[MUSIC PLAYING] Can you hear the shape of a drum?

This question, can one hear the shape of a drum, was famously asked by mathematician Mark Kac in 1966 and eventually resolved in 1992.

When he talked about a drum, he really meant a simple piece of fabric stretched over a fixed boundary without the cylindrical body attached to it.

But before we get to a drum, a two-dimensional vibrating membrane, let's start in one dimension with a vibrating string.

If we start with a string of length L and you pluck it, what happens?

Well, the string makes a wavy shape, and a sound comes out.

Maybe the wave moves like this or this.

But those are complicated.

To understand them, we need to begin with the most basic kind of wave, a sine wave.

Because our string is fixed in place at the ends, it can't make any old sine wave.

It has to have exactly one bump or two bump or three bump or four, five, six.

You get the point.

In these equations, the variable x tells us what point we're at along the string.

The number in front of the x is related to the frequency.

It tells us how many bumps there are on a string of length L. To determine the frequency, we divide it by 2 pi.

So the frequency of these sine waves is 1/2L, 1/L, 3/2L, and so on.

That frequency is what you hear.

It's called a pure tone.

These pure tones, the sound of a single sine wave pulsing up and down, don't really occur naturally.

The computer-generated versions sound like this or this.

You might use a pure tone to tune an instrument or test your speakers or your hearing.

But most sounds you encounter are a combination of pure tones.

When you pluck a guitar string, it produces a bunch of pure tones with harmonically-related frequencies.

The lowest frequency, what musicians call the fundamental frequency, is usually the dominant sound.

The higher frequencies, called harmonics, can also be heard but more softly.

This combination of pure sine waves gives a pleasant musical tone.

To understand how these pure tones connect to what you hear when you pluck a string, let's dive further into the mathematics.

To reduce the amplitude of a sine wave, which makes it quieter, you can multiply it by a number smaller than 1, which changes it like this.

The smaller the number, the further it reduces the amplitude.

We can also add sine waves together.

In that case, you'll hear both pure tones.

If the sine waves are multiplied by different numbers and then added together, you'll hear more loudly the frequency associated with the sine wave that is multiplied by the larger number.

This is basically a recipe for creating more complex wave patterns.

Start with a bunch of sine waves.

Shrink or stretch them to have different amplitudes.

Then add them together.

The sound associated with the wave is a weighted combination of the pure tones.

For example, this wave is made up of these four sine waves.

You'll hear the four different frequencies associated with the four sine waves.

But this one will be dominant because the sine wave is multiplied by the largest number.

Here's the big fact.

This recipe will produce all the waves the string makes.

So for all you violinists, guitarists, cellists, and bassists, when you pluck a string of length L, it wiggles around in different shapes.

But each of these different wavy shapes is just a weighted sum of the basic sine waves with frequencies 1/2L, 1/L, 3/2L, and so on.

This is the idea behind Fourier series, named after the early 19th-century mathematician Joseph Fourier.

Sine waves really are the building blocks for all of the strings' vibrations.

Now, here's Kac's famous question rephrased in one dimension.

Can you hear the length of a string?

Yes, you can.

You can hear the lowest, or fundamental, frequency.

It's going to be 1/2L.

Then you divide 1/2 by the lowest frequency, and out pops L, the length of the string.

This is probably as good a time as any to give a typical mathematician's caveat.

In the real world, a string's density and tension affect its vibrational frequencies.

Not all strings of length L produced pure tones of exact frequency 1/2L, 1/L, 3/2L as we calculated.

We made some assumptions about the string's tension and density that simplified the math.

But adjusting for different physical setups doesn't change the core mathematical ideas.

Speaking of mathematical ideas, I'm going to say some calculus words for the next 30 seconds.

In case that makes you anxious, I'm also going to leave this picture of a kitten on the screen.

There's this handy little thing called the Wave Equation that describes waves.

So the wave produced when I pluck a string satisfies this differential equation.

But we can solve the wave equation in terms of a simpler equation, this one.

