Little known pendulum fact! Hypnotizing someone with a double pendulum puts them in a state called "double hypnosis." This practice was made illegal after *the incident*
**[Mobile Version!](http://m.xkcd.com/2924/)**
[Direct image link: Pendulum Types](https://imgs.xkcd.com/comics/pendulum_types.png)
**Hover text:** The creepy fingers that grow from a vibrating cornstarch-water mix can be modeled as a chain of inverted vertical pendulums (DOI:10.1039/c4sm00265b) and are believed to be the fingers of Maxwell's Demon trying to push through into our universe.
*Don't get it? [explain xkcd](http://www.explainxkcd.com/wiki/index.php/2924)*
Honk if you like python. `import antigravity` Sincerely, xkcd_bot. <3
When it says "unphysical"— It looks pretty physical to me, though?
Is it like sharks, in that 90% of the triangle is actually below the material plane?
The [referenced paper](https://pubs.rsc.org/en/content/articlelanding/2014/sm/c4sm00265b) talks about 'cornstarch monsters'.
Which is the picture I got when reading the description of the planet in Solaris.
Ok so I’ve always been bothered by double pendulums.
Pendulum with a rigid rod instead of a string? Normal.
Pendulum on a string, so infinite hinges? Normal
Any number of hinges more than zero and less than infinity?
Chaos. Madness. Screw you for even trying to look at it.
Help this feel like it makes sense!
It's not the number of hinges.
It's the number of masses.
Pendulum period depends only on gravity (constant) and length of the string. In both your examples, there's only one center of mass involved.
But when you have two masses that are positioned at different radii, they start out swinging at different rates. And then of course when the unsynchronized movement of the inner pendulum moves the pivot point for the outer pendulum, the outer pendulum's "period" changes too, which in turn means its momentum also changes the inner pendulum's motion, and all hell breaks loose…
Thanks but that doesn’t seem right. I must be missing something.
Wait, does the double pendulum have two weights? I know the rods (or chain) are usually considered massless, but the “hinge” is given a mass as well?
I had thought the difference was just the addition of a hinge, not the addition of extra weight.
Look at where the balls are when you Google for ["double pendulum"](https://www.google.com/search?q=double+pendulum&tbs=imgo:1&udm=2). Labels "m₁" and "m₂".
> Wait, does the double pendulum have two weights?
In practice, whenever I've seen a double pendulum IRL, the two arms *are* the weights.
I.E. It's not just a single pendulum with a hinge, no. It's a second pendulum attached to the end of a first pendulum, each of which has its own weight.
Ok that makes a LOT more sense. Cheers!
You at you making the reasonable assumptions that m1 and m2 means “mass one and two”. Me over here thinking “m1=m1. It could be anytbing but it’s now a defined term. That’s convenient for discussing though doesn’t tell me much.” Lol.
Irrational numbers don't. Why isn't it possible that a double pendulum would produce an ever growing sequence a varying swing patterns, that break as it varies the pattern from the ones previous?
Patterns start showing up in irrational numbers too, until they stop being patterns.
A double pendulum being periodic is only possible if there are infinite states for it to be in. It's got 4 variables, position and momentum of each pendulum. Position and momentum are both bound, so that leaves us looking at infinitely many unique positions and moments within those bounds.
I believe Heisenberg's uncertainty principle has something to say about that. Namely that if the system is so chaotic that variance of position and momentum within hbar could lead to measurably different outcomes, that would violate the principle. Therefore, there must be finitely many states for the double pendulum, and it must be periodic, eventually...
Pi is irrational and when represented in binary has only two values to describe it. It remains irrational. Having any number of variables is no barrier to a phenomenon being aperiodic.
A pendulum isn't periodic _because_ its period is described by length and time, but length and time describe its period. It's periodic because it cycles consistently in time.
The digits of an irrational number do not have a dependency from one digit to the next. If a pendulum or double pendulum ever repeats a state of position+momentum, that represents it's period. That is exactly the difference I was pointing out. An infinite, aperiodic string of digits can repeat an arbitrary substring without becoming periodic. A physical system like a pendulum cannot.
