th-cam.com/video/7DHE8RnsCQ8/w-d-xo.html ). Using perfectly out-of-sync propellers will prevent them from trying to get in phase with each other and prevent the whole wind turbine from wobbling. The other case where metronomes won't synchronise is when they're in perfectly opposite phases from the table (this is what happens at 2:10). The table will try to synchronise them both by speeding them up and slowing them down, and the effect will cancel out.
Made a simple program where you can change the frequencies and find out: editor.p5js.org/jrutty/sketches/zOvU7Ehrg Short answer is that if the frequencies are multiples of one another you can get a kind of synchronization, but if they're not then the synchronization effect breaks almost immediately.
I once made a sort of spinning disc strobe with one red and one green led on the disc, each driven by its own oscillator. Spinning the disc at constant speed I could create the illusion of 3 red dots turning clockwise while 4 green ones turned counter clockwise. To my surprise there was a "magnetic" effect such that when the dots were moving slowly in relation to one another they locked. Corresponding, I guess, to f1 and f2 (the oscillators driving the leds) nearly having a small integer multiple, 12. The oscillators must have been weakly coupled by way of sharing the same power supply, both were on the same chip.
If you multiply the difference of the phases in the Sin by two (=C5+$E$1*(SIN(2*(E5-C5))+SIN(2*(G5-C5)))) in C& for example you can get the effect you see with the real metronomes where their phase differs by exactly pi or get in sync, depending on your starting phases
A little disappointed that you went for “tweak freq” rather than modelling the forces directly to see the effect come out of the physics.... Also, modelling if metronome freqs are close bu slightly differ
When the waves are perfectly out of phase, they'll have destructive interference instead of constructive interference, which would then cancel out the resonance and make this "bridge" safer. if there is a physical interpretation of that worksheet, that could be a great safety feature. I also noticed that both times you had all five metronomes, three of the metronomes were exactly in phase with each other and the other two were exactly out of phase of the rest but in phase with each other. The second time, when you let them go longer, they would get into a cycle of going slightly in and out of this pattern. When they were spot on, the box stopped moving (to the eye) which then quit linking them altogether, so they would go off phase, starting the box moving again, which then caused the linkage that set them back toward perfect phase alignment. But the motion of the box was always pretty minimal. This seems really significant and therefore makes me think that I am probably not the first to notice. I bet there's a whole other video to be made on this aspect. I should subscribe!
So basically you applied a function from "the ether of science" to an excel sheet. I'd have found it more interesting to see a (simplified) construction of how the coupling propagates using recursion (from the line above) instead of a formula that just gives you the results.
Zolbat At the risk of sounding a bit ungrateful, I had the same thought. Maybe expanding a bit on the model itself. At this s point, it became a needless spreadsheet exhibition.
This was cathartic. I did my final project in high school years ago on this subject. I used the kuramoto model as well in my paper. I reference that paper you used as well! The sound of those metronome brought back some memories of my own testing.
just for my curiosity: WHAT happens if the metronomes aren't synchronized perfectly? how would that graph look like? is it exponentially more difficult to "code" in excel?
If you're asking if the system would still synchronise if each oscillator had a different period, the answer is yes. This is accounted for in the Kuramoto model with the omega_i term, the natural frequency of the oscillator.
When you say "aren't synchronized perfectly" do you mean they have different resonant frequencies? If so, they try to synchronize but won't stay that way for long. If the frequencies are even multiples of each other then they can eventually reach a sort of synchronization, but it takes much longer. If you want to play around with it I made a very rudimentary program you can play around with: editor.p5js.org/jrutty/sketches/zOvU7Ehrg
An off the cuff hypothesis: the "mistake" is doing the opposite so instead of all parts converging to an in phase state, they converge to be in perfect anti phase. looking at the metronome example, the equation as it should be converges the system to a maximum deflection where as the mistake would converge the system to zero deflection. ie: for two metronomes (pendulums), maximum deflection means both swing at the same frequency in the same direction but for zero deflection, they move in opposite directions.
![[TH] RNT vs SPG - VCT Ascension Pacific - Day 1](http://i.ytimg.com/vi/RUTkf8Detj8/mqdefault.jpg)
6:23 "that's pi, so 1 is a third of the way along"
Engineers: *smile*
Hehehe yeeeeah. We did boys. Accuracy is no more.
@@AeroCraftAviation
mrs obama I've done it, I've stopped accuracy
A parker third?
Well it's 80% accurate.
@@tasherratt ... 80?
more like 95%...
I find the "mistake" to be much more of a fascinating result than the proper synchronization result. I love how it just goes to the maximally divergent phases.
