(dramatic music)
-: Is there a mathematical formula for luck?
What about comedy?
Do we live inside a simulation?
I'm Dianna Cowern and this is "Proof of Concept."
In the ancient Greek legends, Zeus, Hades,
and the rest of the Olympian gods are said
to have actually envied humans for their finite lifespans.
This is because despite all of their power and privilege,
the gods' immortality deprive them
of the greatest gift of all, meaning.
Because a human only exists for a brief flash of time,
every choice we make, every meal, every breath,
every moment of laughter is endowed with the meaning
that comes from being an entry on a finite list.
A list that will end up excluding other experiences
and possible paths we might have followed.
Because our time is finite our choices matter,
but considering the profound impact of our finiteness
makes you wonder if there are any ways
in which we are actually infinite.
Is anything?
This image, for instance,
it's an iconic example of an infinite visual effect.
In fact, it's called an infinity room,
but even this isn't truly infinite.
The effect where you see infinite Diannas is the result
of repeated reflections,
light bounces off one wall and then the other
and back and forth over and over.
But technically it only goes so deep.
Think about the first one second
after I walked into this 10 foot cube.
Light moves quickly.
So it's had enough time in that one second
to create 50 million reflections in the image.
But 50 million isn't infinite, not even close.
Not to mention the fact that the light appears to fade
as you look off into the distance.
That's because it does.
For every reflection,
you lose about one to 5% of the light to absorption,
even in the best mirrors.
So that version of me,
30 layers deep is standing in a room
that's about 50% as bright as this room.
That Dianna is 50% as bright as this one.
With all that in mind,
let's kick off our infinity episode with a game,
a game that isn't technically about infinity,
but instead is about really big numbers.
Here's the challenge.
Put this list of numbers in order from smallest to largest,
the number of different ways a 52 card deck can be shuffled.
The number of people currently living on earth,
the number of distinct paths between the most Northeast
and Southwest corners in Manhattan,
without ever retracing your steps.
The number of ants on earth,
the number of stars in our galaxy.
Enough suspense.
Here's the correct order.
The smallest number on the list
is the current human population of planet earth.
Clocking in at approximately seven billion.
Next up is the number of stars in our galaxy.
There are approximately 43 billion or between six
and seven stars for every person on earth.
So go ahead and name one after yourself.
Name six.
There's plenty to go around.
After that is the number of ants on earth.
Estimates place the number at roughly 100 trillion ants.
It is a shocking, but true fact
that the ants on earth actually outweigh the humans.
100 trillion is a big number,
but from here on the number is veer
into an entirely different realm of massive.
The second biggest number on our list is the number of ways
that a standard 52 card deck can be arranged.
This quantity is commonly known as 52 factorial
and is equal to 52 times 51, times 50, times 49, times 48.
And so on, all the way down to one.
Written out in its entirety,
this number is 68 digits long.
To give a sense of just how big this number is,
it's actually larger than the number of seconds
that have passed since the Big Bang 14 billion years ago.
That means that if someone had started shuffling a deck
immediately after the Big Bang
and changed the order of the cards once per second,
from then, until now,
they would still not have had enough time
to run through every possible ordering.
It also means that if you were to pick up a deck right now
and shuffle it,
the most likely result is
that the exact order your deck is in
has never actually happened before, anywhere, ever.
And as huge as 52 factorial is,
it's dwarfed by the final number on our list.
The number of ways to walk from end to end in Manhattan.
The secret here is the concept of choice.
When you're making choices,
the number of ways things can happen gets really huge,
really fast.
It's what's called exponential growth.
Think how internet memes catch fire.
Let's consider the first three blocks you might walk.
All in all, there are 10 different ways you can choose
to walk them.
By the time you've walked 10 blocks,
there are over a thousand paths you might have taken.
Once you've walked a distance of 30 blocks,
there are over one billion possible paths.
Manhattan is about 250 blocks long,
and about 30 blocks wide.
When you consider every choice you might make going
from corner to corner,
the total number of paths you can take turns out
to be over 80 digits long.
That means that it is one trillion times larger
than the number of ways you can shuffle a deck.
It is seriously big in a way
that's hard to begin to imagine.
And yet we live with these sorts of possibilities every day.
Which of the following is an infinite quantity?
A, the number of different possible chess games?
B, the number of different possible songs under five minutes
in length that can be played on a piano?
