The Banach–Tarski Paradox

Hello, Vsauce. Miguel here. There is a famous way to create chocolate seemingly from 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 this and end up with the same 4 by 8 bar but with a leftover piece, apparently created. of nothing. There is also a popular animation of this illusion. I call it illusion because it is just that. Fake. The final bar is actually a little smaller. It contains much less chocolate. Each square along the cut is shorter than in the original, but the cut makes it hard to notice right away.

The animation is more deceptive, as it attempts to cover up its deception. The lost height of each square is surreptitiously added as the piece moves to make it difficult to notice. I mean, come on, you obviously can't cut up a chocolate bar and rearrange the pieces into more than you started with. Or you can? One of the strangest theorems in modern mathematics is the Banach-Tarski

paradox

. It shows that there is, in fact, a way to take an object and separate it into 5 different parts. And then with those five pieces, just rearrange them. It is not necessary to stretch it to obtain two exact copies of the original item.

Same density, same size, everything the same. Oh really. To delve into what a mind-blowing hit it is and the way it fundamentally questions mathematics and ourselves, we have to start by asking some questions. First, what is infinity? A number? I mean, it's nowhere on the number line, but we often say things like there are an infinite "number" of blah blah blah. And for all we know, infinity could be real. The universe may be infinite in size and flat, stretching for centuries and centuries without end, even beyond the part we can observe or hope to observe. That is exactly what infinity is.

It is not a number per se, but rather a size. The size of something that does not end. Infinity is not the largest number, but 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 an eternity. It is also the number of whole numbers there are, the natural number, the numbers we use when counting things, like 1, 2, 3, 4, 5, 6, etc. Sets like these are endless, 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 last, it is still finite.

The uncountable infinity, on the other hand, is literally larger. Too big to even count. The number of real numbers that exist, not just integers, but all numbers, is uncountably infinite. You literally cannot count even from 0 to 1 in a finite period of time by naming every real number in between. I mean, where to 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 integers that even between 0 and 1, there are more numbers than integers on the entire infinite number line.

Georg Cantor's famous diagonal argument helps illustrate this. Imagine listing all the numbers between zero and one. Since they are uncountable and cannot be listed in order, let's imagine generating them randomly forever without repetitions. Each regenerated number can be paired with an integer. If there is a one-to-one correspondence between the two, that is, if we can match an integer to every real number in our list, that would mean that the countable and uncountable sets have the same size. But we can't do that, although this list goes on forever. Forever is not enough. See this. If we go diagonally through our endless list of real numbers and take the first decimal of the first number and the second of the second, the third of the third and so on and add one to each, subtracting one if it turns out to be a nine, we can generate a new real number that is obviously between 0 and 1, but as we have defined it to be different from every number in our endless list and at least one place is clearly not contained in the list.

In other words, we have used all the integers, all the infinity of them, and we can still get more real numbers. Here is something else that is true but counterintuitive. There are the same number of even numbers as there are even and odd numbers. At first it sounds ridiculous. Clearly, there are only half as many even numbers as all the integers, but that intuition is wrong. The set of all integers is denser, but each even number can be related to an integer. You will never run out of members in 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 infinite. Infinity plus one is also infinite. A good example of this is Hilbert's

paradox

in the Grand Hotel. Imagine a hotel with an infinitely countable number of rooms. But now imagine that there is one person reserved in each room. Apparently it's full, 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 from room number 1 to room number 2. And a guest from room 2 to room 3 and from 3 to 4 and from 4 to 5. etc.

Since the number of rooms is endless, we cannot run out of rooms. Infinity -1 also becomes infinite again. If a guest leaves the hotel, we can transfer all guests to the other side. Guest 2 goes to room 1, guest 3 to room 2, guest 4 to room 3, and so on, because we have an infinite number of guests. It's an endless supply of them. No room will be left empty. It turns out that you can subtract any finite number from infinity and still be left with infinity. He does not care. It's endless. Banach-Tarski has not yet left our sights. All of this is related.

Now we are ready to move on to shapes. Hilbert's hotel can be applied to a circle. The points around the circumference can be considered guests. If we remove a point from the circle, that point disappears, right? Infinity tells us that it doesn't matter. The circumference of a circle is irrational. It is the radius multiplied by 2Pi. So if we mark points starting from the whole, each radius length along the circumference clockwise we will never land on the same point twice. We can count each point we mark with a whole number. So this set is endless, but countable, just like the guests and rooms in Hilbert's hotel.