It kind of makes sense given the model though, rather than each syncing towards each other they're instead syncing away from each other which means the only point they can be stable is if they're equidistant from each other and in fact they move away according to the delta. Interestingly if you set this up so they're overlapping at the start the difference is 0 and they stay balanced, but the instant you add even a tiny difference they move away proportional to the difference between their neighbours which can give some weird results with high K factors(constantly oscillating between two unstable states for example).
As swap of phases produces a result of opposite sign. This is equivalent to setting K to a negative value (because of how sin works). It will force the phases to be spread as much as possible. With 3 metronomes they will separate by 120 deg. But with 4 metronomes, you will get separation by 90 deg, but I suspect there will be a small oscilations around 90 deg. So if for visualization you shift them by 90, 180, 270, or visualize the phase differences they might be never going to zero, depending on the value of K (dampening). It is also sensitive to the integration method. Euler method is known to not be stable for some steps, so you need to be careful. Better method, especially designed for the 'stiff' problems (this is a mathematical term from the theory of ordinary differential equations and ODE solvers), would make it go away, one of the methods to do that is using implicit methods.
Parker synchronization...
One (possibly too) simple yet convincing explanation of what's happening is that the synchronization effect is essentially working to minimize the phase differential. As movax20h put it, subtracting the other way around is the same as using a negative coefficient, so the effect is working in reverse. What's the opposite of minimum phase differential? Maximum phase differential!
Mathematical model for a control system to maintain proper phasing on a 3 phase power system?
The botched function of that first spreadsheet should be called the Parker-Kuramoto Model. :)
Why does this keep happening
@@skyjoe55 Matt attempts to wing it, fails and forgets to edit the goofout of the video.
All the same I appreciate these videos. Thanks Matt.
Almost. Almost Parker-Kuramoto Mode :)
@@gorillaau I wouldn't say he forgets to edit it out, rather than deliberately leaving it in. To show that science doesn't always go right and you can get neat results from mistakes, too.
@@rolfs2165 Good point. Also important in maths to be recognise that you have made a mistake and to back track to find the error. Blindly accepting the answer on the calculator or spreadsheet as being correct will lead to trouble down the track.
9:20 I imagine the (former) student watching this video now, seeing that his paper (from 2005) is getting mentioned and even receives some praise from Matt Parker...
"What's your favorite programming language?"
Normal people: C, C++, Python etc
Matt Parker: Excel
Tom Wildenhain: Power Point
When we were learning Excel on a mandatory computer skills course, I did the assigned exercises, and then passed time seeing what it's capable of. Made a formula to extract the largest common substring of two text cells, and a truly horrible formula to draw Langton's ant.
Still prefer Perl for readability.
That’s the guy who made the (PP™︎TM™︎)™︎, right?
The is only one true programming language and that's IBM Mainframe Assembler. If you can't do it with that then it's not worth doing.
Kyle Hill: Magic: The Gathering
@@Kinkajou1015 Final Fantasy XII
I love the file name of the Excel document is "n-sync" 😂
I don't get it
@@coolmonkey619 It's tearin' up Justin Timberlake's heart that you didn't get it.
@@MrDannyDetail But when the oscillation periods are apart, the others feel it, too (because they're coupled).
I was just coming to check to see if anyone else caught it
That's a reference you don't hear everyday.
The Kuramoto-Parker coupling model you first did is the equivalent of a negative K, which is an inverse feedback, which in turn ends up pushing the phases apart instead of pulling them together. It's easy to model it for N=2 but not sure there is a generic solution for it for arbitrary N.
I can imagine Bryan's (article author) view on this video.
"How many references have you got to the article?"
"Just two. But in 2019 I suddenly got ten thousand likes!"
The "out of phase" one looks like a 3-phase AC transformer where each of the 3 phases is 120 deg out of phase with each other.
And the sum at any point is equal to zero.
I immediately thought about that too :)
Like a phasor?
I thought of three-phase AC too.
It is the same.
"we'll get back to that"
matt: never gets back to that
Mathematically perfectly out-of-sync? The Parker Sync.
*The Parker Sink
It's exactly how tri-phase electrical current works :3
(far from an original observation I might add)
The columns on your spreadsheet in the video are labelled wave "A", "B, and "B".
@@dorothymiles7097 Parker labelling.
Aren't half the metronomes in the at 2:00 locking to a 180 phase shift? Looks like they're all synced, but some of them are backward. That would strongly suggest to me that, whatever the mechanism, there's a very simple physical interpretation of that, somehow.
man makes video about metronomes - forgets he has 0 tolerance for clicking.
