Oh, wow.

Calm down, calm down. Psst!

That is the most efficient calming down I've seen in some time!

In fact it is a very well behaved audience. That is fantastic.

Thank you all very much for coming along in such a polite, orderly fashion.

Thank you very much Deborah — who has vanished! — for introducing me so nicely.

As she said, my name is Matt Parker. I am here to talk to you about Things to See and Hear in the Fourth Dimension.

Before we go racing into that, a little bit about myself –

Oh! actually, most of my information is up there. You can see my name is Matt Parker.

I work at Queen Mary University of London. I actually used to be a maths teacher.

Actually, have we got any maths teachers in?

[A few people whoop]

[some laughter]

All sat down in the front, brilliant. Okay. No surprises there.

So I used to be–

Like, if you've been brought along by a math teacher, like one of your… fine, standard, normal math teachers.

And…

[smirks]

[some laughter]

No math teachers are truly normal, are they?

I used to be a regular math teacher, and then I've gradually drifted into a much more varied career.

So at the University of Queen Mary, which is a university in East London, I work in the maths department.

I am the public engagement in mathematics fellow.

[laughter]

You're laughing at my career, thanks for that. That's...

Don't – don't – don't patronise me.

So we'll get racing into this. And, it is going to be a night of my favourite bits of mathematics.

I'm going to show you my all-time favourite shape.

It is pretty spectacular.

And I am going to hopefully show you some other practical bits of mathematics along the way.

But we will start properly by doing some mathematics. So can everyone who brought their calculator... could you take that out now?

Really?! That is the most movement I have ever seen on "everyone take out your calculators." It genuinely is!

Has anyone got... Okay, I'm seeing a lot of phones. That is fine. Has anyone got an actual calculator?

Really?! Oh, that's so good. Is that a Casio? Very nice.

Can anyone beat a Casio? Another Casio.

They're both from the FX series, it looks like from here. They're standard school-issue ones.

Someone… Another Casio, very nice. No one's brought like a hilarious slide ruler or something?

That has happened.

What have you got? Oh, is that Texas Instruments?! Fantastic! Alright!

This is just my game of guess the calculator!

Okay, so whatever your calculation device is — don't feel intimidated now if you've just got your phone —

— I want you to type into it any two-digit number that takes your fancy.

If, ah…

People that have been to the "Christmas lecture" filming, this is one of the things I do when I've got a lot of time to fill.

So some of you may have seen me do this trick before, but it is one of my absolute favourite maths tricks.

So what I want you to do —

— without talking —

— is to type in your two-digit number, then hit "multiply" and type in the same two-digit number again.

Hit "multiply" and type in the same two-digit number a third time, and then hit "equals".

And what you have done, is you've calculated the cube of your original two-digit number.

So it's two digits, times the same two digits, times the same two digits again.

And what you've done, is the kind of long, tedious part of mathematics.

The multiplying things over and over. This is why we have calculators. Alright?

The fun bit of maths, though, is what you can do with the answer.

Is there anyone who has an answer on their screen?

They've cubed their two-digit number and they're prepared to share it?

Is there anyone who is happy to share?

Okay, I'm going to go with the very keen girl there.

Yes, the one now looking nervous, pointing at herself. Yes. Could you…

Could you read out up there… Could you…

Is someone trying to steal your calculator?

You're like, "Wait, this is too much pressure, gimme that."

Alright. Okay. I think she can do it.

Can you read out, what answer have you got there?

[GIRL] I got 17,576.

17,576. Okay, so you had to very carefully cube twenty…six, to get that.

[applause]

Calm down, calm down.

Who shouted "I did it?" Did someone else do 26?

[CHILD] No.

No? But you did something else! That's excellent. Thanks for joining in.

So… Actually what answer have you got on the screen down there?

Now that we have started what I believe is a conversation.

[CHILD] 6,084!

Okay, 6,084. Did you put in 14 and cube that?

[CHILD] No.

No. Excellent. I regret starting this conversation. [laughter]

So… I will come back to you. In fact, I suspect you're going to feature heavily tonight.

I will come back to you in a moment so just wait there, alright?

So what I'm actually doing, by the way, is not a difficult calculation.

I'm not doing a long, tedious cube root in my head. I'm doing a bit of a trick.

And so I will do a few. I will do three in quick succession, and then if I get all three correct, you go the correct amount of wild.

So is there one person over here who's prepared? Okay, I'm going to go, guy behind the sparkly hat over there.

In fact, if this works you can have the sparkly hat as a prize.

It's disappeared. There you are. You wouldn't believe they convert straight to gas at room temperature.

Okay, so we got you in a second, remain calm.

Okay, one over here… Who have we…

Okay, Miss reluctant lady right in the middle. You don't get a hat, I'm afraid.

You know what? Forage nearby.

And then, who over here? Okay, keen person over here.

Who's taking your glasses off in shock. "Me?!" Yes! Right, okay, so!

So we're gonna do them in reverse order. In a second you're going to call your answer out. So many thousands and then the rest.

And then we will come to... where have you gone. Over here. You're looking more nervous by the minute. It's fine.

And we come over here to you, sir. And if I get all three correct... wild.

What have you got on the screen over here?

[FIRST PERSON] 12,167.

You put in... 23.

[FIRST PERSON] Yes.

[SECOND PERSON] 140,608

52.

Okay, nodding silently is perfectly acceptable. Yes, and over here?

[THIRD PERSON] 704,969

Uh, you've put in 49.

[THIRD PERSON] No.

So… What was the number again?

