Hi, I'm Sean Carroll.

I'm a theoretical physicist

here at the California Institute of Technology.

I've been challenged to explain dimensions

to five different levels.

The idea of a dimension, sometimes in pop culture

is misunderstood, like there's an extra place

you can go, a mystical dimension or something like that.

To a physicist or a mathematician,

a dimension is just the direction you can go in.

Up down, left right, forward backward.

To you and me, we think there's three dimensions around us,

but then physicists start talking about extra dimensions.

How can you hide them?

Where might they be?

I'm hopeful that we're learn something at each level.

[down-tempo music]

We're gonna talk about some science.

Do you like science?

Yes, a lot.

Oh, very good.

You've come to the right place.

So we're gonna think about physics.

Have you heard the word physics before?

Do you know what that is?

Yeah, kind of.

What's your idea what physics is?

Um, I'm not so sure.

Okay.

I just think of physics as the study of everything.

What stuff is, what stuff does.

So today we're gonna talk about space,

and in particular the idea of dimensions.

Have you heard about dimensions?

At the camp

I made a 3D printing one.

3D printing, good.

So I don't choose the size,

all I had to do was the shape.

But do you know what 3D means?

It's three dimensional.

Three dimensional, as opposed to,

what is ordinary printing?

So, ordinary printing would be 2D.

What do you say when something is one dimensional?

What's an example of something that's one dimensional?

Hmm, I think one dimensional might be a circle, I guess,

or maybe a line.

A line is the perfect example

Yeah, a line.

Because it's one thing that's straight, right?

[Hank] Yeah.

So here's some toys.

We're gonna build some dimensions, right?

So what would you say about this?

That's one dimensional.

Exactly.

It's not really one dimensional, right?

So everything has to be one or two dimensional

before it's three dimensional.

And how would you find yourself,

like if someone said where are you,

could you use some words or ideas

to say where you are on that line?

I think I would be maybe there,

since I'm facing it.

But here's what I want you to think about.

If I say I'm at this point on the line,

I could translate that into saying

I'm at the three centimeter point,

if I were here I'd be at the four centimeter point,

the five centimeter point, right?

Yeah.

So every point, every locations on our little line--

Has its own unit.

Has its own unit, has one number.

Yeah.

We need one number to tell you where we are.

That's one dimension.

That's what it means to be one dimensional.

I only need to tell you one number

to figure out where we are.

Unlike three dimensional, you have to tell a lot

'cause if it's like a sphere, you kind of have

to start using points.

There you go. Exactly.

We're gonna build a little two dimensional space.

You wanna do it?

You wanna do the honors here?

Why don't you put those two lines together?

If you make it

two dimensional is this, a corner.

Exactly, there you go.

Another way is if you have space in between

is an angle.

I think you should be in this chair

and you should be explaining this to me.

You're much better at this than I am.

Yeah.

So those are the dimensions.

That's how we think about dimensions.

Remember, we just needed one number

to find ourselves on the line, we need two numbers

to find ourselves on the plane.

I think that would be an X or a Y axis, so--

There you go.

So, do you think we could have more than three dimensions?

3D is the maximum of dimensions for shapes.

Well, as far as we know.

Yeah.

This is why physicists think about things

we don't know about.

We're wondering whether there could be extra dimensions

you've never seen. Yeah.

That are tinier than atoms.

So, okay.

So what have you learned?

What do you know about dimensions now?

How do you think about dimensions

in a slightly different way than you did before?

So at least everything has a certain dimension.

Yeah, do you think you'd be excited

if physicists said that they found

extra dimensions of space?

That would be an amazing discovery.

The news would be spreaded around the world rapidly.

I think so.

I think you're right.

All right Hank, we want you to keep up studying,

learn a lot of math and physics

and help us discover new dimensions some day.

Does that sound like a fun idea?

Yeah.

[playful music]

Do you like science?

Is that something you think about?

Yeah, I do.

[Sean] What kind of science?

I like biology and computer science.

All right, you're in the wrong place.

[laughs] We're not gonna be talking

about biology or computer science.

So we wanna talk about the idea of dimensions.

Do you know what dimension is?

How we would define dimensions?

I guess, I don't know how to exactly define it,

but I know, like the first four.

Right, you know the difference between like one dimension

two dimensions, three dimensions, et cetera.

So let's do a little experiment here.

So there's one dimension.

I'll give that to you.

Now, here's your task.