C, the number of different possible 1000-page novels
written in English?
Or D, the number of different possible Monopoly games?
All of these are huge quantities,
but the rules of chess, the 88 piano keys,
the speed of sound and the approximately 200,000 words
in English, make three of them finite.
While the game of Monopoly can in fact go on forever,
meaning that there is no finite limit
on the number of different ways
a game of Monopoly might unfold.
Speaking of infinity popping up in surprising places.
There's a picture that I wanna show you,
but it hasn't actually been drawn yet,
because the way we're gonna create this picture
is just about as unusual as the subject
of the picture itself.
We're gonna draw it by rolling this die.
Let's go.
This is Jared.
He's a production assistant,
and I'm gonna make him draw dots all day.
Sorry, Jared.
But what you're gonna do is roll this die.
And if you get a one or a two,
you're gonna go from wherever you are on this triangle,
halfway toward A and draw a dot.
And then if you roll a three or a four,
you're gonna go from wherever you are,
halfway toward B and draw a dot.
And if you roll a five or a six,
you're gonna go halfway towards C and draw a dot there.
And you're just gonna keep doing that.
Keep drawing dots as long as you can.
This is gonna make sense, I promise.
So let's try the first one.
You're gonna start right in that center dot.
Roll the die.
And you got a four.
So you're gonna go halfway toward B
and draw your first dot.
Got it?
All right.
Good luck, Jared.
(light upbeat music)
So how big is infinity?
Well, let's consider one specific number
that's infinite in its own special way.
Despite being finite.
I'm speaking of course of pie, 3.14159265 and so on.
And that's all the digits I've ever memorized.
Pie is the ratio of the circumference
of any circle to its diameter.
But what matters for our purposes here is
that its digits go on forever
and they never settle into any predictable pattern.
That combination of infinity
and the lack of pattern has some astounding consequences.
For Shakespeare.
Let's see what I mean.
According to this coding
that lets us translate the digits of pie
from a string of numbers to a string of letters,
the word "proof" would correspond
to the sequence 1618151506.
The phrase "proof of concept" corresponds to this sequence,
which appears in the digits of pie
here around the 89-billionth digit.
So the title of our show is encoded in the digits of pie,
but we're not special.
Humanity has only ever calculated pie
to around 68.3 trillion digits only.
But we do know that pie goes on forever
and every possible finite sequence does occur somewhere.
That means that somewhere in there are
for sure the complete works of William Shakespeare,
as well as a version of Romeo and Juliet,
where they both grow old and have grandchildren
and a word-for-word transcript of this episode.
And the next conversation you are going to have.
Infinity is weird.
And just when you think you might be starting
to wrap your head around it,
you find out there isn't just one infinity,
but in fact,
there are infinitely many different infinities
of different sizes stretching up to--
You get the point.
(upbeat music)
I'm very happy to introduce our special guest today,
Dr. Eugenia Cheng, mathematician, concert pianist,
and author of best sellers,
like "How to Bake Pie" and "Beyond Infinity."
Welcome to the show.
-: Thank you so much for having me.
-: Dr. Chang, what is so cool about Infiniti
that inspired you to write an entire book about it?
-: Well, I just love the fact that small children
can get really excited and compelled thinking
about this idea.
But it took mathematicians thousands of years to figure out
how to deal with it in a rigorous way.
And that huge gap between little kids getting excited
about it and mathematicians pinning it down
is so exciting to me
because it's where we can put ourselves to learn
and discover and dream about all these things
and become more intelligent.
That's what this is about really.
-: Yeah, infinity is so complex.
There are different sizes of infinities.
-: Yeah.
And that is very difficult to get our heads around.
And when we try and get our heads around it,
our brains might feel like they're exploding.
-: Well, to understand how there can be different sizes
of infinity, we have to start by learning how
to compare them.
And to help us figure out how to do that,
we're gonna need a simple but powerful technique
that's at the heart of the following contest, mathematics.
Let's bring in our contestants.
-: I'm Asha and I'm a composer.
-: My name is Jason and I'm an artist and a muralist from LA.
I've never played the counting game before.
And they didn't really explain how to play the game.
They said, we're just gonna figure it out on the spot.
-: I'm Ahmad Dami.
I'm a filmmaker.
I did not know what I was doing.
-: For this competition,
we have tasked our three contestants with a counting race.
So each contestant has a jar of blue objects
and a jar of pink objects.