Reversing the order of subtraction of the phases exactly corresponds to the Kuramoto model with a negative summation. This has the effect of driving the phase of the oscillators away from the mean phase of the system. This can be seen by introduction of the 'mean-field variables' which decouples the oscillators so they satisfy the Adler equation (with some scaling). As for a physical model, nothing comes to mind as this also corresponds to negative coupling strength (K
Quarks.
Also Ising model of antiferromagnetic interaction.
Came down here to say reversing the subtraction is equivalent to negating K, found this comments complete with explanation
Gradient ascent vs. descent, see my comment for physical description :)
How about reversing time? Like theoretically.
Matt: US edition of Humble Pi is coming out in January!
Me: *sitting at home in the US holding my UK edition... "Was I supposed to wait?"
The US version is slightly shorter, less unnecessary U's (honour) and S's (maths).
@@sdspivey Celsius has far fewer letters than Fahrenheit, you know. Just sayin'.
@@Anvilshock But who spells it out? I rarely see anything other than F, C, or K.
You're much better off with the UK version because the English is actually correct
Now get Steve Mould to make a spreadsheet out of metronomes.
i fell asleep in a metronome factory, lost all sense of rhythm.
So you're telling me you aint got rhythm?
@@doublespoonco No i ain't got rhythm.
P&F.
@@robmckennie4203 but look at what you're doing right there
@@rileysteidel7084 need to find out if Matt can send me out a stamp and a book
Imagine the mathematicans back in the old days had some powerful tool like excel to play around with numbers and formulas.
Daniel Röder They might never have developed the powerful tools they ended up with, because there would be no motivation to simplify their calculations.
@@odenpetersen6028 True, and their brains wouldn't be that trained in doing maths when a computer just does it for you. Let's just imagine they have already done their fair part of discoveries and now you just give them this tool for a while to show even more stuff we haven't discovered yet.
Using nothing but a protractor, a compass, and excel prove that you can bisect any angle.
As someone who occasionally does some recreational maths on a computer (mostly in response to one of these youtube videos), simply using a computer is fun but it doesn't get you far. Having a deeper understanding of the maths makes these "mass computation" tools exponentially better.
There's a lot more to maths than algebra. Calculating 32455.23 / 0.9854 (or doing a million "easy" operations) by hand does not make you a better mathematician, it just takes up time that you could be using to think about something more relevant. Computers don't "do maths for you" any more than drills and arc welders "do bridges for you".
“Evenly distributed phases” might be the better description.
I'm wondering what it would look like with more oscillators…
The sign-switched graph reminded me of a juggler, starting off slightly out of phase before everything coming into perfectly synched balance.
As I listen to this on my noise canceling headphones, I wonder more and more about the ‘wrong way around’ maths, and their practical application.
This doesn’t capture the fact that in reality sometimes the metronomes end up 180 degrees out of phase. It seems in your excel model they always end up perfectly in phase. Why is that?
It might have shown it with different start conditions, if not its wrong. The phase relations will be zero or pi radians, as you say. Equiprobable, I believe.
In the Kuramoto model, the anti-phase configuration is an unstable equilibrium, i.e., you can only stay there over time by starting there exactly. I don't know if this is a deficiency in the model, or if in reality the anti-phase configuration is unstable.
EDIT: My guess is that in reality the anti-phase configuration is stable, but the attractor basin around the in-phase configuration is much, much larger.
@@trogdorstrngbd I believe it's a deficiency in the model as the out of phase conditions are very common in physical models and the likelihood of starting one or more oscillators exactly at the counterpoint should be a rare occurrence if Kuramoto is correct. It would only take a small correcting factor to modify the model and make the 180° situation stable as well.
@@khaitomretro Indeed. Watching the video, Matt says two of the five our out of phase, however one of those two was "M0", so actually three of the five metronomes went to the antiphase.
In chapter 1 of the paper is the description of why Christian Huygens considered this phenomenon in the first place. He was sick in bed and noticed his two pendulum clocks on the same wall were always anti-synchronized. When he would restart a clock with a random phase, the clocks would always return to being anti-synchronized over time. So the real world does allow this solution where the model does not.
Place some spheres on the table and a round dish on top of them.
Set the metronomers placed 120 degrees apart, all pointing outwards.
Like a clock, at 12, 4 and 8 o'clock.
So let's see if they behave like your first example!
Or use a lazy susan
It would work if the metronome's travel was tangential to the centre of this circle. Like this I think you atain maximum divergent phases. If they were aligned parallel to the circles circumference it would act similarly to the linear experiment as the forces (rotational force pivoting around the COG) acting on each individual metronome are in line with the travel of the metronome mass
That sound like it should work to put them all in sync
What would happen if you pointed them inwards? The other spread sheet?
