I have been occasionally obsessed with this video for years and I finally sat down to deconstruct it. The video shows a row of 15 pendulums, with each pendulum being slightly shorter than the last. All of the pendulums are started swinging at once and they follow a kind of snaking pattern that devolves into chaos. However, they occasionally line up into two or three groups that swing together before falling out of phase again. After 60 seconds they all line up in one row and the pattern repeats. Have a watch:
How does it work?
So how does this demo work? How can a set of independent pendulums alternate between complete chaos and perfectly aligned rows? There’s a hint on this page:
The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations. When all 15 pendulums are started together, they quickly fall out of sync—their relative phases continuously change because of their different periods of oscillation. However, after 60 seconds they will all have executed an integral number of oscillations and be back in sync again at that instant, ready to repeat the dance.
This quote gives us a decent picture of how to build one of these for ourselves if we wanted to but it misses why this system works in the first place. It does tell us that the arrangement of the frequencies is important so let’s start there.
Let’s start by writing out an equation for where each pendulum is at a given point in time. The standard way to describe the motion of a pendulum is to start with all the forces acting on it, then use the “small angle approximation” and end up with this equation (which I’ve simplified a bit)1
An important property of
Why does the cycle repeat after 60 seconds?
From the quoted description above, each pendulum has a frequency one cycle per minute higher than the previous ranging from 51 to 65 cycles per minute. The equation for the
The description states that the cycle repeats after 60 seconds, and we can verify that by plugging in 60 for
Note that the final term is independent of
What about the wavy rows of pendulums?
Let’s focus on the clearest pattern which is when the pendulums line up in two rows made up of every other pendulum. It seems to happen at roughly halfway through the cycle and we can check exactly when it happens by figuring out when a pendulum lines up with another two steps further down the line. We can accomplish that by setting
This holds true when
To see why there are two different rows instead of a single row at 30 seconds, let’s plug in some numbers for specific pendulums.
So the pendulums with odd indices all line up on one side and all the pendulums with even indices line up on the other.
What about the other patterns? You can see three rows fairly clearly and even four somewhat. In fact, there is a broader pattern here the explains all of these patterns. Let’s look at when two pendulums with an arbitrary gap of
So two pendulums which are
Why do we sometimes see fewer rows than expected?
If we keep investigating, there is something interesting that happens when you plug in
Let’s look at the statement we proved above a bit more carefully. We showed that a pendulum with index
In general this will happen whenever we have a time
An interesting consequence of this is that if
What can we change?
So is there something special about those frequencies that makes this work? Intuitively it seems like you should be able to slow down the video or equivalently slow down all of the pendulums by making them longer and still see the same system. In fact the frequencies and number of pendulums does not matter as long as there is a fixed frequency gap between them. In this case the gap is one swing per 60 seconds but it could be anything you like, and there could be as many pendulums as you like.
Additionally, the only property we used to prove all the statements above was periodicity. Because of that we can construct a system that has similar behavior using any periodic function. One I particularly like is the movement of clock hands around a clock. 5 With this visualization it is much easier to see the patterns for higher values of
How far can we take it?
This system has a limit as it stands. In principle you should be able to see
The closest thing I’ve seen before is Thomae’s function which is commonly used as an example of a function with bizarre continuity properties. However, I want something that operates over both time and space so that we can see where the oscillators are at any given point in time. Thomae’s function is one dimensional and can’t quite accomplish what I want, so we’ll have to extend it a bit.
The best I can come up with is the following:
Essentially this says that if
Send me whatever you come up with for these open-ended challenges.
- Where else can you find this kind of system?
- Implement/build a visualization of this kind of system. Try using another kind of periodic function (e.g. turn signals, planetary systems, something with audio), be creative!
- Implement the above definition for an infinite analog in an appealing way.
- Formulate some alternative definition for an infinite analog to this system.
- Send me an interesting challenge to put here.
https://en.wikipedia.org/wiki/Pendulum_(mathematics) for the standard derivation, if you’re interested in a more thorough look this video by 3Blue1Brown is fantastic ↩
implies and by the transitive property of equality ↩
since it still must line up in one row at multiples of 60 ↩
Actually, the multiplication table for
(integers mod ) is a much better description of what I meant. Thanks, Stephen. ↩