And they simply have to figure out
which jar has more objects
and they can use any of the tools provided here
on the tables. But the goal is to figure out
which jar has the most objects, the fastest.
So you guys ready?
-: Yeah, definitely.
-: All right.
Three, two, one, go!
(upbeat music)
-: Ooh!
I think I see what she's doing.
Dianna: Yeah?
I love that just balls are flying all over the place.
Oh my goodness.
What is Ahmad doing?
-: Looks like this contestant
might actually be counting the blue ones.
Dianna: Oh man.
Individually counting from the different jars.
Eugenia: That's arduous.
Diana: Yeah.
-: I thought I had a plan,
but thinking it through, it wasn't an actual plan.
It was just an idea.
-: This one is trying to weigh them.
-: I chose to weigh the objects.
'Cause I thought that might be faster
than trying to count each object individually.
I was trying to avoid making a mess
so I can actually get an accurate count of the objects.
-: The challenging thing is that there's a lot of different--
-: Different size.
-: There are a lot of different types
of objects with different weights.
-: So it's not really gonna help.
-: No.
(upbeat music)
-: She's going way faster anyway.
-: She's almost done.
Eugenia: Yeah, right.
-: Before we started the game,
Dianna came to me and told me that we were gonna cheat.
And she told me the strategy.
She told me to take out the same amount of balls
at the same time.
So I was doing two at once and basically until the end
and saw how many balls were left over in each glass jar.
-: Yeah, so Asha has been briefed.
-: I see that.
And were the others told not to,
were they told what to do?
Looks like--
-: No.
No, they weren't told anything.
Which technique to use.
-: You stacked it.
-: It was stacked.
Yeah, it's true.
We're gonna have to profusely apologize to Jason
and Ahmad after this.
-: Blue!
-: Oh, look, she's done already.
-: She's done.
We have a winner.
Eugenia: We have a winner by miles.
Director: What would you think if I told you
that we told one of the contestants how to win going in?
So you were always going to lose?
-: All right.
So you told somebody else the actual solution?
-: If one contestant knew how to play the game
or how to figure it out faster,
then that kind of makes it unfair.
-: I felt really good.
I don't feel bad.
I have no remorse for cheating.
-: So how does a chaotic game like this relate
to infinity in any way?
If you were a contestant, is that what you would've done?
-: It is.
Because I know that trick from math,
which is that you don't actually have to know
how many there are in order to know which has more.
The technique that the winner was using
is an important mathematical technique.
'Cause what she was doing,
she didn't actually have to count how many objects were
in each one because--
-: She's just trying the same number.
-: Right, and so that's a mathematical technique
that we can use on infinity
because you can't tell how big infinity actually is,
'cause it's too big. But what you can do is you can try
and take out the same number.
What you're doing there is you're lining them up in pairs.
You try and match them in pairs.
And if you can match them perfectly,
one to one, on both sides, that's called a bijection.
And if you can do that, then you know they're the same size.
And if you can't do that,
then you know that where there's some leftover,
that one is bigger.
And that's how we show that some infinity
is bigger than others,
because we can't ever match them up with each other.
There's always some left,
which means there's no bijection
and that this infinity must be bigger.
-: So even though every infinity goes onto infinity,
some still have numbers left over.
-: Right.
And this was a huge mind boggling development in math.
It was an argument called the Cantor diagonal argument,
but it was basically this game
and it's a very clever way of avoiding
she having to count things
and just seeing who's got some left over at the end.
It is mind boggling,
but that's the whole point.
We boggle our minds and that's how we stretch our minds
and become more intelligent.
-: So the winning strategy was really
a brilliant mathematical technique.
-: Yes.
And that represented an important development
in the study of infinity in mathematics.
-: Brilliant.
At its core,
the winning strategy in the counting race was all
about attempting to create what is called
a one-to-one correspondence
between the two containers of objects.
So as pairs of pink and blue objects were discarded,
they were associated to each other.
And that process proceeded right up
until it couldn't anymore
because there were no more pink objects to pair
with a remaining blue one.
And while we can never count all the way up to infinity,
we can find ways to create one-to-one correspondences
between infinities.
For instance, this one which shows
that the infinity of all whole numbers is the same
as the infinity of just the even ones.
Or this one, which shows that surprisingly there are
as many whole numbers as there are fractions.