Dzhoy Zuckerman
If you focus on just the pendulum it looks the same whether it's pointing inward or outward so it should work the same and synchronize.
However if you consider turning it around to be swapping negatives with positives and vice versa then things pointing inward should be out of synch with things pointing outward by exactly pi. But they'll still be in sync from the perspective of whether they're turning the lazy Susan clockwise or counterclockwise at any moment
The columns on your spreadsheet in the video are labelled wave "A", "B, and "B".
Parker-labels
I suspect he got too excited about the third graph to worry about something as trivial as labeling.
the correct fix is to change the first one to "B".
BTW Matt, it looks like the model that you accidentally first used would be called the "Repulsive Kuramoto Model" and it's apparently interesting enough to get published in PRL!: biocircuits.ucsd.edu/lev/papers/repulsive.pdf
They define the model in the form you show, but with a minus before the sum of sin(theta)'s. But it has the same effect as reversing the indices like you did, since sin(x) is an odd function so -sin(x)=sin(-x).
EDIT: And for those interested in physical systems that this could model, here you go -- "It may serve as a paradigm for many biological networks in which different elements compete against each other. The best known example of such networks are neuronal ensembles with inhibitor coupling [5]. It is well known that in such systems, nonuniform synchronized oscillation patterns may emerge."
Amazing! Thank you
So cool
I love how you investigated the Kuramoto model using just Excel. Even though you said Excel is way under-powered for this sort of analysis, it highlights nevertheless just how powerful Excel is for prototyping, investigating, and even full models. And I was surprised at just how simple the Kuramoto model is as I was expecting some complicated solution to a PDE. Anyway, I have done this with innumerable kinds of models. And when the clunkiness of a spreadsheet is not adequate, I will quickly jump into VBA and bang out a custom function. One could easily create a custom function in VBA for the Kuramoto model that would enable the input of more than three coupled oscillators and output the results as an array. Furthermore, it would be interesting to see the Kuramoto model extended to more dimensions because in your example with the five metronomes, they weren't perfectly in sync which I hypothesize was due to the metronomes not being lined up perfectly.
2:56 Cambridge engineering student here. I walked on this in one of my first year labs. It was fun.
Matt, you've outdone yourself yet again! So many videos report the phenomenon, but no-one I'm aware of works out the physics. Your channel is undisputedly one of TH-cam's finest. Do keep up the awesome work! Cheers, mate!
i've loved the RI christmas lectures ever since i was a kid, it's so cool that you got to be involved, what an honour
Matt, omitting the ω term left out a very interesting feature. In this case, if the platform the metronomes rested on was acted upon by a constant force, that force could be represented as ω. The interesting part is that this constant force would cause the metronomes to go in and out of phase at regular intervals, and if ω/K is rational, then the system is considered phase-locked.
Phase-locking is incredibly fascinating and is fundamental to determining the stability of chaotic systems.
This guy loves spreadsheets more than Christmas 🎅🎅🎅
What are you saying? Spreadsheets are the perfect Christmas gift for the holidays.
@@therabbits69 Especially if you print them out. What a joy on Christmas Eve!
Kupfer Nickel print them and then use them as wrapping paper
Who doesn't?
I'm so glad the sound of those metronomes irritates you. If it didn't we would have had it through the whole video so *thank you so much for stopping them* every time after only a short while.
Some people can really learn from you that once you have made your point you don't have to "hammer" it in. This right there is why I love watching your videos!
The antikuramoto would be useful in sound engineering on any chorus, flanger or phaser effect. You could implement it with a kuramoto for another interesting effect.
It would be interesting to investigate where the waveforms are of different amplitudes
Unfortunately it looked like it pushed all the waves to be completely out of phase with each other so the resultant sum of the waves would be 0. Not ideal for an audio effect 😄
@@tomholton235 That is not how chorus and flanger effects work.
Phillip Siebold I know. Which is why I suggested it wouldn’t make a good chorus etc 😃
@@tomholton235 Choruses work by detuning and delaying and with stereo choruses across multiple channels, so they can obviously work if you use triggers to switch between.
Similarly, they can be used in DAW stations' metronomes (as was shown by Parker here) when you need to offset your channels accordingly. Again, using triggers you switch between the algorithms to get the correct beat offsets. This is especially useful if you are trying to implement swing to your music.