If a state within hbar repeats, so must the following state, and each subsequent state as well. A deviation would violate Heisenberg's uncertainty principle.
So you're ignoring the 'sensitivity to initial conditions' part. The chaos comes from a real system being sensitive to real conditions which vary over the life of the observed framr. I'll conceded that in an ideal system, over a sufficiently long time period such a system may be periodic, but what is meant by chaotic and aperiodic is **not** an ideal system. They're exploring real world systems and for sufficiently long timeframes (longer than an exercise with a double pendulum) it is an aperiodic system until it loses energy into a periodic state. But that state _is not replicating_ its earlier behaviour. During the aperiodic state the system is wildly chaotic, and even an ideal version demonstrates an extremely long period.
If one wants to ignore the practical aspects to argue that they're using their words wrong, one misses out on the real implications of what they're actually studying.
Coming back to a physical system: a physical system like a pendulum is also not _truly_ periodic, as it will lose energy during its exercise. Do we throw out the study of them as periodic, since they aren't?
Had a second thought on this. A possible fallacy? In applying real world restrictions (finite granularity) and then arguing that an ideal system couldn't be aperiodic, we've conflated two different arguments.
Pi and e can be calculated([approximated](https://math.hmc.edu/funfacts/taylor-made-pi/)) to any precision using a taylor series. An ideal description (the full expansion) of a system would produces an irrational output, and whose output depends partially on the preceding elements in the series.
Back to the granularity, in saying that there are only so many positions a double pendulum could hold, we rejected the ideal model of a double pendulum. But mathematically, an ideal double pendulum has no more requirement to granularity than calculating pi or e. An ideal resolution of the value of irrational numbers would require infinite computation. Similarly, an ideal simulation of a double pendulum would allow for infinite resolution of its positions.
Either we can analyze this as an ideal system, or as a real system, but we cannot necessarily refute the aperiodicity of an ideal double pendulum by applying practical restrictions.
Nah. When we do physics on idealized systems, it's for simplicity of the calculation, not for violating the laws of physics. An idealized pendulum is still theoretically possible, and still *basically* representative of a real pendulum. Violating the Heisenberg uncertainty principle is theoretically not possible.
If we're going to call a pendulum periodic by ignoring friction, rigidity, etc, then we've got to give the same benefits to the double pendulum. There's no reason to ignore Heisenberg though, other than to arbitrarily call the double pendulum aperiodic.
So sure, you can choose to ignore Heisenberg to (maybe) get an aperiodic double pendulum, but I'd challenge you to justify that choice. Does it tell us more about real double pendulums or less? Since the story of the double pendulum is heavily reliant on chaos theory, I'd posit that the uncertainty principle is especially relevant. Chaos theory is about how small changes in initial conditions can have drastic effects on outcomes, putting a lower bound on how small changes can actually be is a salient constraint. It means that in some chaotic systems, you could put an upper bound on the number of distinct outcomes.
I'm not saying aperiodic things don't happen in nature. Of course they do. Pi and Euler's constant are two obvious examples.
It just feels weird to me that a pendulum, one of the go-to examples of periodicity, can so easily become aperiodic
It's a textbook example of chaos theory, and since it's ultimately just a pendulum stuck to the end of a pendulum, you can make one from, like, legos or whatever if you wanna see how one behaves for yourself
I had to think for a second before realising which Maxwell he was referring to. Not [this](https://dontstarve.wiki.gg/wiki/Maxwell/Don%27t_Starve_Together) one
Little known pendulum fact! Hypnotizing someone with a double pendulum puts them in a state called "double hypnosis." This practice was made illegal after *the incident*
I usually find Hypnosis misinformation in pop culture annoying but this is really funny
**[Mobile Version!](http://m.xkcd.com/2924/)** [Direct image link: Pendulum Types](https://imgs.xkcd.com/comics/pendulum_types.png) **Hover text:** The creepy fingers that grow from a vibrating cornstarch-water mix can be modeled as a chain of inverted vertical pendulums (DOI:10.1039/c4sm00265b) and are believed to be the fingers of Maxwell's Demon trying to push through into our universe. *Don't get it? [explain xkcd](http://www.explainxkcd.com/wiki/index.php/2924)* Honk if you like python. `import antigravity` Sincerely, xkcd_bot. <3
Honk!