And I'll spare you the details,
but it turns out there's a way to show
that there's no way to create a one-to-one correspondence
between the whole numbers and the full set
of real numbers
or that continuum of numbers you need to fill
in all the gaps between the fractions.
So just like in the counting race,
if you try to pair them up,
you'll always find that there are some numbers left out.
We're talking about problematic ones like pie
and its other irrational cousins like E.
so even though the number of whole numbers is infinite
and the number of numbers on the continuum is infinite,
it turns out that one of these infinities
is actually bigger than the other.
One of these infinities is bigger than the others,
which is it?
A, the number of possible whole numbers?
B, the number of positive whole numbers divisible by 17?
C, the number of different possible circles
that can be drawn?
Or D, the number of seconds between now
and forever from now?
A, B and D are all countable infinities.
But since the area of a circle can in theory be equal
to any irrational number,
there are actually uncountable
many different possible circles that can be drawn.
Speaking of things that go on forever,
let's check in on our dots.
How's it going, Jared?
-: Great.
-: This is actually coming along.
Yeah.
I'm starting to see the patterns.
Do you feel like you're noticing anything?
-: Yeah, definitely a triangular shape.
-: Yeah.
The points on the edge.
Those came from just this rolling process?
-: Yeah.
-: That's so cool.
-: Did not put them there deliberately.
It all happened naturally.
The will of the die.
-: Amazing.
Well, you still have a long way to go.
-: Oh, great.
-: So keep it up.
Good luck.
-: Thank you.
(light upbeat music)
-: Welcome to the "Proof of Concept.
"Infinite Fun with Kyle Hill" segment.
This is Kyle Hill.
-: That's me, baby.
Hello, Diana. Thank you so much for having me.
-: Kyle, it's so great to have you.
-: Hello, my name is Kyle Hill.
I'm a award-winning science communicator
based in Los Angeles, California.
(upbeat music)
-: So excited to have you here.
Before we even get started with the segment,
I wanna play a game with you.
-: Some sort of infinite game?
-: Some sort of infinite game.
-: I'm excited to talk about infinity
because it's such a layered, complex concept.
It can have applications in everything from mathematics
to the number of possible universes
and beyond my excitement for all of that of course,
I love working with my friend, Diana.
(upbeat music)
-: We're gonna play now,
but we're gonna reveal who wins later.
-: Ooh.
-: Okay.
So here's the game, Kyle. We're just gonna get a card.
Kyle: Okay.
-: And we're gonna write the biggest number we can think of.
We're just trying to get a bigger number
than the other person.
-: I don't know, Diana.
I can think of some pretty big numbers.
-: As can I.
-: Well, Diana is the physics girl
and she's very used to big numbers.
Like the number of atoms in the universe
or something like that.
So I expect she's gonna have some large exponent,
but I don't know how large that's gonna be.
Can it go--
Yeah.
-: I don't even know if this makes mathematical sense,
but who cares?
-: I don't even know if this is legible.
-: Why are you writing so much?
-: Biggest number you can think of.
-: Hmm.
In my mind you can write a big number
with just a few numerals.
Well, my strategy,
I know a little bit about mathematics
and big number notations.
So if I do this correctly,
no matter what number Diana gives, I should be victorious.
-: Okay.
You got it?
-: I do. I think I have something spicy.
-: Okay. So go ahead and put it in here.
-: Don't look at it.
-: I'm not looking.
No peeking.
All right, I'm going to put this aside for now, Kyle.
So we'll open those in a bit.
We're about to play some infinite Tic-Tac-Toe, Kyle,
but first I wanna pick your brain about infinity.
-: Yeah.
Get the juices flowing before we play weird games.
-: So Kyle,
what is your go-to example of explaining the concept
of infinity to people?
-: Yeah.
Infinity is inherently a hard concept to even think about
because necessarily it's the most of everything,
but one topic that's particularly salient today
in the public is the idea of infinite universes.
Now, a lot of quantum physicists believe
that the many world's interpretation
of quantum mechanics is the correct one.
That sounds very complicated,
but all it is is every possible interaction,
spawns a new universe, every single possibility happens.
So there's a universe in which I'm speaking Portuguese
to you right now.
There's a universe in which I never existed.
There are an infinite array of possible choices,
and we are only following one of those world lines.
If you want to say it like that.
-: Another universe like ours, but somewhere else.
And another one.
-: Exactly.
-: And another one and on and on to infinity.