Really cool video. This may be obvious to many, but I had to think about it for a second: The equation given for the model is an "equation of motion" (a differential equation that defines d(theta)/dt), but doesn't give you theta(t) outright. So Matt is just doing Euler's method to solve theta(t) and plug that into x(t). Euler's method is the simplest way of numerically solving an ODE when you know theta(0) and d(theta)/dt, by approximating theta(t+dt)=theta(t) + dt*(d(theta)/dt). Of course, there is no factor of dt used here when solving theta(t), but I guess all the scaling here is basically arbitrary.
on the spreadsheet the names of the sin waves were A,B,B also really appreciated the name of the spreadsheet being n-sync
Enjoying the Christmas lectures so far. Seeing my favourite youtubers Hannah Fry, Matt Parker and Tom Scott on the same TV show made my Xmas, and makes me realise we've reached the point now where on-demand culture now dictates what is on traditional TV, rather than the other way around. All three of these people should have their own shows on TV anyway!
Seeing Tom Scott walk out in his red t-shirt was such a good moment, and Matt bringing out Menace again, not to mention his sock/phone sorter (a Parker sorter if you will, lol) :D When I found out Hannah Fry was presenting them I was more than happy, but seeing the guest hosts? It's made the last couple of days so good. Can't wait for the third one :D
Parker Coupling: perfect out of phase :D
Nice one. Damnit, someone else always gets there first :)
Here is my explanation for it! The out-of-phase model is a pretty neat example of gradient descent instead of gradient ascent. The ordinary Kuramoto model tries to put everything in phase, i.e. maximise the energy in the system, which is why there is a plus sign in front of the term with the difference of the phases. This is like gradient ascent. Physically this means the plank will be moving with maximum amplitude possible. When you flip the phases in the Kuramoto equation, because of the property of the sine function, you get an overall negative sign, which is like gradient descent. Essentially, the sinusoidal functions get "balanced," to steal some electrical engineering lingo, meaning the sum of them is 0. There is no energy loss in the system. Physically this means the plank would not be moving at all, but there is still information being conveyed between the metronomes.
12:49 "Oh! What has it done?"
Interesting. Would there be an existing model which more accurately reflects this?
EDIT: Haven't found the answer as of yet, but it seems that Kuramoto also found a "chimera state" where some oscillators desync and act chaotically, while others stay in sync. I don't believe there is an explanation or a full model describing and predicting all generalised possibilities.
In July 1980, I took a riverboat from just south of Singapore into the heart of Sumatra. We entered the Siak River just around sunset and all along the riverbank, about every 10th tree was covered in fireflies - and they were all blinking in sync. Quite an astounding sight - and no noise either.
1:50 AM
Parker upload schedule
7:50 PM in eastern US
@@TobyBW He lives in GB.
He's british so for him (and me) its 0:50 AM
@@daniwalmsley611 Actually, he's Australian, but he lives in Britain. Sorry, ignore me.
Here in Australia, it was 12:50pm on the 28th.
RI Christmas Lectures are about the only thing I ever watch on TV at christmas
13:15 a wild Parker square appears
Parker Synchronization
Repeat performance
Physical model:
A number of N coupled oscillators is represented by N coupled differential equations. This system of equations always has N so-called "normal mode" solutions. All other states (phase differences) can be represented as a superposition of the normal modes.
In the case of 2 metronomes, the normal modes would be: a) both in phase and b) 180° phase difference.
For 3 metronomes, the normal modes you would have a) all 3 in phase, b) 2 metronomes in phase, the third 180° out of phase, c) 3 metronomes with 120° phase difference in respect to each other. The latter is the one settles in when the "order is mixed up" (@17:25)
0:38 Parker Metronomes on sale now
I absolutely love this video, the way he shows his process. Its so much fun watching him work through each step and even discover something new! He doesn't have all the answers, he's here to find them and ask questions!
The physical model for them being perfectly out of phase can be represented by a bad drummer - aka myself.
If you manage to be perfectly out of phase you are an excellent drummer.
I agree with your statement
The bridge thing was so amazing to watch! Amazing!
Matt we love you 💕 and I can’t wait to watch the Christmas lectures as soon as they’re available in the US! Good work out there
Got your book for Christmas and have been absolutely loving it!!!
"Excel is super underpowered for what I'm trying to do."
These works have been spoken by everyone who's ever used excel.
I have a theory of why the swap gave you the opposite result. The Kuramoto Model gives yiu the direction of movement of the angle Theta that gets it in sync.The sine wave is an Odd function, which means a change in the argument’s sign shows up as a change of sign in the output.If Kuramoto gave you the direction towards convergence, changing the argument’s sign changed the direction’s sign, and therefore that function gave us the direction to stay Out Of Convergence :3
The inverted version is actually really useful :) we have a game where a bunch of things happen at random times but at a constant frequency after that and using the parker-kurumoto model, they could be spaced out over time so that the load on the server is constant and not bursty
That's awesome!
What's the game?