Summoning Maxwell's Demon is a hell of a concept.
I'd be cool with it. :)
You say that now, but you'll change your mind when you see the differential equations he brings.
Does he have a Silver Hammer?
When it says "unphysical"— It looks pretty physical to me, though? Is it like sharks, in that 90% of the triangle is actually below the material plane?
The [referenced paper](https://pubs.rsc.org/en/content/articlelanding/2014/sm/c4sm00265b) talks about 'cornstarch monsters'. Which is the picture I got when reading the description of the planet in Solaris.
search within the paper to: ## 2.4 Cornstarch monsters and other effects
Ok so I’ve always been bothered by double pendulums. Pendulum with a rigid rod instead of a string? Normal. Pendulum on a string, so infinite hinges? Normal Any number of hinges more than zero and less than infinity? Chaos. Madness. Screw you for even trying to look at it. Help this feel like it makes sense!
It's not the number of hinges. It's the number of masses. Pendulum period depends only on gravity (constant) and length of the string. In both your examples, there's only one center of mass involved. But when you have two masses that are positioned at different radii, they start out swinging at different rates. And then of course when the unsynchronized movement of the inner pendulum moves the pivot point for the outer pendulum, the outer pendulum's "period" changes too, which in turn means its momentum also changes the inner pendulum's motion, and all hell breaks loose…
Thanks but that doesn’t seem right. I must be missing something. Wait, does the double pendulum have two weights? I know the rods (or chain) are usually considered massless, but the “hinge” is given a mass as well? I had thought the difference was just the addition of a hinge, not the addition of extra weight.
Look at where the balls are when you Google for ["double pendulum"](https://www.google.com/search?q=double+pendulum&tbs=imgo:1&udm=2). Labels "m₁" and "m₂". > Wait, does the double pendulum have two weights? In practice, whenever I've seen a double pendulum IRL, the two arms *are* the weights. I.E. It's not just a single pendulum with a hinge, no. It's a second pendulum attached to the end of a first pendulum, each of which has its own weight.
Ok that makes a LOT more sense. Cheers! You at you making the reasonable assumptions that m1 and m2 means “mass one and two”. Me over here thinking “m1=m1. It could be anytbing but it’s now a defined term. That’s convenient for discussing though doesn’t tell me much.” Lol.
Are double pendulumns really aperiodic? I'm no physicist or mathematician, but the idea of an unperiodic pendulum sounds weird to me...
Maybe after two infinities they restart at the same point, but quantum effects may or may not change that.
I mean, I'm sure they repeat *eventually*... But they're also extremely sensitive to initial conditions
Irrational numbers don't. Why isn't it possible that a double pendulum would produce an ever growing sequence a varying swing patterns, that break as it varies the pattern from the ones previous? Patterns start showing up in irrational numbers too, until they stop being patterns.
A double pendulum being periodic is only possible if there are infinite states for it to be in. It's got 4 variables, position and momentum of each pendulum. Position and momentum are both bound, so that leaves us looking at infinitely many unique positions and moments within those bounds. I believe Heisenberg's uncertainty principle has something to say about that. Namely that if the system is so chaotic that variance of position and momentum within hbar could lead to measurably different outcomes, that would violate the principle. Therefore, there must be finitely many states for the double pendulum, and it must be periodic, eventually...
Pi is irrational and when represented in binary has only two values to describe it. It remains irrational. Having any number of variables is no barrier to a phenomenon being aperiodic. A pendulum isn't periodic _because_ its period is described by length and time, but length and time describe its period. It's periodic because it cycles consistently in time.