I'm going to give you another dimension,

and I'm gonna ask hold those two things

at right angles to each other.

It's easy to do.

Yeah, so there's no trick here.

[laughing] Yeah.

I'm not trying to fool you here. Okay.

Now this is going to be slightly trickier.

I'm going to give you this.

I want you to hold all three of them

at right angles to all the others

at the same time.

Um ...

There you go.

So what that's doing is when you had just two

that was describing a two dimensional plane, right?

Like the two things pick out a plane,

the three things pick out all three dimensional space.

I'm gonna give you one more,

and I'm gonna ask you to hold that fourth one

so that it's at a right angle to all the other three

at the same time.

Um ...

All right, now I am tricking you.

It can't be done.

Yeah. Right?

You can't do it.

So we just experimentally proved

that space is three dimensional.

That's sort of what it means to be three dimensional,

that there are three different directions you can move in

and there's not four or five or six directions

you can move in.

Okay.

There you go.

Three dimensional space, right?

Mm-hmm.

[Sean] So, have you thought about using coordinates

in three dimensions?

Yeah, I was actually doing SAT prep.

There you go.

[laughing]

It showed like the X Y and Z axis as well.

That's right.

So that's exactly what these would be.

Have you heard that there are other coordinate systems

other than XYX?

No.

Well, we could also say how far we are

away from the center, just the distance.

And then the angle that our little line makes

with let's say the X axis.

Yeah.

So that's a different way of giving you two numbers

and locating yourself, and we call those polar coordinates.

It's a different coordinate system.

What we want to do as physicists is look

for extra dimensions.

Can you imagine, can you think of any way

that there could be extra dimensions?

Time?

Time.

Yes, Einstein said that we can think of time

as a fourth dimension, and that's a very fascinating thing

that we could talk about all by itself.

But what about space?

What about the solar system?

Like, if you wanted to tell me where a certain star was

in the sky, do you know how we do that?

I have no idea.

It's exactly the same thing as latitude and longitude

but we put coordinates on the sky.

Oh.

So astronomers call them right ascension and declination,

which are two terrible words,

but basically you've seen on the globe

where you draw latitude and longitude,

what it looks like.

[Juliana] Yeah.

[Sean] And sort of the peels of an orange

kind of thing, right?

[Juliana] Yeah.

So you can define something as, well you know,

how high above a certain locations on earth it is,

but the earth is rotating and revolving around the sun

so we have to define separate celestial coordinates.

So, there's like how many dimension would there be?

We don't know.

You know, the optimistic view is that there are six,

but the thing is, some of them might be

really, really, really, really, really small,

like way too small for us to ever see.

Yeah.

And some of them might be medium sized

that hopefully we can see.

Oh, okay.

So since it's all theoretical,

like this could not be three dimensional.

Absolutely right.

And this is sort of the state of uncertainty

that physicists are stuck living in.

Oh. [laughs]

You know, honestly out there, if you go out

onto campus and talk to the physicists,

half of them will say probably extra dimensions exist,

and half of them say no, that's just nonsense.

Oh. [laughs]

[Sean] We really don't know.

Yeah.

Okay, after all this.

[Juliana] Yeah.

Someone comes up to you on the street and says

what's a dimension?

Oh, man.

I mean, I guess what I've learned today

is just that there are not just three dimensions

or at least we think, I mean everything's theoretical.

It's all just really kind of confusing.

That's right, and you know, if they're still bugging you

you can just like give them some sticks

and ask them to put them together.

Oh yeah.

[Sean] And that would shut them up.

Yeah.

[down-tempo electronic music]

Where do you go to school and what do you study?

I'm gonna be a sophomore at Pomona College,

and I study math and physics.

Oh, math and physics, okay.

What kind of physicist do you want to be?

Do you know?

I really don't know.

I like theory and experimental so it's kind of tough

for me to say.

Usually if it's math and physics,

you end up as a theoretical physicist, right?

Yeah.

Not an experimental one.

So, our theme here today is dimensions.

So this is great that you have some math background

because mathematicians think about dimensions

in a different way.

Hopefully, yeah.

So how would you explain to your friends

who are not math and physics majors,

like what is a dimension, yourself?

My first intuitive thought is what are the coordinates.

Mm-hmm.

So if we are looking at things,

if we're looking at like a dot,

or a line rather, that's one dimensional

because we can only measure it one way,

but then if we look at like a square

then we're increasing like that,

so it's basically what coordinates we can use

to measure something.

