- This is the title sequence from "Rings of Power".
And this is a video that I made in 2016.
So of course when "Rings of Power" premiered recently
I got tons of messages from people saying,
"Hold on a second, are they Chladni figures?"
So to test that hypothesis
I thought I would try to recreate the title sequence
from "Rings of Power" using my Chladni plates.
If you want to know more
about the history of Chladni figures,
then I recommend you watch the original video link
in the card and the description.
But the basic principle is you use a vibrating metal plate
to create a two dimensional version of a standing wave.
I've talked about standing waves on this channel before,
but here's another example.
If I jiggle this rubber tube
at just the right frequency,
in other words, one of the resonating frequencies,
then I get a standing wave.
The points that aren't moving up and down are called nodes
and the points that are moving up and down
are called anti nodes.
If I jiggle the tube faster
until I find the next resonating frequency
or the next harmonic,
then I end up with more nodes and more anti nodes.
For another example of standing waves
check out my flame tube video.
This video is also my best attempt to explaining resonance,
link in the card and the description as well.
So if you pluck a string or waggle a tube,
you get standing waves.
And a string is essentially a one dimensional structure
when analyzed in this way.
So it's natural to ask the question
what would a standing wave look like
on a two dimensional surface?
And that's what Chladni plates do.
Strings and metal plates are quite different,
so let's add a conceptual stepping stone between the two.
The main difference between a string and a metal plate
is a string is floppy and a metal plate isn't.
So let's put something in between,
something that is still one dimensional but isn't floppy.
For example, this strip of metal.
I can vibrate the thing at one end
and when I get the frequency right,
look, there's a one dimensional standing wave.
You've got nodes and anti nodes.
So really a Chladni plate
is a two dimensional version of this
as opposed to a two dimensional version of a string,
but it's all standing waves.
And by the way,
there's some beautiful mathematics going on here
because what we're doing is we're going up a dimension.
We're going from a one dimensional string
to a two dimensional surface.
So we might expect other aspects of the phenomenon
to scale up in a similar way.
So take the nodes of this standing wave, for example.
They are points in space,
and points in space are zero dimensional.
So when we go up a dimension
we might expect the nodes
to go from being zero dimensional to being one dimensional.
That is to say we should expect the nodes to become lines.
And look, that's exactly what we see
on these Chladni plates.
The lines formed by these grains
show us the parts of the plate that aren't moving.
They show us the nodes.
And that makes sense,
because grains that find themselves
on vibrating parts of the plate
will be jiggled around until they eventually find themselves
on a part of the plate that isn't vibrating,
and that's where they'll settle down.
They'll settle down on the node lines.
So to recreate the title sequence from "Rings of Power,"
we need to look at the specific shapes that they make
and figure out how to make those.
And to do that, we need to delve into the math a little bit.
Look at the symmetry, for example.
In this frame you can see four lines of symmetry.
In this frame you can see two lines of symmetry
but the symmetry is kind of broken here and there.
Here there is only one line of symmetry.
This has rotational symmetry only, no mirror symmetry,
which I don't think is possible with Chladni figures
which makes me think there must be
at least some CGI involved in the title sequence.
Maybe the whole thing is CGI,
but if it is, I reckon they're using simulation software
for at least some of it to produce those Chladni figures.
And finally, these transitions
between different Chladni shapes.
I want to try and capture that as well.
For the most part though, there's a combination
of two lines of symmetry and four lines of symmetry.
And that implies a square Chladni plate
like the one from my original video.
See for example,
this Chladni figure clearly has four lines of symmetry.
For comparison, I got a pentagonal Chladni figure made.
And look, the shapes that I can produce on this one
have five lines of symmetry.
So let's start with the square.
In my original video, I showed the wave equation
for the plate when it's vibrating in this mode.
And look, there's an animated plot
of the wave moving up and down.
And also look at 2016 me just sticulating
to the top left corner of the screen
where 2016 me forgot to put a graphic.
You can also work out the equation for the node lines.
That equation looks like this.
You don't really need to get your head around that
except to notice that we have two variables, n and m,
that we can set to any positive whole number value.
And you get different patterns on the plate
depending on what values of n and m you choose.
And by the way, that's another beautiful bit of math
as we scale up to the next dimension.
Like, for a one dimensional string
there's only really one variable n
that you can set to different values
to give you different standing waves,
and that just tells you how many nodes you have.
