The Banach–Tarski Paradox


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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