- This episode was made possible by Brilliant.
Hey there, welcome to up "Up and Atom"
I'm Jade and I want you to imagine
that you're an ancient mathematician,
trying to figure out the circumference,
the outer edge of this wheel.
Now you can't exactly use a ruler
and measuring tapes don't exist yet.
So, what can you do?
Well, one way would be to roll it one full rotation
and measure the distance it traveled.
You're feeling extra curious today
and you want to try the same thing
with the differently sized wheel
but then you notice something interesting.
There is actually a smaller circle inside the larger wheel
which seems to have traveled the same amount.
So, does that mean it has the same circumference?
Well, that doesn't seem right.
Intuition tells you that the smallest circle
should have a smaller conference
and therefore should have traveled a shorter distance.
So, what's going on?
This is a paradox known as Aristotle's Wheel
and it originated
in an ancient Greek text called "Mechanica".
It tortured mathematicians and philosophers for centuries.
And in this video, we're going to solve it.
One attempt to figure out what was happening
was made by a French mathematician named Gilles de Roberval.
Now I'm way too Aussie to say that
without probably offending some French people.
So, I'm gonna get my French husband to say it.
- Gilles de Roberval.
- And he focused on the path
traced out by the point on the circumference
of a rolling circle.
(light-hearted music)
This distinctive shape is called a cycloid.
He then wanted to see what happened
when we traced out the path made by the inner circle too.
The line traced out by the smallest circle
is kind of stretched out.
In fact, the smaller the inner circle is
the more stretched out the path seems to get.
So, what does this mean?
While you think about that,
let's take a look at another attempt to solve the problem.
This puzzle also captivated the mind
of the great astronomer Galileo
and to figure out what was happening,
he decided to look at a wheel made of hexagons.
Just like with the circles,
the wheel has a large outer hexagon
and a small inner hexagon.
When you roll the large hexagon,
it travels a distance equal
to the sum of the length of its sides.
Just like a circle rolls out
the length of its circumference.
Now let's take a closer look
at what the inner hexagon is doing.
We can clearly see it lifting off the path,
even though it travels
the same distance as the bigger wheel,
it's sides aren't touching the entire time.
I wanted to make this extra clear for you guys,
so we're going to mark out the track made by the wheels.
(light-hearted music)
There, a nice clean skip.
Galileo then imagined what would happen
if you repeated the experiment with a shape
with even more sides, like an octagon.
Here we see that the skips are shorter and more frequent.
He then imagined a circle as a polygon
with an infinite number of sides.
He concluded that the inner circle
must be making an infinite number of skips.
While Galileo wasn't exactly right,
it does get us closer to the modern day solution.
When we put Galileo and Roberval's conclusions together
we get only one possible answer.
The inner circle must be slipping.
Unlike the hexagon and the octagon
the inner circle is not skipping
as it never leaves the surface.
So, the only possibility is that it's getting dragged along
with the outer circle.
This seems like a pretty simple answer.
So, why did it baffle some of the greatest minds of history?
Well, first, this isn't how we generally think of slipping.
Usually we think of it as something like this,
where one point on the circle
stays in contact with the path,
but anytime the outer circle rotates,
the inner circle rotates by the same amount.
The outer circle never pauses
to let the inner circle slide along to catch up.
It's all happening in the same continuous rolling motion.
Another way to say this is that each point of contact
between the circles and the path is unique.
But how do two circles of different sizes
both make unique points of contact
with the surface through the entire rotation?
Do the two circles have an equal number of points?
As one scholar put it,
"This is where the problem begins, not where it ends."
Let's talk about what it really means
for one thing to be equal to another.
If I have this many objects,
how do I know if I have an equal number of each?
Well, one way is to pair the objects together
and see if there are any leftover.
If there aren't any leftover,
I have an equal number of each.
This is called a one-to-one correspondence.
And it was thought that if two sets of objects
have a one-to-one correspondence, they're equal.
Seems legit.
Now let's try the thing with circles.
To simplify things, let's roll them out
into their circumferences.
A circle is just a rolled up line, after all.
Now we have what would appear to be
two differently sized lines.
Let's see what happens when we try to make
a one-to-one correspondence between them
to make sure that we really are
choosing unique points on the line.
We have to make a triangle.
Draw a line from the tip of the triangle,
through the shorter and longer line.
No matter what angle the line is drawn
it will always intersect each line at a unique point.