Something you didn't touch on: The Kuramoto model allows for each oscillator to have their own increment/frequency/omega parameters. That's quite useful for real-world models where there is going to be some expected variance in the frequency of oscillators. For example, your metronomes will likely move in/out of phase on their own over long periods of time, despite your attempts to set them identically.
I completely agree with you. I tried making the spreadsheet with each independent omega parameters. It seems to work.
1drv.ms/x/s!AuCyus6wr0yslisdlCyjwfW3P3vf?e=wpuOWF
Having the phases equally spaced would be amazing for building a walker. Maybe somehow measure the position of the occilators and couple it to the position of servos attached to legs.
The leg motion could be the coupling force.
I'd look up Strandbeest by Dutch artist Theo Jensen if you aren't already familiar.
So happy to see Humble Pi Audiobook finally available in the US! Thank you and Happy New Year!
This chanel helped me find love for math
As an electrical engineer I found the equally spaced version so much more satisfying than the in-phase one.
The evenly distributed phases cancel each other out, leaving out a null wave if you summed them.
It's like the 3 phase system in AC, all of them some degrees apart from each other so they sum to 0.
wish i could ever just sit, chill and toss around some fun maths in spreadsheet as a fulltimejob just like this gentleman
I saw this in my sub feed. It said it had been posted "14 seconds ago." I found that a bit odd. It is the earliest I've been to a newly posted video (since I clicked on it right away). It currently has no views and 3 likes, which is also odd. It also has five comments... but none have appeared on the screen. Another odd thing. Well, Happy New Year!
Also, at 14:23, Matt tries to get the Sine waves perfectly in phase but instead makes them perfectly out of phase... Parker phase?
Views are only counted if you watch the video far enough (people like before watching) and they need to be more accurate in the end (because the money YT has to pay to the creators depends on it), so they make sure it's done well. That doesn't mean it's always 100% up-to-date, but that they take the time to verify each view to be a genuine view and not e.g. generated by a bot. However, views, being the most common events on YT, are cached a lot more than likes or comments, which are (relatively) rare occurances in comparison.
Two of those are odd, but the first one is even
That is the way TH-cam works. Many servers need to be in sync, and as you can see from this video, that takes a while.
The first minutes of a TH-cam video being public often have mismatched numbers like that. I think it has something to do with different TH-cam datacenters taking time to talk to each other and get a final version of the various counts.
a new video is the best christmas present i could ask for
I would preorder the US release but i couldn't wait and got the UK one. Had to go through the whole thing and cross off all the extra U's but totally worth it.
As someone who needs those extra u’s for words to look normal, you disgust me.
I pre-ordered the ebook and the waiting is killing me. I'm US but I'm fluent in UK and i don't mind the extra u's and plural "maths", but none of the ebook platforms have the original version available.
I'm sure it's a publisher thing. Still, books shouldn't be region locked.
@@jerry3790 do you prefer your Armor colored or flavored?
I'm so looking forward to around the beginning of February when the awesome Lectures will be available to watch around the world :)
The real question is, at what value of K does the kuramoto equation appear in Tupper's self-referential formula?
The rate at which metronomes go into phase (the strength of the coupling).
Hi, Matt! I’m not much of a mathematician in any way, but I do find mathematics fascinating, especially as a musician and have always really enjoyed your videos! This particular one has gained my interest because of my familiarity to a similar topic in music. You should check out Steve Reich, a minimalist composer who uses the compositional technique of “phasing” in some of his music. Having played one of these pieces, “Drumming,” I became very familiar with the concept and was excited to see it in mathematics. In music, two (or more) players play the same rhythm, whether in unison or offset, and one player stays steady in tempo while another gradually speeds up, going out of phase, and through that eventually lines up on a different partial of the rhythm to create different interlocking patterns using the same rhythm through this process of “phasing.” This also often achieved using electronics and was how it was discovered, and was originally thought to be almost impossible for a single musician to do on their own. It’s a really awesome thing to listen to and creates an extremely satisfying moment when all the voices eventually sync back up in unison. You can find this in other works of his like “It’s Gonna Rain,” “Violin Phase,” “Piano Phase,” and more. He also takes this gradual process and makes it immediate in a process called “phrase shifting.” This takes out the gradual speeding up and at a certain moment a voice changes where exactly the rhythm starts like in his piece “Clapping Music.” You should definitely check out his music if you found this metronome phase cool!
The clickity clack is kind of calming
this is more education about basic functions of excel than I have ever received.
You could have just made K negative. Also would be interesting to see how much you could perturb the angular frequencies before the synchronisation stops working
+
I know right. Probably his brain was too filled up with Christmas brandy.
This is interesting. I was looking for a way to code a simulation and the best I found was this. Great job.