That's exactly right, and so you've heard

of spacetime-- Yeah.

[Sean] Being four dimensional, right?

Mm-hmm.

Now, in some sense that's kind of trivial to you and me,

because of course you have space,

which is three dimensional, you have time

which is one dimensional, so spacetime is four dimensional.

But it didn't turn out, it didn't occur to anyone

that that was a sensible way to talk

until really relativity.

This is the crucial thing, right?

Is that what Einstein realized is that

sure there's both time and space, but how we divide

spacetime into time and space can be different

for different people, and really there's a real sense

in which four dimensional spacetime

is kind of a generalization of three dimensional space.

And I think to really explain this

we're gonna need a blackboard.

Okay.

Let's bring one in.

[playful music]

All right.

The thing about relativity

is that they really want you to think

of spacetime as one four dimensional thing, right?

It's kind of like space.

It's not just three dimensions of space

and one dimension of time.

Why though?

Why?

This is a very good question.

So, let's just start with space, right?

We know a little bit about space.

So here's my simple minded way of drawing space

two dimensions, because that's how many

I can draw on the blackboard.

Let's say X and Y.

And what is special about space, there's many things,

but one is that if I have a curve or a path

between two points, there's a distance

that you can calculate, right?

And the distance between those two points

doesn't depend on your coordinates.

It doesn't depend on whether you're in radial coordinates

or Cartesian coordinates, or whatever.

I'm allowed to imagine a curve that does something like this

between those two points, and if I were a person

walking on that curve, I would have an odometer

with me maybe, and I would know, you would know,

even without having done that,

this path is always gonna be longer than that path.

There's a formula, Pythagoras' Theorem

that tells you what the shortest distance path is.

That's the point.

The physical-ness of what is real

is the distance along a certain curve.

So spacetime is like that.

That's why it is useful to think about spacetime.

Okay.

So let me draw spacetime.

Here's how we usually draw it.

I'll just say X, but all of space is condensed

in this one direction, and this is time, okay?

So if you're a little person

and you started some event, so you start,

you're located somewhere in space,

somewhere in the three coordinates of space,

and whether you like it or not, you're moving

through spacetime just by getting older.

Yeah.

When people ask me can you travel through time?

I say yes, yesterday I traveled 24 hours into the future

and here I am, a day later.

So that's just this.

Okay, you're moving through time, like it or not.

So what Einstein says is look, I can travel

through spacetime in different ways,

like I could hop in a rocket ship

and fly out and then fly back,

and then I could meet you there.

So this is a different trajectory through spacetime, right?

Yeah.

And it's almost exactly like the space story.

The space story says there's a distance,

distance is different along different curves.

Einstein says there's something that measures

the length of these curves,

and we call it the proper time.

It is literally the time that you would read

on your wristwatch.

So it's kind of like our fundamental time?

Or our base time?

Well kind of.

What Einstein wants to get across is there's no such thing

as fundamental, like there's the universe's time,

this big letter T that might tell you how old

the universe is, but then every individual

has a clock with them, and they measure their own time

depending on how they're moving through the universe.

I see.

And the crucial difference is

that time is not the same for this person who stayed behind

and sat in their chair,

and this person who zoomed out there.

Why is that, that this one's shorter then?

There is what we call a metric on spacetime,

and when we talk about Euclidean space

versus a curved space, versus a sphere or something,

that's a different metric.

And spacetime has its own metric

which says the following thing,

that the path between two events in spacetime

that is a straight line, will always be the longest time.

I see.

Oh, okay.

[Sean] That's the difference.

All right.

So when Einstein had this idea,

oh gravity could be related to the curvature

of spacetime, he did some equations,

so he got it, that's a long story,

we'll put that aside.

What he figured out was that rather than gravity

living on top of spacetime, it's a manifestation

of the curvature of spacetime

so when you have like the earth, the sun, the moon,

they cause a gravitational field,

they're actually warping the spacetime around them.

They're giving it a different geometry.

Would it be if I had like a spring

or not a spring, but like a sheet,

and I dropped a book in the sheet, curves down?

Yeah, exactly.

If you had a sheet that was originally flat,

and you set a marble on it,

it would go in a straight line,

but then if you put something on it

so it warps it, that marble is now gonna be deflected.

I see.

Einstein says that gravity is just like that.

I see.

But there are no straight lines,

because spacetime itself is curved.