-: And it gets so granular that one universe,
I did this with my hands and another universe I did that.
Now, if you extend that out to absolutely everything
that could possibly be from, you know,
someone else won World War II to my hair color is different.
You get an infinite array of universes out there
where you are doing a different thing,
or you may not even exist.
Or I could have hair as nice as yours,
but we don't live in that universe.
-: That's your go-to explanation for infinity?
-: It's a bit long-winded.
-: Do you think that infinity truly exists
physically somehow in our universe?
-: Now I know I kind of just talked
about having an infinite number of universes,
but in terms of just living in our own
and what we can see, what we can observe,
what we can experiment with,
it seems unlikely to me that there's an infinity
within this universe.
And that stems from the fact that
from the very moment you begin to exist,
there are a number of constraints.
There's only a set number of particles in the universe,
as far as we know,
and with all those constraints
on what could possibly take place
inside of our own experience, in our universe,
it seems unlikely to me that there's
a true physically infinite thing happening here.
-: If I tell you, you can count one, two, three, four, five,
but you can't count in between, some constraining,
you can still have infinite numbers.
-: Yes.
Conceptually, I could count as high as I wanted to,
but not before I die.
-: Right.
-: So I can't.
-: There are physical human constraints.
-: Yes.
-: Yeah.
-: While infinity is a mathematical concept
that we know about,
that we can work with, that's useful in mathematics.
I couldn't give this interview an infinite number of times.
-: It's hard to imagine a way
you could physically experience infinity.
-: Yeah. And I don't even know if
that's the right way to frame infinity as a concept.
-: That's the type of explanation in infinity
that people just freeze and say,
"Oh my God, this concept is beyond my comprehension."
-: Yeah.
-: It really is.
-: No, yeah, yeah.
-: But I think most infinities are beyond our comprehension.
-: No, absolutely.
I think a lot of us, when we were kids,
at least maybe learning mathematics for the first time
had that experience with,
I'm plus one better than you, I'm plus two better than you.
And then at some point when we're learning, say like,
well, I'm infinity better than you.
-: Yes.
-: And that seems like some unbeatable concept
or number where the conversation kind of stops.
And I think the fact that so many of us have had
an experience like that,
I think it shows that this as a concept
is inherently interesting.
And then the math nerd was like,
"Well, I'm Infinity plus one, which is technically bigger."
And you're like, "Go, go read a book."
-: The fact that this conversation's happening
on elementary school playgrounds
shows you how it's such a human concept to think of
and to conceptualize infinity,
even when we can't fully grasp it.
Right?
-: Yeah, it's kind of analogous to, you know,
how long humans have been thinking about what zero means,
what nothing means
in the same way at the other end of the spectrum,
what does everything mean?
What does infinity and it's yeah--
It's probably just a product of how our brains evolve.
That it's just fascinating, no matter what.
-: It's so hard for computers
to deal with the concept of infinity.
And yet kids arrive at this on the playground.
-: Yeah.
Computers today still have to use
when they're randomly generating numbers, for example,
they still have to use seeds.
It's not an actually infinite number of things.
Computers can't do that.
And yet the human mind very quickly goes
to what is the biggest, what is everything?
And that's occupied human minds
for as long as we were humans with minds.
So yeah, it's infinitely interesting.
-: It is, it is in fact.
There are conceptually infinite things
about human experiences.
Like there are games that are infinite.
-: Sure.
-: Some that are not,
like there's a finite number of checkers games.
-: Yes.
-: But there's an infinite number of poker games.
-: Yeah, it's interesting.
Once you insert the human element into it
and you have all these outside variables,
then infinity comes into play.
Like there's a set number of cards that you could have,
a set number of hands you could be dealt,
but once you have the human element inserted into it
in decision and choice,
then it can become an infinite array of things.
And that's true of many card games
like "Magic: The Gathering."
Many computer games are infinite in that way as well,
just because human choice is not necessarily rational
or predictable.
-: All right, Kyle.
So speaking of infinite games,
we are going to now play an infinite game of tic-tac-toe.
-: Like, how long is that gonna take?
Because I have lunch with a very important person.
It's not you.
(upbeat music)
-: All right, Kyle, we are going
to play infinite tic-tac-toe.
Obviously this board is not infinite.
-: That's true.
-: But it's big enough for our purposes
because the way you play this game
is you wanna get five X's in a row or O's in a row
instead of the normal three.
