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  • Hey, Vsauce. Michael here. There's a famous way

  • to seemingly create chocolate out of nothing.

  • Maybe you've seen it before. This chocolate bar is

  • 4 squares by 8 squares, but if you cut it like this

  • and then like this and finally like this

  • you can rearrange the pieces like so

  • and wind up with the same 4 by 8

  • bar but with a leftover piece, apparently created

  • out of thin air. There's a popular animation of this illusion

  • as well. I call it an illusion because

  • it's just that. Fake. In reality,

  • the final bar is a bit smaller. It contains

  • this much less chocolate. Each square along the cut is shorter than it was in

  • the original,

  • but the cut makes it difficult to notice right away. The animation is

  • extra misleading, because it tries to cover up its deception.

  • The lost height of each square is surreptitiously

  • added in while the piece moves to make it hard to notice.

  • I mean, come on, obviously you cannot cut up a chocolate bar

  • and rearrange the pieces into more than you started with.

  • Or can you? One of the strangest

  • theorems in modern mathematics is the Banach-Tarski

  • paradox.

  • It proves that there is, in fact, a way to take an object

  • and separate it into 5

  • different pieces.

  • And then, with those five pieces, simply

  • rearrange them. No stretching required into

  • two exact copies of the original

  • item. Same density, same size,

  • same everything.

  • Seriously. To dive into the mind blow

  • that it is and the way it fundamentally questions math

  • and ourselves, we have to start by asking a few questions.

  • First, what is infinity?

  • A number? I mean, it's nowhere

  • on the number line, but we often say things like

  • there's an infinite "number" of blah-blah-blah.

  • And as far as we know, infinity could be real.

  • The universe may be infinite in size

  • and flat, extending out for ever and ever

  • without end, beyond even the part we can observe

  • or ever hope to observe.

  • That's exactly what infinity is. Not a number

  • per se, but rather a size. The size

  • of something that doesn't end. Infinity is not the biggest

  • number, instead, it is how many numbers

  • there are. But there are different sizes of infinity.

  • The smallest type of infinity is

  • countable infinity. The number of hours

  • in forever. It's also the number of whole numbers that there are,

  • natural number, the numbers we use when counting

  • things, like 1, 2, 3, 4, 5, 6

  • and so on. Sets like these are unending,

  • but they are countable. Countable means that you can count them

  • from one element to any other in a

  • finite amount of time, even if that finite amount of time is longer than you

  • will live

  • or the universe will exist for, it's still finite.

  • Uncountable infinity, on the other hand, is literally

  • bigger. Too big to even count.

  • The number of real numbers that there are,

  • not just whole numbers, but all numbers is

  • uncountably infinite. You literally cannot count

  • even from 0 to 1 in a finite amount of time by naming

  • every real number in between. I mean,

  • where do you even start? Zero,

  • okay. But what comes next? 0.000000...

  • Eventually, we would imagine a 1

  • going somewhere at the end, but there is no end.

  • We could always add another 0. Uncountability

  • makes this set so much larger than the set of all whole numbers

  • that even between 0 and 1, there are more numbers

  • than there are whole numbers on the entire endless number line.

  • Georg Cantor's famous diagonal argument helps

  • illustrate this. Imagine listing every number

  • between zero and one. Since they are uncountable and can't be listed in order,

  • let's imagine randomly generating them forever

  • with no repeats. Each number regenerate can be paired

  • with a whole number. If there's a one to one correspondence between the two,

  • that is if we can match one whole number to each real number

  • on our list, that would mean that countable

  • and uncountable sets are the same size. But we can't do that,

  • even though this list goes on for

  • ever. Forever isn't enough. Watch this.

  • If we go diagonally down our endless list

  • of real numbers and take the first decimal of the first number

  • and the second of the second number, the third of the third and so on

  • and add one to each, subtracting one

  • if it happens to be a nine, we can generate a new

  • real number that is obviously between 0 and 1,

  • but since we've defined it to be different

  • from every number on our endless list and at least one place

  • it's clearly not contained in the list.

  • In other words, we've used up every single whole number,

  • the entire infinity of them and yet we can still

  • come up with more real numbers. Here's something else that is true

  • but counter-intuitive. There are the same number

  • of even numbers as there are even

  • and odd numbers. At first, that sounds ridiculous. Clearly, there are only half

  • as many

  • even numbers as all whole numbers, but that intuition is wrong.

  • The set of all whole numbers is denser but

  • every even number can be matched with a whole number.

  • You will never run out of members either set, so this one to one correspondence

  • shows that both sets are the same size.

  • In other words, infinity divided by two

  • is still infinity.

  • Infinity plus one is also infinity.

  • A good illustration of this is Hilbert's paradox

  • up the Grand Hotel. Imagine a hotel

  • with a countably infinite number of rooms. But now,

  • imagine that there is a person booked into every single room.

  • Seemingly, it's fully booked, right? No.

  • Infinite sets go against common sense.

  • You see, if a new guest shows up and wants a room,

  • all the hotel has to do is move the guest in room number 1

  • to room number 2. And a guest in room 2 to room 3 and 3 to 4 and 4 to

  • 5 and so on.

  • Because the number of rooms is never ending

  • we cannot run out of rooms. Infinity

  • -1 is also infinity again.

  • If one guest leaves the hotel, we can shift

  • every guest the other way. Guest 2 goes to room 1,

  • 3 to 2, 4 to 3 and so on, because we have an

  • infinite amount of guests. That is a never ending supply of them.