So do you think if you had to explain relativity

what would you say?

I think I'd go with kind of the train paradox.

Let's say I'm stationary, and someone's moving past me

on the train, they think they're stationary on the train.

Like they think that they're not accelerating,

but if they start walking through the train cars,

then they are accelerating in their frame,

but then from my outside frame

where I'm completely removed, I see they are accelerating.

So I guess that relativity is all about perspective,

I guess in a way.

Yeah, that's right.

And it goes exactly back to what we drew on the board

where how those two people in the train and on the ground

would divide spacetime up differently

to space and time. That was pretty good.

[Sean] Yeah.

I learned a lot.

[Sean] It's a lot of fun stuff to talk about.

[down-tempo electronic music]

So observational cosmology, so like what do you look at?

So, I work on two ground-based surveys in the optical,

and we're basically trying to make huge maps

of the universe so that we can study dark energy.

I'm sure you've heard about extra dimensions

a little bit.

I've heard it, yeah.

Well I'm thinking about the idea

that there might be more than three dimensions of space.

What is your impression of theoretical physicists

who think about things like extra dimensions of space

that they haven't ever seen?

I get a little scared,

[laughing]

because I think how can you prove these theories?

Right.

One theory I've heard of, I don't know if this fits in

with that, is bubble universes.

Mm-hmm.

Is that an extra dimension?

Does that fit into that, or is that something?

It does.

In fact, one way that different universes

might be created and be different from each other

is that different universes could have

effectively different numbers of dimensions.

Like we have three dimensions around us,

but there's people out there, aliens, who could live

in five dimensional universes.

And are each of those dimensions,

are they governed by the same laws of physics,

or is there like a separate Lagrangian for each universe--

It's a, yeah.

[Bela] Or how does that work?

We think that it would be,

no all of this is incredibly speculative,

and we don't know for sure.

But the idea is that there is some deep down

underlying laws that are universal and the same,

but they show up differently,

so they appear different.

So the specific particles and forces and masses

would be completely different in different parts

of the multiverse.

Okay.

Why in the world would you think that there

are extra dimensions, right?

So you've heard of string theory?

I have.

So string theory is basically a theory

of quantum gravity.

Mm-hmm.

So we have quantum mechanics, right?

The theory of atoms and so forth and how those work,

and then we have gravity, and gravity doesn't

seem to fit in.

It's the one force of nature that we can't really

easily fit into this quantum mechanical framework,

so string theory is one of the best ideas.

Right.

That's the good news.

The bad news is that it only seems to work

if spacetime is ten dimensional.

[chuckles]

So you would say well then it's wrong.

It can't be right.

Spacetime is not ten-dimensional.

Spacetime is four dimensional.

We've observed that.

But instead, we say if spacetime looks four dimensional

to us, but string theory, which might be the best theory

we have of quantum gravity says it must be ten dimensional,

maybe we can hide those extra six dimensions somehow.

So here's how we could get lucky.

We could imagine that there are extra dimensions of space

that are curled up somehow

that are so small that we can't see them.

This is actually an old idea.

It goes back to Kaluza and Klein

right after general relativity was invented in 1915.

But there's a more recent idea that says

there could actually be relatively large extra dimensions.

There could be extra dimensions that are actually

that big, like a millimeter across

that you would not have noticed.

But here's the new, exciting idea.

So let's imagine, okay.

This is a piece of paper.

But let's imagine this is our entire world.

So, in other words, our real world is three dimensional,

but let's imagine that we're just idealizing it down

to two dimensions, so we all live here.

You and I live here.

But let's imagine that we're embedded in this bigger space.

So there are extra dimensions that are actually big, okay?

And let's imagine that.

When I say we live on this three dimensional world

what I mean is imagine that the particles

that you and I are made of, right, the quarks, the leptons,

the electrons and everything, all the forces we know about,

electromagnetism, the weak nuclear force,

the strong nuclear force,

imagine they can't leave this surface.

This is what we call a brane.

B-R-A-N-E.

Have you heard this word before?

I've heard it, yeah.

[Sean] Do you know where it comes from?

No.

From membranes, you know.

We have lines, one dimension.

We have two dimensional surfaces.

So if you have a line that is a vibrating physical thing,

you call it a string, right?

Right.

If you have a two dimensional surface that vibrates

and is a physical thing, we call that a membrane.

Membrane theory goes back, it never was as popular

as string theory, but it's been around for a while.