[THIRD PERSON]: 704,969

Oh, 704 thousand?! [laughter]

Uh… I suspect, at a guess, it's going to be 89?

[laughter]

[applause]

I don't entirely feel like I earned that.

So, right. Now what I'm doing is, like I said, it's just a crude… I've got two rules.

One rule that tells me the first digit of the original number. And the second rule that tells me the second digit of the original number.

And I'm not doing anything other than just bluntly applying those two rules.

So as you read it out, I'm listening to the digits and applying those two patterns to work out what they are.

So if you cube something that's not a two-digit number, then I won't know.

I will just take whatever the answer is and put it through the same algorithm.

And if I mishear it… I thought you said 104 thousand, and I wasn't smart enough to know that it's not the cube of a two-digit number.

So I still applied the same two rules.

Now, the two rules, by the way, are incredibly easy. And if you knew what they are, you would not be impressed at all.

Which is why I'm not going to explain them. [laughter]

[disappointed murmurs] But… If you wanna learn them…

If you wanna learn them, when you get home tonight, get a calculator, get some paper, get a pen…

Maybe invite some friends over… have a maths party.

And you can serve triangle shaped snacks.

If you cube a lot of two-digit numbers you will very quickly see the patterns come out, right?

So start cubing two-digit numbers, look for the patterns, and sooner or later you can learn what this is.

But, to do something slightly more difficult… now this is the warm up over. Just something slightly more difficult.

I've brought with me… And you can put your calculators away.

I have brought with me one of my favourite maths toys. I have it here.

It is the Rubik's Cube.

I am gonna attempt to solve this for you live on stage.

Before I do that, because people will accuse me of pre-arranging, I'm going to give it to you.

Would you mind taking that, and mixing it up so that it's in a random arrangement.

I'm gonna attempt to solve for you the Rubik's Cube.

Oh, has anyone got a timer? Could one person get their timing device back up, who have I not interacted with yet. Yes, Miss.

Okay, would you mind when I start…

If you're solving that, you're in so much trouble! [laughter]

Mix it up, alright?

I better get all the stickers back.

And so, Miss, when I start solving it, you start timing. When it's solved…

Thank goodness; gimme that, gimme that.

[laughter]

Thank you very much.

Okay, so, when I start solving...

Are you just texting a friend?!

"You wouldn't believe what's in the show tonight." Alright, so you start.

Have you opened a timer? You're putting your phone away now?

It hasn't got a timer? Alright, so you said "No problem I'll time you" — you didn't even pretend!

You could have just counted the seconds.

Okay, so you've now stolen a phone. Okay, does this one have a timer?

[GIRL] Yes.

Okay. To be fair, this show is involving alot more admin than I expected.

Could you, when I start solving it, start the timer, alright?

At the end, once I've got it — once it's finished…

When I finish it, no one react, alright? At the very end, could you call the time out?

If you deem that suitably impressive, you can start applauding.

Okay, alright, here we go. So.

And the time starts…

[GIRL] Now.

[audience laughter]

Did you start? [GIRL] No. Good! Alright!

Only if I say "now", you start — but not that — you get the…

[audience laughter]

The time starts now!

Okay, right. So, people might think that a Rubik's Cube is famously very difficult to solve.

So how on Earth can I solve it in a timely fashion?

And to be honest, if I don't get this done in the first 20 to 30 minutes, we'll just pack up and go home.

[audience laughter]

I'd just call the show off at that point!

You can learn how to solve a Rubik's Cube.

You can learn what combinations of twists, move different bits of it around…

And the current world record for solving a Rubik's Cube is 23.19 seconds.

How long have I been going for?

Thirty! Oh, several people are timing me!

[audience laughter]

Why am I not surprised?

So anyway, I've been going for over half a minute, so… A more impressive me would be done by now!

But if there are any Rubik's Cube experts in — and believe me, it is very likely —

— you will know that 23.19 seconds is actually not the current world record for solving a Rubik's Cube.

That is the current world record for solving a Rubik's Cube while you're blind-folded!

So… Yeah! Think that through, alright?

Someone was given a Rubik's Cube and proceeded to solve it in under half a minute.

And in fact, there are all sorts of ridiculous world records for solving Rubik's Cubes.

There's a world record for solving a Rubik's Cube while underwater — 19.01 seconds.

There's a world record for solving a Rubik's Cube solely with your feet, which is currently 23.14 seconds.

And I have solved it in…?

One minute 18.95?

[applause]

I'll take it!

Alright, so!

What I absolutely love about the Rubik's Cube, and the reason I could solve it, is…

Actually, I can bring it up on the screen, here, so if you can't… There we go!

That's a lot less flattering than I was hoping for!

The Rubik's Cube, what I love about it, is that it's a perfect example of a 3D cube.

'Cause it spins in three different directions — it spins that way, it spins that way, and it spins that way.

And when you're solving the Rubik's Cube,

the middle bits — those middle squares — never move.

They remain perfectly still.

They are effectively the 3D axis of the Rubik's Cube.

There we go. And... Pause.

Perfect, right, so...

So as I move it around, all I'm doing is, with reference to those points that never budge,

I've learned what twists in the three different directions move all the different stickers — or, well, different pieces, actually — into the correct positions.

And with a bit of practice — it turns out all the instructions are on YouTube — with a bit of practice, you can learn to do it reasonably quickly.

For proper nerd cred, though, under 30 seconds. And my best time is… I can just break a minute.

I'm not brilliant. But I tend to solve it while socializing, so that's my attempt at a record.