But when you scale up to two dimensions,
you have two variables that you can change
to get different patterns.
Using Desmos, I can quickly run
through the various combinations of n and m,
and see the patterns that they would give.
So in theory,
these are the patterns that I can create on a square plate,
and those seem to match pretty well
with these patterns from the title sequence.
Let's see if I can actually produce those in real life.
By the way,
the way you can select different patterns in practice
is by forcing certain parts on the plate to be nodes.
Like if I want to produce this pattern
then I should pinch the plate at this point here.
By doing that, I'm saying
that a node line has to touch this point.
(Chladni plate rings in high pitch)
(Chladni plate rings in low pitch)
(Chladni plate rings in high pitch)
(Chladni plate rings in low pitch)
(Chladni plate rings in higher pitch)
(Chladni plate rings in lower pitch)
Interestingly,
when the symmetry breaks down on the pentagonal plate,
you go from having five lines of symmetry
to having just one line of symmetry, like this.
(Chladni plate rings in high pitch)
And that somewhat matches this part of the title sequence.
I really wanted to create this part of the sequence
but I definitely can't do that with this plate.
And that's because the boundary conditions are wrong.
What do I mean by boundary conditions?
Well, let's go back to the one dimensional example.
The boundary conditions
are just whatever we are forcing the system to do.
Like, this end of the rubber band
or this boundary of the rubber band, if you like, is fixed.
It can't move.
This end is forced to vibrate
at the frequency and amplitude that we drive it at.
Those are the boundary conditions.
Compare that to this strip of metal.
This boundary is free to move.
That's a boundary condition too.
In that sense, boundary conditions also
describe what you are not forcing the system to do.
But the point is you get different standing waves
at different frequencies
when you change the boundary conditions.
Now, this is the way
that Ernst Chladni originally made Chladni figures
by fixing a plate in the center and then bowing the edge.
And so we have a boundary condition here.
We have a plate that is fixed in the center,
which means that any pattern we produce
will have lines that go through the center.
The center must be part of a node line
because it can't move.
And you can see that here
with these choices that are available to me with n and m.
And this is a problem, because in this shot,
there are no lines passing through the middle.
So we need to change the boundary conditions.
Instead of the plate being fixed in the middle,
we're gonna say, let's vibrate the plate
from the middle instead of bowing it at the edge.
And I can do that with this setup here.
This plate is fixed to a vibration generator
and I can send any frequency I want,
and see what happens.
When I find one of the resonating frequencies of the plate
I get a nice pattern.
See this one for example
there are no node lines passing through the middle.
So is it possible to get something
like this image from the title sequence?
Well, with these new boundary conditions
the equation for the node lines changes to this.
So these are the sorts of patterns I can get in theory.
This one is probably the closest
to the one from "Rings of Power".
"Rings of Power" is missing the straight lines
which to me is more evidence that there's some CGI involved.
But anyway, I swept through a huge range of frequencies,
found a handful that worked really well,
and then I coded something up
so I could quickly switch between those good frequencies.
To be honest,
most of the patterns that I can create with this plate
and the center fixed plate,
don't match any of the patterns that I can create
by playing around with the equations in Desmos,
and vice versa.
Most of the patents I can create in Desmos
I don't see on the plate.
At least some of them seem to match up,
but it's interesting.
There's obviously more to this
than I've been able to uncover.
This one is probably the closest
that I'll get to that clip from the title sequence.
- My precious.
- No, it looks nothing like it.
- I thought it was good.
- Find a better one.
- But I can't.
- Of course, the other thing
that makes the title sequence from "Rings of Power"
different to my video from 2016 is cinematography.
Like, back then, I didn't have nice lenses.
I didn't have a track.
I didn't have a clue what I was doing.
Whereas these days I have some of those things.
Like, clearly this shot here, for example, is on a track.
In other words,
the camera is moving along tracks in a nice smooth way,
so I could set up something like this.
And so here it is all put together.
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This video is sponsored by Jane Street.
Jane Street sponsored my videos
because like me and presumably like you,
they enjoy figuring out how things work.
Specifically, they enjoy figuring out how things work
in the financial world.
So much so that they're looking
for more people to do just that.
In fact, applications are now open
for their paid summer internship, 2023.
Internship roles include things
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quantitative research, and business development.
They have offices in London,
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So if you like the sound
of figuring out how things work in the financial world,
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It's a really good program.
If you like what you see, then you should apply.
The link is on the screen.
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