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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

• 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

• of the