If people wanna have a play with the Rubik's Cube afterwards, please do come down and have a look at it.

And the fact that it is a 3D cube, I think is absolutely brilliant.

And my idea behind the talk today is eventually, via some practical maths,

I'm going to work my way up to showing you the 4th dimension.

In fact, my all-time favourite shape only works in 4 dimensions.

So, this is our 3D cube. I'm gonna try to work our way up to shapes that exist in more dimensions than we have.

But it's fine; we're not going to go racing straight into it.

We're going to gradually warm up, and we're gonna finish with a 4D version of the cube.

So, before that, I have promised some practical mathematics. So I thought we'd just get that out of the way now.

I'm gonna show you the mathematical way to tie your shoelaces.

And this… I mean, you can save an incredible amount of time if you tie your shoes the maths way.

So if I bring this camera around — because I'm well aware you can't all see my shoes, right —

– but if I set a camera up down here - I'll put it over here.

So, in theory, if I get that there... Okay.

So, if I put my foot in front of that, can you all...

[CROWD] Yeah.

Oh, and you can see the one behind... Oh wait... Yeah, there you go...

[audience laughter] [RANDOM CHILD] Cool!

It's like the world's nerdiest chorus line. Look at that!

Anyway, so, you can see my... [repeated laughter]

This show is going in a lot of unexpected directions!

Okay, so you can see my shoelace here, in fact if I angle that down in a second...

It'll be a little bit less... there we are.

Alright, so, I'm going to tie my shoe the mathematical way, and the way this works is:

You start with the two laces... With a foundation knot… And you just hold them…

And most people do some kind of ridiculous moving the laces around one knot, right.

In reality, what you can do is once you've done the foundation knot...

If you just cross the two laces over, they'll tie themselves. Hence…

[weak audience reaction]

That's the correct amount of amazed!

[laughter]

Would you like to learn?

[AUDIENCE] Yes!

Alright, so, I'll do it again. I'm gonna give you a few seconds.

If you've got shoes with laces, choose your favourite shoe, undo the laces.

If you haven't got laces, odds are the person next to you does.

[laughter]

And they're not using at least one shoe. So everyone get some laces within reach.

[CROWD MURMURS]

Okay, here we go. Are you ready?

[CROWD] Yes

Okay, so, under the laces, do that foundation knot like what you can see. I've got tied on my shoe so far.

Then, if you have a close look at the foundation knot,

one lace is going backwards and one is going forwards.

So take the one that goes backwards, curl it up and over so it's going forward,

and hold it on the way down like that.

And then take the one that's already going forward and curl it backwards and hold it on the way there.

Now, all you have to do is take the two bits you're holding, put them under the other loop,

swap hands, pull tight, and you've got a knot.

[CROWD MURMURS]

Is there a delay on that?

[CROWD] Yeah

Oh, sorry, so I'll do it again.

If you missed it, just very quickly look slightly to the left, alright, and you'll see it again.

So, here we go. I'm going to do it one last time.

I'll do it again with instructions on the screen, and then I'm going to give you about 30 seconds to have a go,

and then we'll carry on with the talk. Okay here we go.

So I'm going to untie my laces here. We have picture and picture instructions.

So, you start with the back one curling forward - hold it on the way down -

The front one curling back - hold it on the way down -

and then you cross them both under the other loop, swap hands, pull tight, and it's done.

The great thing about doing that: Not only will you save seconds of your life every day,

but the knot you end up with is mathematically the same as the standard knot you get doing it the really long way.

And so you're not sacrificing knot quality, but you are doing it an awful lot quicker,

and I can say that it is mathematically the same because there's an area of maths called knot theory.

There are mathematicians called knot theorists — which is the best name ever — but they are theorists.

They just look at knots. [laughter]

And so… There may be a few puns in this section.

And so knot theorists study knots, and sadly at the moment, humankind's understanding of knots is terrible.

Humans are just... We are not good at knots whatsoever.

And knot theorists do not have a reliable way to work out how to undo a knot.

And so, for example, if I show you this knot up on the screen here — in fact, this is a picture from my book — that knot...

Oh, and the convention when you draw a knot is if the string goes underneath, then you just leave a little gap.

So, where it goes underneath, it disappears briefly and comes out the other side.

So they're not just little bits of string that go underneath each other and the rest.

We have no idea what the best way is to undo that knot. No idea.

And the way that we like to undo knots is:

You can choose one bit where the string goes underneath and you can cut it from underneath,

bring it on top and join it together.

We call that a crossing switch, because you take one bit which crosses underneath and you switch it to cross on top.

And at the moment, the world record for undoing this knot is three crossing switches.

So, if you swap it in three places, it will come completely undone

and you will get a loop of string with no tangles in it whatsoever.

No one has ever found a way to undo this with two crossing switches.

No one has ever managed to prove you can't undo this knot with two crossing switches.

It is an unknown, open bit of mathematics. It is called the ten-eleven (10₁₁) knot.

I put it in my book, because what I want people to do is make it out of string and then have a go,

and it won't be in this arrangement, because people have tried all the crossing switches in this arrangement,

but if you make it in that arrangement, and then you pick it up and move it around so different bits cross eachother,

and then you make two crossing switches and take a photo of you pointing at them

– because if it works and you can't recreate it, no deal –

but if it does work, if you take a photo of you pointing at two bits,

you then make crossing switches at those two points and it becomes undone,

and you then send me an email, mathematical fame and fortune are yours...

...for a very narrow definition of fame and fortune.

So, there are other ones. This is just the easiest one I've come across.