Why do you do this to me Matt? I was about to go to sleep and now I have to stay up another 20 minutes!! (Not complaining though!)
Same, 2am here in Germany
Amen
Since sin(-x)=-sin(x), you get dphi/dt being the negative of the normal dphi/dt. So what previously was an unstable equilibrium turns into a stable equilibrium and vice versa. And equally spaced phases being an unstable equilibrium for the Kuramoto model makes sense.
Would this also work in 3 Dimensons? Like pendulums swinging in different directions while hanging on a board that's also hanging? Or would this just lead to a Chaotic pendulum ?
Excel absolutely is powerful enough for this! You could easily expand it to arbitrary amounts of metronomes with a couple of macros
Read "Spreadsheet", clicked instantly
The inverse phase one makes total sense, if you think about it. In the correct model the "energy" of the system is lowest if the system has the lowest total phase shift (all oscillators are in sync). If you inverse the coupling (sin(x) is an odd function, so sin(-x) = -sin(x)), the energy of the system gets lower, if the phase shift gets larger. Therefore the system will tend towards maximizing the total phase shift.
Sidenote: Total phase shift does not mean sum of all phase shifts directly, but the sum of the sin-s of all phase shifts.
Hopefully we on youtube get to see the lecture Hannah did with you. Also, digging the scruff. 😍🔥
As far as I understand, you can see them on BBC4 (and their Mobile app), and on the Ri TH-cam channel at the end of January (i.e., four weeks later).
Yeah, they mention the TH-cam channel it will be on at the very end of the video
(And a link in the description)
This is the second video where you have talked about your book (humble pi) at 3:14.
It is a good book I have enjoyed reading it
What happens if you, instead of putting the oscillators on a "linear coupling board", put them on a "circular coupling board", like on the tips of a huge fidget spinner/ball bearing?
As the metronome ( placed correctly) will jerk the spinner clockwise and anticlockwise, instead of left-right, it will be the same as in linear coupling, only in a circle.
I'm not (necessarily) a math kind of person but I am a sort of "intuitive engineer/physics observer". As such, I think I understand that the movable platform is the (ϴj - ϴi) part of the equation. If put the right way around, the board is finding the difference between the two movements (result of the combined but differently timed momentum of the pendulums) and over time canceling that difference with it's own opposing movements. Kind of like using bent knees to cancel your own bouncing on a trampoline. If placed the wrong way around (not sure but I think that's only a mathematical option) the movement of the board still finds the difference but applies it in an additive way - like working a paddle ball where, when done properly, the motion of the paddle is directly opposite to the motion of the ball except it amplifies the speed and impact of the ball. The elastic band is finding the difference of movement and adding it to the motion of the ball and paddle maintaining a 180° out of phase movement. So ... right way around, the platform moderates or filters the movement(s) - wrong way around the platform controls the movement(s). That's my spit-ball evaluation.
I will get the spreadsheet, complete with the A, B, and B sine waves.
I think this is my favourite TH-cam video ever!
Three-sided coin pls
That is a nice generalized solution. As an engineer, I like a more specific solution that is based on physical properties. This would be more fun to build as a dynamic analysis in an FEM model.
Are we going to talk about the fact that he named the spreadsheet n-sync
I decided to learn more (beyond just GCSE) about maths and computing at 33, I'm currently taking a proper run at learning excel fluently as a stepping stone to learn more elsewhere - thanks for giving us a fun brief/scenario to explore the tools!
Parker Synchronizing: doesn't exactly sync, in fact it makes stuff as out-of-sync as possible, perfectly out of sync!
The perfectly out-of-sync sine waves at 13:15 (aka. "The Parker Sync") reflect metronomes that never will go in sync with each other. Another physical analogy is the blades of a propeller or the glass tubes of a centrifuge. Here's why:
Start out with two (or more) metronomes with the same frequency and a random phase shift, just like at the start of the video. If you add up the sine waves of their positions, you will get another sine wave with the same frequency, and the average of their phase shifts as its own phase shift. This is actually the movement of the table, and is also the phase shift all the metronomes will end up with once they're in sync.
However, if you start with phase shifts that are perfectly out-of-sync ("The Parker Sync"), once you add up all the sine waves they will cancel out and give you 0. The table will not move and the metronomes will not synchronise. Coincidentally, these are the phase shifts (or angles) you would use for the blades of a propeller or the glass tubes in a centrifuge (a Numberphile video on that topic here: th-cam.com/video/7DHE8RnsCQ8/w-d-xo.html ). Using perfectly out-of-sync propellers will prevent them from trying to get in phase with each other and prevent the whole wind turbine from wobbling.