But if you had extra dimensions of space,

then you can three dimensional vibrating things,

and four dimensional vibrating things.

So how do these strings give rise to things

like mass and charge, and basically give us

the properties of the particles that we see?

Well basically the strings are the particles that we see.

It's exactly the same thing as we said for the straw,

if you look at it far away it looks one dimensional.

A little loop of string,

so a little one dimensional circle that is vibrating.

If you look at it from very, very far away,

it just looks like a particle.

So in string theory, an electron is a little string.

A photon is a little string.

So, is string theory part of what they call

the Grand Unified Theory?

Is it supposed to be the last thing

that sort of unifies all the forces together?

Yes, string theory's even better.

So the phrase Grand Unified Theory was coined in the 1970s

for theories that joined electricity and magnetism.

So a good string theory is gravity plus

the Grand Unified Theory.

Okay.

It's even better.

It's the theory of everything.

What would you tell a friend of yours

if they asked you what dimensions are,

what extra dimensions are, what a brane is?

So we have three spatial dimensions.

A brane is sort of the next level.

So a brane is a higher dimensional object

that vibrates through space.

That's right.

And we could live there.

The world we see around us, the three dimensions

of space around us could reflect the fact

that we are somehow stuck on a three dimensional brane

trying to escape.

It was really cool to learn about strings and branes

and how looking at gravity on small scales

is actually connected to what I do

in looking at gravity on these cosmological scales,

and it's definitely something I'm gonna think about

in my research.

[atmospheric electronic music]

You're a string theorist, so tell us what kind

of string theory you do, what it means

to be a string theorist.

One of the things that's key

in the whole story of string theory

is the piece of it that talks about

quantum theories of gravity.

So I'm very excited about what happens to spacetime,

what does it even mean at the quantum level.

Cool, so do you think a lot about extra dimensions

in your everyday life?

Uh, yes I do.

And so when you think about extra dimensions,

you put them together with brains and different fields

wrapping around the extra dimensions and so forth, right?

Yes.

You know, a lot of people, a lot of string theorists,

they care a lot about all the different ways

in which we could hide the extra dimensions.

As someone who cares about cosmology,

I wanna start asking why are the extra dimensions

small at all?

How did that happen?

Is this something you think about yourself?

Yes, well ultimately we'd like to understand

the observable universe.

If string theory turns out to be the thing

that the universe cares about,

we'd like to know, with all of these possibilities

that are in string theory, how do we get the one

that looks like the world we live in?

The thing I want to talk about is this paper I wrote

with Matt Johnson and Lisa Randall

where we realize there's another way

to compactify extra dimensions spontaneously, dynamically.

If you imagine starting with this big piece of paper

you couldn't wrap up everything,

but within some region of space you could make a tube.

Okay. Right?

And down there in the tube it looks like

your long thing that is compactified in one direction

and infinitely extending in the other direction,

so there's one less macroscopic dimension of space.

Hmm. Okay, that's nice.

[Sean] You think that sounds plausible to you?

It sounds like fun.

What I would immediately ask is

where do the large dimensions come from in the first place?

Is that something you address, or you just assume

that all the dimension are large as a starting point?

Yeah.

So we certainly assume that they're all large

as a starting point, and it's worse than that.

So in our paper we imagine you start

in de Sitter space.

You start in a universe with no matter,

no anything like that, just empty space

but with an energy that is positive

and all the dimensions are large.

But where do they come from?

It was always there.

Why?

Why not?

Okay.

[laughing]

Well you know, I think that this is again, this is--

You're replacing one prejudice with a different prejudice.

I would say that we shouldn't be prejudiced

one way or the other.

Just a large, ten or 11 dimensions

with a cosmological constant

doesn't seem like a fundamental starting point.

It sounds like there needs to be something

to underlie that.

But it's a fun scenario.

Yeah, so let me tell you a little bit more

about the scenario.

So you know there are black holes, good?

Right?

I've heard of them.

[Sean] There are black branes as well.

Indeed.

Why don't you explain to us what a black brane is?

Well in some ways, it's very much like a black hole

in that you have far away from it

you have these large just flat dimensions.

As you move in, there's something in the interior.

It has, itself, higher dimensions,

and so because physicists love to joke

they thought, well it's like a membrane

but it can have many different dimensions.

Let's use P to stand in for those number of dimensions,

and so they called them P-branes.