Oh, I've had one email so far! The book came out in October; someone sent me and said,

"I did the knot!" I'm like, oh my goodness, so I invited a friend around.

We actually had a little knot party; we got some string

and it turns out the knot they started with wasn't quite the ten-eleven.

It was a different knot. We spent all evening. Eventually, we worked it out.

I've left myself very open to string-based trolling, it has now occured to me.

So, please... only if you think you've got it... oh goodness... uh... send me an email and I will check.

And hopefully, sooner or later, someone will do it.

And mathematicians are working on having a much more systematic way of going about this.

Instead of just having a go, trying to work out a way that given any knot,

you can calculate what crossing switches will undo it.

And at the moment, bacteria are better at undoing knots than humans are. And that's not good!

Because when bacteria reproduce, they unzip their DNA and their DNA gets very tangled.

And what they do is they have little enzymes that go around and perform crossing switches.

They will snip the DNA on one side, move it around another strand, and join it back together again.

And those enzymes in bacteria are more efficient at undoing knots in DNA than anything humans can do.

And at the moment, biologists are working with mathematicians to try and work out first of all,

what the bacteria are doing, and secondly how we can stop them from doing it,

because if we know what they're up to, we can some how impede their ability to untangle themselves.

And there's a whole future wave of medical ways that we can combat that bacteria if we had a better mathematical understanding of knots,

and I think that's absolutely amazing.

It's a fantastic ongoing area of maths research.

Umm. But up next is a slightly more fun maths you can try at home.

So if you get sick of playing with string, I'm going to show you a fantastic activity you can do.

Oh! I've brought some other things for show and tell!

Before I get to... Knots, right, are great, but there's a related area of links.

And links are when you have more than one loop that go through each other.

And there's a famous set of links called the Borromean rings.

And they're pictured on this beer can.

This is a beer can from the 1950s.

I spent ages stalking it on eBay.

I eventually, uh, managed to buy one.

It's the one in the picture here. You can come and have a look at it if you want to afterwards.

And the reason I really wanted it, is the logo is a mathematical set of rings, they're called Borromean rings.

And the way they work, the way those three links go through each other, if you break any one of those three loops, the other two will come apart as well.

So it only holds together as long as all three are intact.

You break any one, the rest separate.

So here you can see they put purity, body, and flavor.

Because if you remove any one of those from your beer the beverage just falls apart.

And, I thought that's an amazing mathematical logo.

And you can do it for more links, you can link four links.

Such that if you cut any one the other three come apart, and it works for any number.

For any n links, you can arrange them in such a way that if you cut one, the other n-1 come apart as well.

And I had heard rumours about this can; I finally managed to buy one; I was so pleased.

Recently, I was in New York doing some maths talks, and I went for a bit of a walk before one of my talks I was doing.

And I walked past a billboard for the same beer. Ballantine's beer.

They still make this beer.

And I saw it; I took a photo of me with it.

I was so… I was so excited; I had no idea this beer still existed.

And so you can see how happy I am.

But then, suddenly, my excitement turned to distress.

Because [laughter]

they have changed the logo!

[audience boos]

They've taken the crossing markings off the logo!

Right, and so I did the talking and in the talk I said right we have to get a bottle of this beer

afterwards we walked around several bars in New York demanding do you have this beer eventually

we found one that does. I smuggled the bottle back to the country there you are.

So if you wanna I didn't declare it in the customs cause I have to put what you have brought in

an empty beer bottle full of disappointment...

and a.... so if you look up cause I brought the old can with me to the bar.

So you can compare and contrast the two. I took a photo the next day in better light.

Right so they had taken all the markings, people have tried to say that it's okay Matt

they have just turned it into a Venn Diagram. I am not buying that, right, they have ruined

all the maths, all that links, so I am very upset. So if you'd like to come afterwards

you can have a look of the original and the changed mathematical logo of the beer.

Right, so. On to an activity you can try yourself at home. I'm gonna show you now

my all time second favourite shape. This is not the number one favourite shape; we'll work our way up to that

but my all time second favourite shape, and to make this

Oh, this is not my second favourite shape, by the way.

This is a rectangle.

[laughter]

I mean, don't get me wrong, big fan, but we can do better than just a straight up rectangle!

What you can do is, if you get a rectangle that is sufficiently skinny, and you join the two ends together

You get a loop.

The technical maths name for this is a boring shape. [laughter]

You can make it an exciting shape by taking the two ends apart, turning one of them over, so when you

stick them together, you get a loop with a twist in it.

Now many of you -looks like the sort of people who come along to these things- would have seen one of these before.

It was invented by a guy called Mr. Möbius in the late 1800s.

And he was so excited when he made what we now know as the Möbius loop

that he proceeded to name himself after the shape. [laughter]

And, uh, the great...

Not everything tonight is a fact; I should make that very clear.

So, the craziest thing about the Möbius loop, it has got all sorts of unusual properties.

So, some of you would have heard that is only got one side it also has only got one edge, which is amazing.

And there's all sort of strange things; but the best thing you can do with a Möbius loop is an activity when you've got two of them,

which are exactly the same length

and to get two the same length, you can see I have used paper reasonably wide.

If I make a single snip in the center there and because I originally taped it all together at once,

I cut through the tape, I'm gonna get two loops precisely the same length.

Okay, so I get back to where I started, and so now I have cut that in half...

I've…

What?!

[laughter]

I've just blown your minds!

Alright, so the Möbius loop can't be cut in half.

I genuinely cut it in half, like I properly cut right down the center, there's no funny business there.