This is solving for the eigenvalues and eigenfunctions of the Laplacian.

Anyway, the solution to this simpler equation is the sine waves we talked about before, the ones that made up the more complicated waves.

In music, when you break down a note into its fundamental frequency and upper harmonics, you're actually breaking down a calculus problem into a simpler one.

OK.

The calculus part is done.

Thanks, kitten, you've done your job.

Now we're going to bump the whole thing up to two dimensions.

Instead of thinking about plucking a string, we'll think about tapping a drum.

We're only talking about the flat surface of a drum, without any type of cylindrical body attached.

Again, I'll give the mathematician's caveat.

To simplify the math, we are making some assumptions about the drum's density and tension.

The situation in two dimensions isn't that different than one dimension.

When you tap a drum, it undulates and produces waves which are made up of simpler component waves.

You can hear the frequencies of those component waves.

In one dimension, we used the length of the string to compute the frequencies it could produce, 1/2L, 1/L, 3/2L, and so on.

In two dimensions, we can use the shape of the drum to figure out what frequencies it produces.

The tones it makes are combinations of these frequencies, so we can determine what sounds the drum makes.

But Mark Kac asked the reverse question.

He wanted to know, if you know the exact frequencies that a drum produces, can you figure out the shape of the drum?

Remember, in one dimension, the answer was yes.

The lowest, or fundamental frequency, is always 1/2L, so 1/2 divided by the fundamental frequency will give you the length.

But the shape of something in one dimension is simple.

It's just the length of the string.

There's so many more possibilities in two dimensions.

Our drum could look like this or this or this.

But here's another way to phrase Kac's question that might make it a little easier to get at the answer.

Are there two differently shaped drums that produce exactly the same frequencies?

He knew that if they did exist, they have to be pretty similar, like they have to have the same area and perimeter length.

And 26 years after Kac posed the question, three mathematicians did find two drums that vibrate at exactly the same frequencies-- these two.

These drums aren't wildly different.

You can kind of easily rearrange one into the other.

So the answer to Mark Kac's question, no.

Even a robot with perfect pitch can't hear the shape of a drum.

Bummer.

But you can hear a drum's area and perimeter length.

You can even hear how many holes a drum head has in it.

It turns out that the sound of a drum can teach you a lot about its shape, which is pretty cool.

We'll see you next time on "Infinite Series."

Hello.

This week I'm recording from beautiful Munster, Germany, which is not where the cheese comes from.

That's in France.

Let's talk about the poison wine puzzle.

If you have 100 bottles of wine, it'll take seven rats to figure out which one was poisoned.

If you have a million bottles, it will take 20.

Way too many folks answered this correctly for me to acknowledge everyone.

But here's a few people I want to give a shout out to.

Toby Vijamaa for being the first person to answer, Christopher Bocksel for giving the most succinct answer, and Andrew Kozma for using the most exclamation marks.

A lot of people pointed out that for n bottles of wine, it takes log base 2 of n many rats to figure out exactly which one is poisoned.

This is because 2 to the power of the number of rats you have is the maximum number of bottles you can figure out whether are poisoned or not.

Interestingly, if the number of bottles is a power of 2, like 4 or 32, do you need to start numbering the bottles with 0 instead of 1 to figure out exactly which one is poisoned?

Check out the comment thread started by Gavin Claugus for details.

The set-ups for math problems can be pretty far fetched.

And plenty of you pointed out unrealistic aspects of our puzzle.

Travis B. suggested that we just dump all the wine together and dilute the poison.

Kemptonka did some awesome calculations to show that half our rats would die from alcohol consumption anyway, regardless of whether they drank the poison or not.

The comments also brought up some great questions, like do all 10 rats have an equal probability of dying?

What if there's infinitely many rats?

What if exactly two bottles are poisoned or exactly three?

I'll leave it to you to ponder these questions or maybe discuss at a holiday party.

We're off next week, so from us here at PBS "Infinite Series," have to great new year.