And just like those guests, even if one has left, we can move the rest. Move them counterclockwise and all the rooms will fill. Point 1 moves to fill the hole, point 2 fills the place where point 1 used to be, 3 fills 2, and so on. Since we have an endless supply of numbered points, there won't be any gaps left unfilled. The missing point is forgotten. Apparently we never needed it to be complete. There is one last necessary consequence of infinity that we should discuss before addressing Banach-Tarski. Ian Stewart proposed a brilliant dictionary. One he called Hyperwebster. Hyperwebster lists all possible words of any length made up of the 26 letters of the English alphabet.

It starts 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", etcetera, etcetera, until the end. entry in an infinite sequence of "z." Such a dictionary would contain every word, every thought, definition, description, truth, lie, name, story would be in that dictionary, as well as every thing. only thing that didn't happen to Amelia Earhart. Everything that could be said using our alphabet. Obviously, it would be huge, but the company that publishes it could realize that they could take a shortcut if they put all the words that start with a in. a volume titled "A", they would not have to print the initial "a".

Readers would know to simply add the "a", because it is the "a" volume. The editor keeps every "a" word except the first one. "a", which surprisingly has become every possible word. Only one of the 26 volumes has been broken down into everything. Now we are ready to investigate this video. titular paradox. What would happen if we turned an object, something 3D, into a Hyperwebster? Could we decompose parts of it into the whole? Yes. The first thing we need to do is give each point on the surface of the sphere one name and one name only. A good way to do this is to name them based on how they can be reached from a given starting point.

If we move this starting point across the surface of the sphere in steps of just the right length, no matter how many times or which direction we turn, as long as we never go back, it will never end up in the same place twice. . We only need to turn in four directions to achieve this paradox. Up, down, left and right around two perpendicular axes. We will need all possible sequences that can be formed from any finite length from these four rotations. That means we'll need left, right, up and down, as well as left left, left up, left down, but of course not left right, because, well, that's going backwards.

Going left and then right means you are the same as you were before you did something, so there is no left right, no right left, no up, down or down. Also notice that I'm writing the rotations in order from right to left, so the final rotation is the leftmost letter. That will be important later. Anyway. A list of all possible sequences of allowed rotations that have a finite length is, well, huge. In fact, countably infinite. But if we apply each of them to a starting point in green here and then name the point we land on based on the sequence that got us there, we can name an infinitely countable set of points on the surface.

Let's see how, say, these four chains on our list would work. From right to left. Well, rotating the starting point like this gets us here. Let's color code the point according to the final rotation in its chain, in this case it is counterclockwise and for that we will use purple. Then up, down, down. That sequence takes us here. We call the point DD and color it blue, since we end up with a downward rotation. RDR, as this point will be called, takes us here. And for a final rotation to the right, let's use red. Finally, for a sequence that ends with up, let's color code the point orange.

Now, if we imagine completing this process for each sequence, we will have a countably infinite number of named and color-coded points. That's great, but not enough. There are an uncountable number of points on the surface of a sphere. But don't worry, we can pick a point we missed. Any point and color it green, making it a new starting point and then run each sequence from here. After doing this on an infinite and uncountable number of starting points, we will have named and colored each point on the surface only once. Except for the poles. Each sequence has two rotation poles.

Locations on the sphere returning exactly to where they started. For any sequence of clockwise or counterclockwise rotations, the surveys 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 colored in more than one color. For example, if you follow some other sequence to the north or south pole, any subsequent right or left will be equally valid names. To fix this, we'll simply count them out of the normal scheme and color them all yellow. Each sequence has two, so there is a countably infinite number of them.

Now, with each point on the sphere having a single name and one of six colors, we are ready to disassemble the entire sphere. Each point on the surface corresponds to a single line of points below it to the center point. And we will drag the line of each point along with it. We will leave the only central point aside. Well, first we crop and extract all the yellow poles, the green starting points, the orange up points, the blue down points, and the red and purple left and right points. That's the whole sphere. With just these pieces you could build everything.

But look at the piece on the left. It is defined as a piece composed of each point, which is accessed through a sequence that ends with a turn to the left. If we rotate this piece to the right, it's the same as adding an "R" to the name of each point. But the left and then the right are a setback, they cancel each other out. And see what happens when you reduce them. The set becomes the same as aset of all points with names ending in L, but also U, D and every point reached without rotation.