If you cut a Möbius loop in half, you end up with one piece, a single loop.

And that just freaks me out.

[laughter]

But it gets worse...

If you take a longer rectangle to start with,

this time when you start with your zero twist cylinder,

instead of doing one twist, if you do one, then two, then three, you gets what's called a three twist mobius loop.

And if it seems at all familiar, it's because it is the recycling logo, there you are.

[laughter]

You know, the next time you're walking down the street and you see the recycling logo, you can go

"Hey! That is a three-twist Möbius loop!"

[laughter]

That only works if there is someone next to you.

[laughter]

You'd look insane otherwise.

Anyway, right so, a three-twist Möbius loop, I'm going to do exactly the same thing again

and I'm going try and cut it right down the center.

In your own minds,

without calling it out, I want you to see

if you can guess once I cut right down the middle of a three twist mobius loop,

how many pieces of paper will I end up with?

Will it still be one loop?

Will I get two? Or will I get three?

In your own minds decide...

Okay, turns out I get -

that

give it a second - here we go

Is it one, is it two, is it three? Is it two?

Turns out it is one, it is a single loop but now there is a knot tied in it.

Right, so, I started with a loop of paper with no knots, I cut it down the center and it tangled itself,

at no point did I undo the loop.

So somehow a knot got it there, oh and for any knot fans in that is a trefoil.

And so (some of you will be here!), I can't get rid of it, it is like a trivial knot - it is properly in there.

In fact it takes one crossing switch to get rid of it.

And actually that is how the knots get in DNA. Because DNA is a very very twisted, very long strand,

when it reproduces it unzips down the middle, just like I did you cut it right down the middle

and by cutting a twisted loop in half it ties itself in knots.

Some bacteria and apparently some human cells (not a biologist) have circular DNA, so it ends up very very knotted.

Even if the two ends are not joined together, by cutting a tangled thing in half it gets incredibly knotted.

And so that's where the knots come from when DNA reproduces, and you have to do crossing switches to get rid of them.

But the fact that by cutting a loop you get a knot really messes with my head.

But it gets worse...

[laughter]

I skipped right over our friend the zero twist, because this is quite a boring shape,

if I was to cut this in half right in the center I would just get two zero twists.

Not that exciting.

I'm gonna make one slight change, I have got another cylinder here

I'm gonna stick both of them together at right angles.

I'm them going to cut them both in half while they are stuck together.

Oh, and if any of you are going to try this at home afterwards (and a lot of you are!),

make sure you put plenty of tape on both sides because if you don't tape it together sufficiently, it will become undone.

while you're cutting through them.

there we go, right, so again in your own mind

without calling it out without ruining it for anyone else

see if you can guess how many pieces of paper I'm gonna get.

And, if you think you've got a good guess of that…

Have a guess what shape or shapes those pieces of paper will be.

Right through the first one, we get that...

Right through the second one...

And I end up with…

A square!

[audience is delighted]

Yeah!

[applause] Yes!

[applause]

So good!

SO good.

[laughter]

Most of the audience who are down here -

- and you are all my favourites

Anyone arriving late or other people who now look shocked

Are sat at the very top over here

And occasionally people have been popping in, particularly people who work at the RI

Because they're like what is this maths comedy thing all about

So they've been coming in

It didn't happen, I was really hoping someone would come in RIGHT then.

'Cause all they would see is me going, "Hey guys… A SQUARE!"

[laughter]

And you're all like "A square?! Oh my God, round of applause for the…"

[laughter]

Wait 'til you guys see a triangle! My goodness!

That's some advanced maths tonight!

Anyway!

I think that is amazing, that two loops both cut in half give you a square.

In fact you can reverse that to see why.

If I put these two ends back together like that

If I identify those two edges, I get a strip of paper with a loop at each end

And if I then put those two loops back together

So that they line up

That's my original two cylinders at right angles to each other

You cut that one in half and you get that

You cut that one in half and you get a square

But it gets worse.

[laughter and murmurs of shock]

Two Möbius loops stuck together at right angles! [excited murmurs]

I am gonna cut!

This is the last one, there are no more down there!

I am going to cut -

- without talking -

- both of these, in half, at the same time

And then I want you to try and guess, in your own minds -

- without calling it out -

- how many

[audience talks]

- How many pieces of paper and what shape or shapes do you they will be?

Right, I have cut all the way through the first one

And I've got... that

Right ok, here we go

It will take me a second to untangle these when I'm done by the way

I'm cutting through the second one

And I end up with

[audience "aww"s]

Two hearts that go through each other!

[audience excited]

[applause]

Too late!

Alright so I held these up and you were like "nah..."

And then there was just silence and one person went wait a minute...

We applauded the square...

[laughter]

If we don't applaud this we're officially dead inside

It's too late though

So that is absolutely brilliant, right?

You now know what I gave my girlfriend for Valentine's Day last year!

[laughter]

Two Möbius loops and a pair of scissors!

[laughter]

And instructions!

Oh, and we got married! Last July! So, you know — boom! It works!

[applause]

That is the most expensive maths proof I have ever done.

The wedding ring is iron-nickel meteorite

It's a very different talk... See me afterwards.

What a substance!

Anyway, so, that is absolutely brilliant!

If you want to make one of these

There's a secret to bear in mind

When you make a mobius loop, you're actually faced with a choice

You can either twist it to the right

Or you can twist it to the left

And you get a right handed or a left handed mobius loop

They're mirror images of each other

They're different shapes

For the hearts to work

You gotta have the right handed attached to a left handed

If you attach them both, and they're both the same twist, then you don't get two hearts.