-: Hmm.
So there must be some mathematical reason to that.
And this is practically infinite for our purposes,
as you say.
-: It is practically for our purposes
and you can play anywhere you want.
So even if I start here,
you can play over there while I win on this side of course.
-: Don't be so sure, Dianna.
So what do you wanna be?
-: I'll be X's.
-: That makes me O's.
-: You'll be O's.
All right.
Go ahead and start.
-: I'm gonna go first.
I think everyone in their life has played some version
of Tic-Tac-Toe, but I've never played a version like this.
That's so much larger.
That's practically infinite for the game that we're playing.
So I have no strategy going into it.
I hope something comes to me as we start playing.
I'm gonna go right here.
-: Ooh.
All right.
My territory.
All right.
I will go--
I'm gonna go close to you.
Right here.
-: I feel like if I stay there,
it's just like a regular game of Tic-Tac-Toe.
So I'm gonna mix it up a little bit and I'm gonna go here.
Now don't be fooled, dear viewer.
It looks like I'm trying something fancy
with this giant board,
but I have no idea what I'm doing.
Instead what I'm trying to do is
if there's something to a giant Tic-Tac-Toe board,
it must be going outside
the normal three by three box, right?
So I'm just gonna try to distract Dianna over here
while I advance my plans over there.
Okay.
So I guess that's starting to force me to do something,
but I also wanna start laying my own plans.
-: Kind of like normal Tic-Tac-Toe so far,
just a lot more squares to think about.
Gotta get in there, Kyle.
Well, obviously.
-: Yes, but I'm forcing you to waste your moves.
-: It's true.
Are you though?
-: I don't know.
Interesting.
Most interesting.
-: If I didn't go right here.
-: Yes.
-: You would win.
-: That would be--
-: That would be unwise.
-: I mean, this is already more interesting
than normal Tic-Tac-Toe.
Interesting.
-: I made him say, "Hmm,"
he thinks I have a strategy.
-: I don't want you to start doing either of those.
-: I like how this just sort of shot off.
-: I feel like the--
This is my whole strategy.
-: Yeah.
-: Diagonals.
The creeping worry that I have while playing this game
is that because I have no experience with it
and we're kind of just winging it
is that I am completely missing out on something
that should be deducible about this game.
Well, if you have this in this position,
then you just force them to do X or O in that position.
I'm not seeing any of that so I could lose it any moment,
it feels like.
(upbeat music)
-: This is the longest game of Tic-Tac-Toe I've ever played.
-: Oh, I should have put it there.
Dianna!
-: Kyle Hill.
-: Eliminated.
-: Do I get to cross it?
-: Yes, you always get to do the cross if you get it.
Okay.
-: There it is.
Good game, sir.
-: Yeah, I guess.
Well, I think what's fun about games like this is
that you can have a sudden realization
of what the strategy is and how to play it well,
especially if you're playing it for the first time.
So the fact that Diana suddenly realized that
in a certain orientation of her X's,
basically her victory was assured.
So even though losing doesn't feel great,
especially to a colleague of mine
who I very much wanted to embarrass on this show,
I think it's fine.
We learned something.
-: The rules of this game demonstrate the balance
that exists between the finite and the infinite.
Two finite humans playing a potentially infinite game
where boundless possibility is constrained
by just a bit of strategy.
-: Well, that was certainly interesting, wasn't it, Diana?
-: That was fun.
That was fun, Kyle.
-: I can confidently say I've never done anything
like that before, and I'm still confused.
-: You'll have to keep practicing.
But we have one more thing to do.
-: Oh, that's right!
-: We have to reveal who won the infinite game.
-: Yeah.
After all this thinking about big numbers
and infinity and playing around with it,
I wonder where our heads were at
when we started this conversation.
-: Yeah, okay.
You hold my card and I hold your card.
And then on three, two, one, we flip it over.
All right.
Three, two, one, go.
-: So Diana wrote 99 to the 99 to the 99 to--
So I think this is the biggest,
I mean, this is bigger than a Google.
This is bigger than anything.
What I wrote was the subset of all Diana's numbers
that she gives plus one.
-: He did indeed.
He did indeed say my number.
-: I tried to mitigate.
-: Plus one.
This is like the playground game, Kyle.
This is the classic playground trick
Kyle played on me and I never learn.
-: I know some nerdy viewers of you
at home might disagree with my actual notation
of that number, but the idea was there.