If you don't believe me try finding
a unique point on either line
and drawing back up to the tip of the triangle.
A unique point on one line will always correspond
to a unique point on the other.
So, then there does exist a one-to-one correspondence
between two lines of different length.
This complicates our idea of what it means
for two things to be equal.
In fact, if we take a segment of the line
we can still draw a one-to-one correspondence
between any point on the line and any segment of the line.
So, any segment of a line has a one-to-one correspondence
with the line itself.
So, does this mean that all lines are equal?
That even a segment of a line
is the same size as the line itself?
Well, actually it's just the opposite.
All of this leads to the very strange conclusion
that a one-to-one correspondence
does not always mean that two things are equal.
The implications of this run very deep
when we consider that numbers can be represented on a line.
But wait, let's take a step back.
What about these guys?
We showed that a one-to-one correspondence
between them meant they were equal, right?
So, why does it work for these guys and not these guys?
Can you spot the difference?
Well, these objects are discreet,
you can split them up into neat individual packages
while these objects are continuous,
uninterrupted, you can't split them up.
Let's see how this fares
when we talk about a very specific type of object, numbers.
Take all positive whole numbers.
Whole numbers are discrete
because there exists spaces between them.
They're better represented as dots rather than a line.
There are an infinity of them
because no matter how high you count,
you can always add one more.
Now, take all the negative whole numbers, same deal.
No matter how low you count, you can always go lower.
Every positive whole number can be paired
with a negative whole number in a one-to-one correspondence.
So, the number of positive whole numbers
is equal to the number of negative whole numbers.
The sizes of their infinity are equal.
This might seem like a weird thing to say, but just hold on.
We can also count them
even though we would never stop counting,
we can at least start counting.
Now, let's start adding numbers.
Let's add the fractions, the decimals,
the irrational numbers, the transcendentals.
In fact, let's add every number
until there are no more gaps.
This makes up what is called the real number line.
Unlike the whole numbers which are discreet,
the real numbers lie on one continuous line
from negative infinity to positive infinity.
If you point to any way on the real number line,
there will always be a number there.
Now let's do something.
Let's take a segment of the real number line, zero to one
and then let's take the segment one to 1 million.
Here's the thing, just like with
our two differently sized lines from before
we can always find a one-to-one correspondence
between any point from zero to one
and any point from one to 1 million.
This leads to the conclusion
that there are equal amount of numbers between zero to one
as there are one to 1 million.
In fact, there are an equal amount of numbers
from zero to one as zero to infinity.
To really hit this home, let's say what happens
if we try to count the numbers between zero and one.
What is the first number after zero?
Well, let's just ball park it and say it's 0.01
but then we can just add another zero
to make 0.001 or 0.0000000000001.
In fact, the question, "What is the next number after zero?"
is impossible to answer because for any number we choose
there will always be a number that is closer to zero.
So, we can't even count the numbers between zero and one
let alone zero and infinity.
The amount of numbers that lie between zero and one
and zero and infinity are both uncountably infinite.
This is very different to the infinity of the whole numbers,
which we can count and list.
So, here we have two different types of infinities,
countably infinite, and uncountably infinite.
The surprising result that there are different kinds
of infinities was the work of Georg Cantor.
I've made a video which goes more into this idea
and gives you the proof of it if you're interested.
The paradox of Aristotle's Wheel
took over two millennia to solve.
While the physical answer turned out to be pretty simple,
it led to some very strange ideas
about what it means for two things to be equal
and the strange nature of infinity.
It's a reminder of the mysteries that surround us
in everyday life if you just take a closer look.
If you're curious about the world around you,
but are not sure where to start,
one resource I use is Brilliant.
So, Brilliant is a problem solving based website and app
with a hands-on approach.
It's math, science and computer science content
help guide you to mastery by taking complex concepts
and breaking them up into bite-sized understandable chunks.
I find that it's a really good way to learn
because it forces you to think hard
by answering questions as you go.
In fact, if you liked today's episode,
you'll like their course on number theory.
It goes into more detail about the nature of infinity
and best of all you can work through it at your own pace.
They have over 60 courses and are always adding more,
including an entire one dedicated to infinity coming soon.
I will definitely be checking that out
as I'm sure there is still so much to learn.
If you'd like to go ahead and sign up,
the first 200 people to sign up with this link
will get a 20% discount off premium membership.
Just go to brilliant.org/UpAndAtom.
Great, thank you for watching and I'll see you next time.
Bye.
(upbeat music)