You get, like…

…a boat.

[laughter]

And a thing, I dunno.

And they're not even joined together! It is a world of disappointment!

Oh, teachers!

I've just planned your next lesson back!

This is so good, particularly with sixth formers

Because if you do this with sixth formers

You do the whole lesson like that -

- If you send me an email, I'll send you the worksheets -

And as you go along at the beginning of the lesson you say

"Alright class, make sure to pay very close attention and follow all the instructions very carefully"

And they're all like "pfft, please!"

"Follow instructions?!"

And then you don't tell them about the twist

And a third of them by accident will do it right, and get the two hearts

And the rest will get the boat and the thing

And then you walk around going: "I told you to pay attention!"

They'll do it again, and they'll do exactly what their friend's doing

And it worked for their friend, but it won't work for them

Then their head will explode

[laughter]

I still miss teaching sometimes...

It's absolutely great fun

But I wanna get to my favourite shape

Which is in the fourth dimension

I'm now going to show you what the fourth dimension looks like

We're gonna come running at this

A lot of people are vaguely familiar

With dimensions from an area of maths you might not expect

So when you're at school you will have been forced to draw charts

You will have been forced to plot things on graphs

And when you were doing that, you were actually working with dimensions

And so in two dimensions, you've only got two directions you can move in

You've got up and down, and you've got left and right

And that's an x-y axes graph, where you plot things onto it

And infact you can start plotting a variety of co-ordinates

And I've put on here

All the combinations of zero and one

And you do this quite a bit

You do it in maths because it's fun

You do it in science, because if you've done an experiment

And there's two different things you're measuring

Then you can plot two bits of data simultaneously as data points on an x-y co-ordinate

Infact

More people should use two dimensional plots than they do

Most stick to one dimensional plots, which I think is ridiculous

For example the football

The football premier league

Have a league table with only one dimension

It's just straight up and down

You could, if you wanted, plot all premiership teams in two dimensions

And you'd get a 2d league table

It would look a little bit like this

I've plotted net wins

That's the number subtract the number of losses

None of this two points ridiculousness

And then on this one here, I've plotted the number of goals scored subtract the number of goals conceded

As you can see some teams have scored a negative number of goals

So now at the end of the season you just calculate which team is the greatest distance from the origin

And they win, it's all very straightforward

I can't believe they haven't implemented this!

You can use arguments to solve arguments

If you're trying to spot your favourite team, there you go

I think that

[audience excited]

You've already gotten more sport than you were expecting

Don't be disappointed

You can plot great things in two dimensions

In this case you've done all the combinations of zero and one

Then what you've actually done is plotted the co-ordinates of the corners of the square

We can do the same thing in three dimensions

We've now got three different directions in which we can move in: Up, down, left, right

And now you've got out

This is the reality we live in

We have 3 directions we can move, up and down, left and right

And out and back.

And then combinations of those, so we can go across and out.

Or we could just take a shortcut there.

And if you put on every combination of zeros and ones when you've got 3 coordinates

And so in science this happens quite a bit

you do an experiment with 3 different types of data

You can plot it on a 3D plot

But if you join together all the combinations of zeros and ones, on a 3D plot

You get the corners of a cube.

You can now do this one dimension higher.

If you had a 4D plot, and the problem here is we can't imagine that.

Because for a 4D plot you'd have left and right, up and down, out and back,

And then another direction.

Another direction that we just can't even try and visualize.

Which is a real shame.

But, it would be very useful.

In science, you'd do experiments where you have 4 different things you're measuring.

If you've got 4 different types of data you wanna plot on the same graph

You need a 4D plot.

And you can do this, but sadly you can't see it easily.

So you lose that visual link.

But, there is still a few, well, there are a few ways you can visualize what a 4D cube will look like.

And, I'm going to try to show you tonight what a 4D cube would seem like

And to do that, what's really handy is,

To imagine how we would show a 3D cube... to a 2D creature.

So, if you had a 3D cube and you wanted to show it to someone who is perfectly flat, right

You could take a cube.. one way is to take your cube and then

unfold it into it's net

So that is one way to go down a dimension. You can take a 3D cube and unfold into it's net.

In fact, what I have here is a video of a net folding and unfolding into a 3D cube

Oh! I am not trying to patronize you

But, are we all happy, that is a video of a 2D net folding into a 3D cube?

[Audience: gentle whisper] Okay. Wow, mixed, okay.

Because it's not.

That is not a video of a 2D net folding into a 3D cube

That is a video of the shadow of a 2D net folding into a 3D cube

Right. At the top there, I have got the 3D cube

I can then unfold it into it's 2D net

All I was showing you before was the bit underneath..

You can see there is a light above it and is casting a shadow onto that flat surface

But your brain is so used to things being in 3D

That when I showed you the bottom bit.. your brain imagined the top bit

Oh! You are, ofcourse, now watching this projected on a wall

But don't think about it for too long. Right! [Laughter]

And so, Just take the essence, okay! Right, so [Laughing]

What you are looking at the bottom there is the shadow of what's happening above it

But you can reconstruct the situation above

just by watching the shadow underneath

And what's great is, just like 3D objects

can cast 2D shadows..

4D objects can cast 3D shadows

So, even though I cannot show you an actual 3D object, what I can show you is the.. the 4D Object (corrected)

What I can show you is the 3D shadow of that 4D object

And so, what I've got here...

This is our 2D net of a 3D cube

What I have next to it is the 3D net of a 4D cube

And just like before, you could watch the shadow of this one

folding up one dimension higher

I am now gonna show you the 3D shadow

of the 3D net folding into a 4D cube

And it looks a little bit like this...

