[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.