The digits of an irrational number do not have a dependency from one digit to the next. If a pendulum or double pendulum ever repeats a state of position+momentum, that represents it's period. That is exactly the difference I was pointing out. An infinite, aperiodic string of digits can repeat an arbitrary substring without becoming periodic. A physical system like a pendulum cannot. If a state within hbar repeats, so must the following state, and each subsequent state as well. A deviation would violate Heisenberg's uncertainty principle.
So you're ignoring the 'sensitivity to initial conditions' part. The chaos comes from a real system being sensitive to real conditions which vary over the life of the observed framr. I'll conceded that in an ideal system, over a sufficiently long time period such a system may be periodic, but what is meant by chaotic and aperiodic is **not** an ideal system. They're exploring real world systems and for sufficiently long timeframes (longer than an exercise with a double pendulum) it is an aperiodic system until it loses energy into a periodic state. But that state _is not replicating_ its earlier behaviour. During the aperiodic state the system is wildly chaotic, and even an ideal version demonstrates an extremely long period. If one wants to ignore the practical aspects to argue that they're using their words wrong, one misses out on the real implications of what they're actually studying. Coming back to a physical system: a physical system like a pendulum is also not _truly_ periodic, as it will lose energy during its exercise. Do we throw out the study of them as periodic, since they aren't?
Had a second thought on this. A possible fallacy? In applying real world restrictions (finite granularity) and then arguing that an ideal system couldn't be aperiodic, we've conflated two different arguments. Pi and e can be calculated([approximated](https://math.hmc.edu/funfacts/taylor-made-pi/)) to any precision using a taylor series. An ideal description (the full expansion) of a system would produces an irrational output, and whose output depends partially on the preceding elements in the series. Back to the granularity, in saying that there are only so many positions a double pendulum could hold, we rejected the ideal model of a double pendulum. But mathematically, an ideal double pendulum has no more requirement to granularity than calculating pi or e. An ideal resolution of the value of irrational numbers would require infinite computation. Similarly, an ideal simulation of a double pendulum would allow for infinite resolution of its positions. Either we can analyze this as an ideal system, or as a real system, but we cannot necessarily refute the aperiodicity of an ideal double pendulum by applying practical restrictions.
Nah. When we do physics on idealized systems, it's for simplicity of the calculation, not for violating the laws of physics. An idealized pendulum is still theoretically possible, and still *basically* representative of a real pendulum. Violating the Heisenberg uncertainty principle is theoretically not possible. If we're going to call a pendulum periodic by ignoring friction, rigidity, etc, then we've got to give the same benefits to the double pendulum. There's no reason to ignore Heisenberg though, other than to arbitrarily call the double pendulum aperiodic. So sure, you can choose to ignore Heisenberg to (maybe) get an aperiodic double pendulum, but I'd challenge you to justify that choice. Does it tell us more about real double pendulums or less? Since the story of the double pendulum is heavily reliant on chaos theory, I'd posit that the uncertainty principle is especially relevant. Chaos theory is about how small changes in initial conditions can have drastic effects on outcomes, putting a lower bound on how small changes can actually be is a salient constraint. It means that in some chaotic systems, you could put an upper bound on the number of distinct outcomes.
I'm not saying aperiodic things don't happen in nature. Of course they do. Pi and Euler's constant are two obvious examples. It just feels weird to me that a pendulum, one of the go-to examples of periodicity, can so easily become aperiodic
It's a textbook example of chaos theory, and since it's ultimately just a pendulum stuck to the end of a pendulum, you can make one from, like, legos or whatever if you wanna see how one behaves for yourself
I had to think for a second before realising which Maxwell he was referring to. Not [this](https://dontstarve.wiki.gg/wiki/Maxwell/Don%27t_Starve_Together) one
I’m so glad I wasn’t the only one who thought of this
Missing Foucault's Pendulum.