And when you're talking about infinity, the idea,
the concept is what's important.
I feel very good about beating Diana in this game
because I lost in infinite Tic-Tac-Toe,
which was extremely embarrassing.
As you know, my entire brand is related
to being a Tic-Tac-Toe champion.
And so this felt good.
-: What would've happened if you had done mine plus one
and I had done yours plus one.
We would've broken the game.
-: We would've never left these seats.
You know, even though I didn't win the main game
of this episode,
if true infinity exists somewhere, if there are,
for example, an infinite number of possible universes,
then there is a universe in which I both won that game
and I'm the host of the show.
Thanks for watching.
Does that--
That doesn't work?
Okay.
That's cool.
-: So far we've been talking about the infinitely big,
but it's just as important to consider the infinitely small.
The Greek philosopher Zeno noticed a long time ago
that infinity can be found in surprisingly mundane places.
And in many cases it can't be avoided.
For example,
he pointed out that even when doing something
as simple as trying to walk from one spot to another,
we have an infinite number of things we have to do
before we can get there.
So before I can get from over here to the other side
of the room, I have to of course, walk halfway there.
And now that I'm here, I have to get halfway again.
And now that I'm here,
I have to get halfway from this spot to the end of the room.
But those first three halfways are only three items
on an infinite to-do list
that includes getting halfway there over and over,
infinitely many times.
But somehow we make it.
How is that possible?
The details of why we make it have something
to do with a calculus concept called limits
and infinite series that basically show
how an infinite to-do list like this one gets done
in a finite amount of time.
As long as the steps take less and less time
and get shorter quickly enough.
Well, that's one solution.
Some people think that sort of explanation
isn't even required
because they endorse theories of physics,
like certain formulations of quantum gravity,
where space and time are discreet.
In other words,
those theories propose a universe where space time
comes in tiny indivisible units
beyond which no meaningful notion
of when and where can even be defined.
This vision of the universe doesn't allow
for Zenos' paradox.
Since you can't keep cutting the distance
in half indefinitely.
You'll eventually hit the basic unit of length
that can't be divided.
But while it solves one paradox,
it potentially introduces another that's even weirder.
Pay close attention,
because I'm about to prove that in a discrete universe,
assuming you can't skip over spots and teleport,
either nothing moves at all or everything would move
at exactly the same speed.
Check this out.
If L is the smallest unit of length
and T is the smallest unit of time,
consider the following speed, L over T.
That is the speed of moving one unit of length
per one unit of time.
If something were to move faster than one L per one T,
then the length of time it would take to move one L
would be less than one T, which is impossible.
Since we said one T is the smallest possible amount of time.
But then if something were to move slower
than one L per one T,
then the distance it would travel in one T would be less
than one L, which is impossible.
Since we agreed one L
is the shortest possible amount of distance.
Therefore everything that moves would move at the same speed
L over T.
One L per one T.
Kind of trippy, right?
So in a quantized version of the universe like this,
everything would move at the same speed.
And as weird as that seems,
we can get even weirder by asking the following question,
is it possible everything's not only moving
at the same speed,
but perhaps everything's also moving the same direction?
Might not be quite as crazy as it sounds.
One of the basic rules you might remember
from geometry class says that parallel lines never cross.
This rule, called the parallel postulate
allows for a little wiggle room.
We can change it slightly to say
that parallel lines do cross,
but they only do so at a special point,
that happens to be infinitely far away.
Adding these points at infinity and saying
that infinitely many parallel lines can meet there,
gives you what's called non-euclidean geometry.
And because in real life,
we can't ever reach those points at infinity,
we aren't entirely sure what sort
of geometry best describes the universe that we live in.
Stay with me now, because this is the extra cool part.
According to Einstein's theory of relativity,
light waves don't experience time the same way
that matter does. In fact, you might even say
that light doesn't experience time at all.
So if one day in the unimaginably distant future,
there comes a time when the last bit of matter reverts back
to being energy in the form of electromagnetic waves
or light,
there would be nothing left in the universe
that experiences time, at least not in the usual way.
At that moment, if the shape of reality is non-euclidean
all of the light would simultaneously meet up
at that point at infinity.
But what's a moment if there's no time
and how can things be simultaneous if there's no time?
Anyway, a whole universe worth of energy,
interacting at the edge of space and time
where being as far away from each other as possible
suddenly loops back into being
as close together as possible.
