The Banach–Tarski Paradox

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

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

of the

normal scheme and color them all yellow. Every sequence has two,

so there are a countably infinite amount

of them. Now, with every point on the sphere given just

one name and just one of six colors, we are ready to take the entire sphere

apart. Every point on the surface corresponds to a unique line of points

below it

all the way to the center point. And we will be dragging

every point's line along with it. The lone center point

we will set aside. Okay, first we cut out 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 entire sphere. With just

these pieces you could build the whole thing. But take a look at the left piece.

It is defined by being a piece composed of

every point, accessed via a sequence ending

with a left rotation. If we rotate this piece

right, that's the same as adding an "R" to

every point's name. But left and then right

is a backtrack, they cancel each other out. And look what happens when you

reduce

them away. The set becomes the same

as a set of all points with names that end with L,

but also U, D and every point reached

with no rotation. That's the full set of starting points.

We have turned less than a quarter of the sphere into nearly three-quarters

just by rotating it. We added nothing. It's like the Hyperwebster. If we had the right

piece

and the poles of rotation and the center point, well, we've got the entire sphere

again, but with stuff left over. To make a second copy,

let's rotate the up piece down. The

down ups cancel because, well, it's the same as going nowhere

and we're left with a set of all starting points, the entire

up piece, the right piece and the left piece, but there's a problem here.

We don't need this extra set of starting points. We still haven't

used the original ones. No worries, let's just start over.

We can just move everything from the up piece

that turns into a starting point when rotated down.

That means every point whose final rotation is up. Let's put them

in the piece. Of course, after rotating points named

UU will just turn into points named U, and that would give us a copy here

and here. So, as it turns out, we need to move

all points with any name that is just a string of Us.

We will put them in the down piece and rotate the up

piece down, which makes it congruent to the up right

and left pieces, add in the down piece along with some up

and the starting point piece and, well, we're almost done.

The poles of rotation and center are missing

from this copy, but no worries. There's a countably

infinite number of holes, where the poles of rotations used to be,

which means there is some pole around which we can rotate this sphere such that

every pole hole orbits around without hitting another.

Well, this is just a bunch of circles with one point missing.

We fill them each like we did earlier. And we do the same for the centerpoint.

Imagine a circle that contains it inside the sphere

and just fill in from infinity and look what we've done.

We have taken one sphere and turned it into two identical spheres

without adding anything.

One plus one equals 1. That took

a while to go through, but the implications are huge.

And mathematicians, scientists and philosophers are still debating them.

Could such a process happen in the real world?

I mean, it can happen mathematically and math allows us to abstractly predict and

describe a lot of things in the real world

with amazing accuracy, but does the Banach-Tarski paradox

take it too far? Is it a place where math and physics

separate? We still don't know.

History is full of examples of mathematical concepts developed in the

abstract

that we did not think would ever apply to the real world

for years, decades, centuries, until eventually science caught up and

realized they were totally applicable

and useful. The Banach-Tarski paradox could actually happen in our real-world,

the only catch of course is that the five pieces you cut your object into

aren't simple shapes. They must be infinitely complex

and detailed. That's not possible to do in the real world, where measurements can

only get so small

and there's only a finite amount of time to do anything, but math says it's

theoretically valid and some scientists

think it may be physically valid too.

There have been a number of papers published suggesting

a link between by Banach-Tarski

and the way tiny tiny sub-atomic particles

can collide at high energies and turn into more particles

than we began with.

We are finite creatures. Our lives

are small and can only scientifically consider a small part of

reality. What's common for us is just

a sliver of what's available. We can only see so much of the electromagnetic

spectrum.

We can only delve so deep into extensions of space.

Common sense applies to that which we can

access. But common sense is just that.

Common. If total sense

is what we want, we should be prepared to accept that we shouldn't call infinity

weird or strange.

The results we've arrived at by accepting it are valid,

true within the system we use to understand, measure, predict and order the

universe.

Perhaps the system still needs perfecting, but at the end of day,

history continues to show us that the universe isn't strange.

We are.

And as always,

thanks for watching.

Finally, as always, the description is full of links to learn more.

There are also a number of books linked down there that really helped me

wrap my mind kinda around Banach-Tarski.

First of all, Leonard Wapner's "The Pea and the Sun." This book is fantastic and it's full of

lot of the preliminaries needed

to understand the proof that comes later. He also talks a lot about the

ramifications

of what Banach-Tarski and their theorem might mean for mathematics.

Also, if you wanna talk about math and whether it's discovered or invented,

whether

it really truly will map onto the universe, Yanofsky's

"The Outer Limits of Reason" is great.

This is the favorite book of mine that I've read this entire year. Another good one is E.

Brian Davies'

"Why Beliefs Matter." This is actually Corn's favorite book,

as you might be able to see there. It's delicious and full of lots 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 cannot more highly recommend Matt Parker's

"Things to Make and Do in the Fourth Dimension." He's hilarious and this book

is

very very great at explaining some pretty awesome things.

So keep reading, and if you're looking for something to watch,

I hope you've already watched Kevin Lieber's film on

Field Day. I already did a documentary about Whittier, Alaska over there.

Kevin's got a great short film about putting things out on the Internet

and having people react to them. There's a rumor that Jake Roper might be doing

something on Field Day soon.

So check out mine, check out Kevin's and

subscribe to Field Day for upcoming Jake Roper action, yeah?

He's actually in this room right now, say hi, Jake. [Jake:] Hi. Thanks for filming this, by

the way.

Guys, I really appreciate who you all are.

And as always,

thanks for watching.

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