In it goes and over

[Audience: wow] That's not so bad..

Now, if you watch the one that you're used to

If you start feeling a bit queasy.. in a bit "Dimension Sick"

Right! Look at this side. Because this one's okay. This is safe..

And if you think about it

When the lid turns over on top of the cube

on this one, we are used to it

It looks like, in the shadow, it looks it stretches around

And we know it's not stretching around

We know it's just turning over

But when it turns over one dimension higher, it's shadow looks like it's stretching

So over here, that purple cube at the top is not stretching over

It's turning over

But when a cube turns over in 4D, it can look like stretches around in 3D

And the actual moment it's folded together

It looks like this, here

That is the 3D shadow.. of a 4D cube..

But because this is a shadow, it's been cast in such a way

that it's got "Perspective"

So, that blue cube in the middle is actually the same size as the red cube on the outside

Except, it's further away in 4D

And when things get further away in 4D

they look smaller... In 3D, just like

if I showed you this.. You all know the blue square is smaller than the red square

You go.. Well, ofcourse the blue square is smaller than the red square

it's just.. it's the same size. It just looks smaller because it's further away

So you know red square, blue square, same size

But one looks smaller in it's shadow because of perspective

In fact, if you set this cube rotating, they take turns being the big one

Right! The blue one's big... And then the red one's big..

And it looks like they are going through each other

But they are not actually going through each other. They are going in front and behind

But if you just look at what's happening on the screen

The red and blue are at the same place quite a bit, they look like they are going through... there

And then they take turns getting big and small

I can do the same thing

I can take a 3D shadow of a 4D cube

And then I can set the 4D cube rotating

And (laughs), it looks like this

So now they are taking turns being bigger and smaller

But they are not actually bigger or smaller

They are both still the same size all the time

They are just getting closer and further away in 4D

And they look like they are going through each other

But they are not going through each other

They are going in front and behind, one dimension higher

When they are going front and behind in 4D

from our.. you know.. pathetic 3D point of view

It looks like they are going through each other

At this point, if your head's not hurting, you are not paying attention.

[Laughter]

Unfortunately, it now gets worse.

So, urm, oh! If you want to try this yourself at home

You can build high dimensional cubes out of straws

I.. I... strongly recommend you do this

In fact, if you want to start with a 1D cube

To build a 1D cube out of straws

Pick up a straw and... You're done!

[laughter] There you are.

1D cube, commonly referred to as a line. I'm gonna use a red for the first dimension

If you then add some more straws, you can make a square

So I have still got the red going one way, the blue now goes the other way

If you bring in a third colour, I have got yellow

You end with a cube

And if you then get another cube and join it together

at all possible corners to this one

You get a... Oh! oh sorry

That is the same thing.. I forgot.. I mean this is a new model I just made

Right! That is the same cube, but I have done it with perspective

So that's a perfectly flat shape, but it looks like

A 3D cube from perspective

Because what you can on dimension higher

Is you can make the wire-frame model of a 4D cube

That's without perspective and then, this is with perspective

So I have shrunk down one of the cubes and popped it in the center.

And so you end up with this..

All the edges and all the corners everything is there of a 4D cube

But built out of straws

Oh! "Pipe-cleaners". Use pipe-cleaners for the corners

You just twist two of them together, cut off the extra bits

You've got a nice way to hook

It's all explained in great detail in a book I can highly recommend

[Laughter]

Aah.. Now, the other way you could look at higher dimensional shapes

is what.. is if you drop them into a lower dimensional universe

This is what happens if you take a 3D square and drop it into a 2D world.. Does that, right!

And if you were living in that 2D world

And someone dropped a 3D square on you

It would look like.. ah.. 3D cube - 3D square, same thing. Right!

It would look like a square appears and then disappears

So it comes in.. and then it goes out.

And all you would see is suddenly a 'square' and then 'no square'

You know that's not very exciting.

But we can do better.

If you drop the 3D cube edge first, looks like

a rectangle come in and then goes out again

So it's 'rectangle in', 'rectangle out'

And if you were a 2D creature, you would see a rectangle appear and then go away again

And what you are seeing is a kind of series of 2D slices through a 3D cube

And... ahh.. and

But if you were a 2D creature, you would have no idea what's actually happening.

All you would see are the slices come in and go out

You would have no idea what's going on over here You'd see that.

From that, you've got to recreate what a 3D cube looks like.

From those 2D slices

I can now show you what it would look like

if someone took a 4D cube

And dropped it onto our 3D world, in a passed through, edge first

Looks a little bit like this..

In in comes, around it goes and out.

[Audience: shocked]

If you're ever walking down the street..

And you see this, a triangular prism appear in front of you, distort and go back out again

Someone is now throwing 4D cubes at you.. [Audience: laughter]

I recommend running (laughs)..

Alright! But it could be worse. Because you can drop things corner first

Here is a 3D cube going through 2D world corner first

You get a triangle that comes in, distorts and goes out.

Again, a 2D creature will have no idea what going on over there

See, why that triangle comes in one way?

It goes through the space that's filled by hexagon

and goes back out the other way.

Alright! We can do the same thing one dimension higher

If you take a 4D cube, when you drop it corner first onto a 3D universe

It looks like this.

In it comes, around it goes and out. [Audience: Wow...]

[Audience: whispers]

If you're ever walking down the street (laughs)

And you see THIS come in, distort and go out.. alright!