Something like that could be the perfect recipe
for a brand new big bang,
creating a cycle of infinities inside of infinities.
Like I said, unproven and hotly debated,
but definitely cool to think about, which reminds me--
(light upbeat music)
Let's take a closer look at one example
of the sort of counterintuitive things
that can happen when we try to calculate with infinity.
Consider the number 0.9999
where the nines go on forever and ever.
Surprisingly, it turns out
that this number is actually equal to one.
I'm not just saying that it's very close to one.
It actually is one.
Here's a simple proof of this fact using basic algebra.
Say that 0.999 equals X,
then 10X equals 10 times 0.999 or 9.999.
So 9X equals 10X minus X equals 9.999 minus 0.999
equals nine.
So 9X equals nine.
And lastly 9X equals nine
of course means that X equals one.
But we started out by saying that 0.999 equals X.
So 0.999 equals one.
That's a proof.
Upon close inspection we see that infinity
is actually playing a subtle but fundamental role here
because the algebra only works out this way
if there really are infinitely many nines.
For instance, if instead of infinitely many nines,
there were just 10 nines then,
where the rest is just zeros forever.
So 9X equals 10X minus X equals--
So the proof can't continue in the same way.
Now that was a 100% legit math, but watch out
because fooling around
with infinite calculations can get tricky real quick.
And if you aren't careful,
you might end up thinking you've proven something,
which isn't true at all.
Like this old trick,
which looks like a proof that one equals zero.
But of course it isn't.
One equals zero?
Turns out infinity is the problem here.
That sum is an example of what's known
as a divergent series and taken on into infinity,
can't be manipulated using normal algebra.
Sometimes infinity is just a bit too much to handle.
(light upbeat music)
Oh my gosh, Jared, this looks amazing.
I think that you've done enough dots.
I think you're done at this point.
How does it feel?
-: Satisfying to see it goes so far.
It actually looks really good.
-: It looks amazing.
You really can see all of the triangles popping out
of the pattern.
It's very obvious that you followed the directions
because you would not have gotten this pattern
if you didn't draw like this using the die.
-: Of course.
-: Yeah.
Amazing.
So you may or may not be familiar with this picture,
but this is an approximation of something known
as the SierpiÅski triangle.
And I say it just an approximation
because the real SierpiÅski triangle
is a theoretical object.
That's technically impossible to draw
as it would require following this process
or something like it for infinitely many steps.
So you'd be here for forever.
That being said, with enough time on your hands,
this process would yield the image
to whatever level of complexity
or resolution you're looking for.
The process used to draw this image is known
as fractal encryption.
It's an alternative method of coding information
that rather than saving an entire image pixel by pixel,
instead saves a rule or a process that a computer
or you can follow to recreate the image from scratch.
The technique isn't just for images.
In theory, it could be applied
to almost any sort of information.
Using this method, something like the SierpiÅski triangle,
which is technically too large of a concept
to ever save as a traditional photo
can be recreated from scratch by an algorithm
to whatever level of resolution is required
and is saved in a single line of code.
It's a lot smaller.
This method has been used to make image storage
way more efficient for things like rendering 3D graphics
for video games and digital sound effects.
But most compelling of all with this concept
are the early indications
from cutting edge neuroscience research
that our brains may be using something analogous
to fractal encryption to store memories,
allowing for some combination of saved information
and then algorithmic processes
to be followed in our minds that come together
to recreate memories.
whenever we need to recall them,
rather than pulling up a saved file,
which would be more akin to looking at a photo
or watching an old video.
Which of the following geometrical oddities actually exists
in theoretical math?
A, a snowflake with a finite area
enclosed by an infinite boundary?
B, a funnel that can only
hold a finite amount of liquid,
but would take infinite paint to color?
C, a sphere that can be cut into pieces
and rearranged into two spheres of exactly the same size?
Or D, all of the above?
The Koch Snowflake, Gabriel's Horn
and the Banach-Tarski paradox are just a few
of the strange things that can happen
when infinity runs wild.
If our minds are in fact using a form of fractal encryption,
that would mean that we carry a version of infinity around
inside of ourselves,
that our capacity for creativity
and invention may in fact be unlimited.
But it makes you wonder which infinity are we?
The first one or one of the really big ones?
This has been "Proof of Concept."
Thanks for watching.
(upbeat music)