  • No room will be left empty. As it turns out,

  • you can subtract any finite number from infinity

  • and still be left with infinity. It doesn't care.

  • It's unending. Banach-Tarski hasn't left our sights yet.

  • All of this is related. We are now ready to move on

  • to shapes. Hilbert's hotel can be applied

  • to a circle. Points around the circumference can be thought of as

  • guests. If we remove one point from the circle

  • that point is gone, right? Infinity tells us

  • it doesn't matter. The circumference of a circle

  • is irrational. It's the radius times 2Pi.

  • So, if we mark off points beginning from the whole,

  • every radius length along the circumference going clockwise

  • we will never land on the same point twice,

  • ever. We can count off each point we mark

  • with a whole number. So this set is never-ending,

  • but countable, just like guests and rooms in Hilbert's hotel.

  • And like those guests, even though one has checked out,

  • we can just shift the rest. Move them

  • counterclockwise and every room will be filled

  • Point 1 moves to fill in the hole, point 2

  • fills in the place where point 1 used to be, 3 fills in 2

  • and so on. Since we have a unending supply of numbered points,

  • no hole will be left unfilled.

  • The missing point is forgotten. We apparently never needed it

  • to be complete. There's one last needo consequence of infinity

  • we should discuss before tackling Banach-Tarski. Ian Stewart

  • famously proposed a brilliant dictionary.

  • One that he called the Hyperwebster. The Hyperwebster

  • lists every single possible word of any length

  • formed from the 26 letters in the English alphabet.

  • It begins with "a," followed by "aa,"

  • then "aaa," then "aaaa."

  • And after an infinite number of those, "ab,"

  • then "aba," then "abaa", "abaaa,"

  • and so on until "z, "za,"

  • "zaa," et cetera, et cetera, until the final entry in

  • infinite sequence of "z"s. Such

  • a dictionary would contain every

  • single word. Every single thought,

  • definition, description, truth, lie, name,

  • story. What happened to Amelia Earhart would be

  • in that dictionary, as well as every single thing that

  • didn't happened to Amelia Earhart.

  • Everything that could be said using our

  • alphabet. Obviously, it would be huge,

  • but the company publishing it might realize that they could take

  • a shortcut. If they put all the words that begin with

  • a in a volume titled "A,"

  • they wouldn't have to print the initial "a." Readers would know to just add the "a,"

  • because it's the "a" volume. By removing the initial

  • "a," the publisher is left with every "a" word

  • sans the first "a," which has surprisingly

  • become every possible word. Just one

  • of the 26 volumes has been decomposed into the entire thing.

  • It is now that we're ready to investigate this video's

  • titular paradox. What if we turned an object,

  • a 3D thing into a Hyperwebster?

  • Could we decompose pieces of it into the whole thing?

  • Yes. The first thing we need to do

  • is give every single point on the surface of the sphere

  • one name and one name only. A good way to do this is to name them after how they

  • can be reached by a given starting point.

  • If we move this starting point across the surface of the sphere

  • in steps that are just the right length, no matter how many times

  • or in what direction we rotate, so long as we never

  • backtrack, it will never wind up in the same place

  • twice. We only need to rotate in four directions to achieve this paradox.

  • Up, down, left and right around

  • two perpendicular axes. We are going to need

  • every single possible sequence that can be made

  • of any finite length out of just these four rotations.

  • That means we will need lef, right,

  • up and down as well as left left,

  • left up, left down, but of course not

  • left right, because, well, that's backtracking. Going left

  • and then right means you're the same as you were before you did anything, so

  • no left rights, no right lefts and no up downs and

  • no down ups. Also notice that I'm writing the rotations in order

  • right to left, so the final rotation

  • is the leftmost letter. That will be important later on.

  • Anyway. A list of all possible sequences of allowed rotations that are finite

  • in lenght is, well,

  • huge. Countably infinite, in fact.

  • But if we apply each one of them to a starting point

  • in green here and then name the point we land on

  • after the sequence that brought us there, we can name

  • a countably infinite set of points on the surface.

  • Let's look at how, say, these four strings on our list would work.

  • Right up left. Okay, rotating the starting point this way takes

  • us here. Let's colour code the point based on the final rotation in its string,

  • in this case it's left and for that we will use

  • purple. Next up down down.

  • That sequence takes us here. We name the point DD

  • and color it blue, since we ended with a down

  • rotation. RDR, that will be this point's name,

  • takes us here. And for a final right rotation,

  • let's use red. Finally, for a sequence that end with

  • up, let's colour code the point orange.

  • Now, if we imagine completing this process for

  • every single sequence, we will have a countably infinite number of points

  • named

  • and color-coded. That's great, but

  • not enough. There are an uncountably

  • infinite number of points on a sphere's surface.

  • But no worries, we can just pick a point we missed.

  • Any point and color it green, making it

  • a new starting point and then run every sequence

  • from here. After doing this to an

  • uncountably infinite number of starting point we will have indeed

  • named and colored every single point on the surface

  • just once. With the exception

  • of poles. Every sequence has two poles of rotation.

  • Locations on the sphere that come back to exactly where they started.

  • For any sequence of right or left

  • rotations, the polls are the north and south poles.

  • The problem with poles like these is that more than one sequence can lead us

  • to them.

  • They can be named more than once and be colored

  • in more than one color. For example, if you follow some other sequence to the

  • north or south pole,

  • any subsequent rights or lefts will

  • be equally valid names. In order to deal with this we're going to just count them out