That's the full set of starting points. We have turned less than a quarter of the sphere into almost three quarters just by rotating it. We don't add anything. It's like Hyperwebster. If we had the correct piece and the rotation poles and the center point, well, we would have the entire sphere again, but with leftover things. To make a second copy, let's rotate the top piece down. The down-ups are canceled because, well, it's the same as going nowhere and we are left with a set of all the starting points, the entire up piece, the right piece and the left piece, but there is a problem here .

We don't need this additional set of starting points. We haven't used the originals yet. Don't worry, let's start again. We can simply move everything from the top piece which becomes a starting point when turned down. That is, each point whose final rotation is above. Let's put them in the piece. Of course, after rotating the points called UU will simply become points called U, and that would give us a copy here and here. So as it turns out, we need to move all the points with any name that is just a string of Us. We'll place them on the bottom piece and rotate the top piece down, making it congruent with the top right and left pieces, add the bottom piece along with some top and the starting point piece and, well, We are almost done. .

In this copy the rotation poles and the center are missing, but don't worry. There are an infinitely countable number of holes, where the rotation poles used to be, which means that there is some pole around which we can rotate this sphere so that each pole hole orbits without colliding with another. Well, this is just a bunch of circles with one point missing. We filled each one as we did before. And we do the same for the center point. Imagine a circle containing it inside the sphere and just fill it in from infinity and look what we've done.

We have taken a sphere and converted it into two identical spheres without adding anything. One plus one equals 1. It took a while to do it, but the implications are huge. And mathematicians, scientists and philosophers still debate them. Could such a process occur in the real world? I mean, it can happen mathematically and mathematics allows us to abstractly predict and describe many things in the real world with astonishing precision, but does the Banach-Tarski paradox take it too far? Is it a place where mathematics and physics are separated? We do not know it yet. History is full of examples of mathematical concepts developed in the abstract that we didn't think would ever be applied to the real world for years, decades, centuries, until science finally caught on and realized they were totally applicable and useful.

The Banach-Tarski paradox could happen in our real world, the only catch of course is that the five pieces you cut your object into are not simple shapes. They must be infinitely complex and detailed. That's not possible to do in the real world, where measurements can only be small and there is only a finite amount of time to do something, but mathematics says it is theoretically valid and some scientists think it may also be physically valid. Several papers have been published suggesting a link between Banach-Tarski and the way small subatomic particles can collide at high energies and become more particles than we started with.

We are finite creatures. Our lives are small and scientifically can only consider a small part of reality. What is common to us is only a small part of what is available. We can only see a limited part of the electromagnetic spectrum. We can only delve so far into the expanses of space. Common sense applies to what we can access. But common sense is just that. Common. If what we want is total meaning, we should be prepared to accept that we should not call infinity rare or strange. The results we have reached by accepting it are valid, true within the system we use to understand, measure, predict and order the universe.

Perhaps the system still needs refinement, but in the end, the story continues to show us that the universe is not strange. Are. And as always, thanks for watching. Finally, as always, the description is full of links to learn more. There are also a series of books linked in there that really helped me understand Banach-Tarski a bit. First up, "The Pea and the Sun" by Leonard Wapner. This book is fantastic and is filled with many of the preliminaries needed to understand the test that comes next. He also talks a lot about the ramifications of what Banach-Tarski and his theorem could mean for mathematics.

Also, if you want to talk about mathematics and whether it has been discovered or invented, whether it will actually map out in the universe, Yanofsky's "The Outer Limits of Reason" is great. This is my favorite book I've read this entire year. Another good book is "Why Beliefs Matter" by E. Brian Davies. This is actually Corn's favorite book, as you might be able to see there. It's delicious and filled with a lot of great information about the limits of what we can know and what science is and what mathematics is. If you love infinity and math, I can't recommend Matt Parker's "Things to Do and Do in the Fourth Dimension" more.

It's hilarious and this book is very, very good at explaining some incredible things. So read on, and if you're looking for something to watch, I hope you've already seen Kevin Lieber's film about picnic. I already made a documentary about Whittier, Alaska, there. Kevin has a fantastic short film about posting things on the internet and how people react to them. There's a rumor that Jake Roper might be doing something soon at Field Day. So check out mine, check out Kevin's, and subscribe to Field Day for the next Jake Roper action, okay? In fact, he's in this room right now.

Say hello, Jake. Hello. Thanks for filming this, by the way. Guys, I really appreciate who you all are. And as always, thanks for watching.