The other case where metronomes won't synchronise is when they're in perfectly opposite phases from the table (this is what happens at 2:10). The table will try to synchronise them both by speeding them up and slowing them down, and the effect will cancel out.
Now I'm really interested to see what would happen if you made them coupled but with different frequencies. What would the end result be
Made a simple program where you can change the frequencies and find out: editor.p5js.org/jrutty/sketches/zOvU7Ehrg
Short answer is that if the frequencies are multiples of one another you can get a kind of synchronization, but if they're not then the synchronization effect breaks almost immediately.
Thanks m8
@@SirPhysics Amazing program. Thank you very much
I once made a sort of spinning disc strobe with one red and one green led on the disc, each driven by its own oscillator. Spinning the disc at constant speed I could create the illusion of 3 red dots turning clockwise while 4 green ones turned counter clockwise. To my surprise there was a "magnetic" effect such that when the dots were moving slowly in relation to one another they locked. Corresponding, I guess, to f1 and f2 (the oscillators driving the leds) nearly having a small integer multiple, 12. The oscillators must have been weakly coupled by way of sharing the same power supply, both were on the same chip.
The “even-spaced out-of-phase” plot is an example 3-phase AC power (US) and how it looks.
Least interesting video title award for 2019 goes to Matt Parker
Strongly disagree
If you multiply the difference of the phases in the Sin by two (=C5+$E$1*(SIN(2*(E5-C5))+SIN(2*(G5-C5)))) in C& for example you can get the effect you see with the real metronomes where their phase differs by exactly pi or get in sync, depending on your starting phases
A little disappointed that you went for “tweak freq” rather than modelling the forces directly to see the effect come out of the physics....
Also, modelling if metronome freqs are close bu slightly differ
I love the way you are fascinated by what you are doing and eventually get to see there.
looks like a Parker beard has grown :)
Nah, that's a Parker Shave right there
When the waves are perfectly out of phase, they'll have destructive interference instead of constructive interference, which would then cancel out the resonance and make this "bridge" safer. if there is a physical interpretation of that worksheet, that could be a great safety feature. I also noticed that both times you had all five metronomes, three of the metronomes were exactly in phase with each other and the other two were exactly out of phase of the rest but in phase with each other. The second time, when you let them go longer, they would get into a cycle of going slightly in and out of this pattern. When they were spot on, the box stopped moving (to the eye) which then quit linking them altogether, so they would go off phase, starting the box moving again, which then caused the linkage that set them back toward perfect phase alignment. But the motion of the box was always pretty minimal. This seems really significant and therefore makes me think that I am probably not the first to notice. I bet there's a whole other video to be made on this aspect. I should subscribe!
So basically you applied a function from "the ether of science" to an excel sheet. I'd have found it more interesting to see a (simplified) construction of how the coupling propagates using recursion (from the line above) instead of a formula that just gives you the results.
Zolbat At the risk of sounding a bit ungrateful, I had the same thought. Maybe expanding a bit on the model itself. At this s point, it became a needless spreadsheet exhibition.
This was cathartic. I did my final project in high school years ago on this subject. I used the kuramoto model as well in my paper. I reference that paper you used as well! The sound of those metronome brought back some memories of my own testing.
just for my curiosity:
WHAT happens if the metronomes aren't synchronized perfectly?
how would that graph look like?
is it exponentially more difficult to "code" in excel?
You mean if they have a different period?
If you're asking if the system would still synchronise if each oscillator had a different period, the answer is yes. This is accounted for in the Kuramoto model with the omega_i term, the natural frequency of the oscillator.
@@MrJoepenn Would the in-phase frequency simply be an average of the different frequencies?
When you say "aren't synchronized perfectly" do you mean they have different resonant frequencies? If so, they try to synchronize but won't stay that way for long. If the frequencies are even multiples of each other then they can eventually reach a sort of synchronization, but it takes much longer. If you want to play around with it I made a very rudimentary program you can play around with: editor.p5js.org/jrutty/sketches/zOvU7Ehrg
some things are every easy in excel, other hard, depends on what you want to do. For more difficlut programms in excel you can use VBA.
Woo more spreadsheets... I actually was also doing spreadsheet for fun this Christmas.
The way he writes x stresses me out.
}{ marks the stress.
An off the cuff hypothesis: the "mistake" is doing the opposite so instead of all parts converging to an in phase state, they converge to be in perfect anti phase.
looking at the metronome example, the equation as it should be converges the system to a maximum deflection where as the mistake would converge the system to zero deflection.
ie: for two metronomes (pendulums), maximum deflection means both swing at the same frequency in the same direction but for zero deflection, they move in opposite directions.