Yeah, I've been trying to explain this

to lower levels, and they always look at me like,

are you pulling my leg here a little bit?

Or is this the real thing that they use?

Yes, it is the real thing they use.

We study these branes in de Sitter space,

and so what we found is that there are black brane solutions

which instead of having a singularity inside

are non-singular, and stable, and compactified.

Is this a familiar thing?

I know things like this have been discussed, right?

Yeah. Yeah.

So, a lot of the branes that are relevant for studying

things to do with anti-de Sitter spacetimes

in various dimensions actually start out

as these non-singular type brains,

and then you find in the core,

there's actually an anti-de Sitter spacetime.

Right.

Imagine that rather than going down to a point,

it just asymptotes to some fixed radius

that continues infinitely far down there.

But then you can ask, okay what about the rest

of the universe?

You have some sphere, some two dimensions compactified,

and the balance between the cosmological constant

and the electromagnetic field keeps it

at a fixed distance.

Right. Good, good.

But then the other dimensions,

the transverse dimensions, don't need to be anti-de Sitter.

They can have either positive, zero,

or negative cosmological constant.

So you can basically get any sort of cosmological solution

times a compactified sphere.

Is it that these have the lowest action

of all the things that could possibly,

all the possible solutions?

I guess.

I mean, maybe you'd think otherwise.

Again or maybe you could change my mind,

I think that as long as it can happen

it will happen some of the time, right?

Okay, that's fair.

I mean it's one of the things that could happen, so

I mean it obviously,

You know, we know-- It's quantum, baby.

It's quantum.

[laughing]

It's a loud, it's of a loud

it's gonna happen.

I still laugh from Keanu Reeves I'm afraid.

[laughing]

Someone probably wrote it for him.

So it's possible that if you do have this starting point

of an empty de Sitter space

with positive cosmological constants and fields lying around

you inevitably get a multiverse.

It just happens, you know,

it's just part of quantum nucleation to different things,

and then maybe you need to explain

why we live in this universe

rather than some other one

via the anthropic principle or something like that.

So how does our kind of, you know, low-tech

old fashioned way of making new universes intersect

with the kinds of things you'd think about.

Well, it's interesting because there's a lot

of activity going on inspired by things we learned

from working in anti-de Sitter space

where the cosmological constant has the wrong sign.

So you will find that there are a number of papers

as many of which you've probably read,

probably you've read more of them than I have,

which try then to take the different kinds of brains,

these extended solutions, and they're many different kinds

doing different things, we understand well enough now

that we can put them together in various ways.

They intersect, they dissolve inside each other,

they wrap each other.

They overlap, they wrap around.

They overlap, and you do them in all sorts

of different ways and show that you can construct

four dimensional universes with a cosmological constant

of your desire, depending upon how you did the construction.

I know not everyone buys that, right?

Not everyone buys it.

There's a big discussion in the literature right now.

And if we put aside for the second

the concerns of our colleagues in other parts of physics,

and it's just amongst us chickens,

what do you think about the future of cosmology

and string theory, and the dynamics

of these extra dimensions?

I think it's gonna get way more interesting

than we're currently able to grapple with right now,

and I think there are hints of it.

There could be a description of the physics

that might be our early universe

that is like that molecular description.

And then from that emerges flat spacetimes

maybe four, maybe some mechanism tells you

it's four and not something else.

I think that's where we're going.

If the viewers wanted to know more about how spacetime

can emerge from quantum mechanics,

they could read the graphic novel you just wrote.

Right, yes.

I wrote and drew a graphic novel called The Dialogues:

Conversations About the Nature of Universe.

And I just wrote a book

called Something Deeply Hidden

about many worlds, quantum mechanics,

and how spacetime can emerge from it.

Plenty of reading material for the audience out there.

Excellent.

[down-tempo electronic music]

So that was fun.

I hope that everyone did learn something.

I know that I did.

It was very nice to see how the idea of dimensions

and space resonates in different ways

with different people, all the way up to talking

to Clifford about the forefront of modern research.

I really do think that we're not done

understanding how dimensions work.

There are three dimensions of space, why that?

Why not two?

Why not 27?

I think that we really don't yet have the data

or ideas to think about this, but we're creeping up on it.

I'm optimistic about progress in the near future.

[atmospheric electronic music]

I'm a professor at the University of Waterloo.

And I'm a sleep scientist at UCSF.

Today I've been challenged to explain lasers--

To explain the topic of sleep

at five different levels.