Someone's out there, throwing 4D cubes at you, 'pointy' end first

[Audience: Laughter]

Somehow you've annoyed them.. (laughs) Right!

Definitely run. Possibly put in a hat. Sparkle's optional

Oh! That's why you wore it. You're like, "Well! Safety first with 4D cubes." (laughs)

Alright! So that is absolutely incredible.

Oh! And it's a bit like the triangle, one dimension lower.

But here, it's a tetrahedron.

It's a triangular pyramid that comes in, distorts and goes out facing the other way.

Absolutely amazing. But, you ask yourself.

Is this Matt's all-time favorite shape?

And, no. It is not.

Right! My All-time favorite shape is the 4D equivalent to the Mobius loop.

You can make a Mobius loop...

Oh, by the way, what I love about doing family shows at the RI --

because everyone's here in families, so it's all family units --

a lot of parents-and-kid...

My game, my play, while I'm talking is Guess Who Dragged Who Along?

[Audience: laughter] So good.

Often you see the parent, elbow go out, "Look! I told you: learn it!"

And often you see the parents just go, "I'm going to have to buy so much stuff when we go home."

Anyway, so my all-time favorite shape...

(It's okay, I'll finish on this, right?)

My all-time absolute favorite shape...

To make a Mobius loop... A Mobius loop only works in 3D.

It is a 2D surface but it requires three dimensions to join it together.

And we can extend the pattern into a higher dimension.

So when I made a normal Mobius loop

you could imagine that I start with a square

and then I join the two ends together.

And you can see here I've indicated with arrows how to join the ends together.

So I want you to join the two ends together so that the arrows match.

We call them matching instructions.

And if you try and do that with a piece of paper that you keep on a flat surface,

you can't get the arrows to line up.

If you try and stretch it around and put them together, they don't match.

If they matched, you'd get a cylinder with no twists. Right?

To get them to line up, what you need to do

is take one end, lift it up and turn it over, and then put it down.

And to do that you've had to pick it up into the third dimension and put a twist into it.

And now the two ends do line up.

Oh, and if you are a 2D creature looking at this...

you'd go, "Hang on! This is cheating..."

"...because it looks like this edge here just magically goes through the middle."

The 2D creature is like, "You can't just put an edge in the middle of the face. That's ridiculous."

And we're like, "Oh, no-no-no, it doesn't go THROUGH. It goes OVER."

But when it goes OVER in 3D, it looks like it goes THROUGH from a 2D point of view.

Only in 3D can you fully appreciate the wonder of the Mobius loop.

Now to do it one dimension higher!

What you do is you start with a piece of paper, but this time with more matching instructions.

I've got two sets of arrows: I've got the red ones and the blue ones.

And the red ones are easy. I can join those together and I get a Cylinder.

I then have got to join the blue ones together.

And to do that I've got to stretch it around, I've got to line them up...

Now, if they matched, and I joined them together,

you would get a tube that joins up with itself.

That's the shape that in Mathematics we call a 'Torus'...

...which, in Physics, they call a 'Doughnut'.

[Audience: laughter]

Physicists, man!

And so it is an amazing shape, the Torus (Doughnut)!

And so I've got a model of it here. It is an amazing shape.

You can do things on a torus that you can't do on a flat surface.

There is a famous problem in Maths.

If you want to try this afterwards, it's a great puzzle.

It's called 'the Utilities Puzzle',

where you've got three houses and you want to join them all up to three different sources of utilities.

You've got the Gas Company, Power Company and the Water Company.

And you need to join them up such that the pipes don't cross.

So you've got to draw a line from each house to all three utilities

with no lines crossing each other.

I called it a puzzle. But what mathematicians call a puzzle, in this case, is impossible.

Which is proof that mathematicians can be jerks.

You cannot solve this on a flat surface, but...

...if these houses were on a doughnut...

you CAN solve it.

In fact, I've got this mug which has this printed on it,

because it's possible to solve it on a mug because you have the handle.

A mug is the same shape as a doughnut because you have the extra bit stretched around.

Oh! any Maths people in: if you're familiar with the Four Colour Problem...

any flat surface will need four colours or fewer to colour in any combination of regions,

so that no two contacting regions have the same colour.

On a torus you need seven colours.

And so I designed a mug with all seven colours on it.

I've got seven different regions and each one contacts all six other regions to show that they all have to be different colours.

I've brought it along.

I you want to have a look at this afterwards, come down and have a play with it.

There it is, there. There is my seven-colour torus.

Come down and have a look at it, have a play with it.

Absolutely amazing..

Torus, wonderful, love it.

It's a 3D shape.

We've all played with it before. You may have eaten one.

We can do better.

What would a 4D twisted doughnut look like?

And for that to happen, you need to join the two ends of this tube together, so they face different directions - so they twist different ways.

But how do you turn over the end of a tube? You can't rotate it, 'cause it would still be clockwise or anti-clockwise.

The only way -

to join the two ends of this tube together, so the arrows match,

is to shove one side through the other one, then fold it back, then join it up.

And then you get this shape. This is a twisted doughnut.

It is a shape called the 'Klein-Bottle', it only properly works in 4D

because this bit here looks like it goes through the side - in fact i got a 3D Model of it

it looks like this tube goes through the surface

in 4D, it doesn't. In 4D it goes around, it's as good as a donut in four dimensions - it doesn't go through itself

Annoyingly, in 3D - this is the 3D Shadow of it - it always looks like it goes through itself

Very disappointing, but what are you gonna do ?

Because in 3D, you can at least make it, you can make 'em out of glass.

4D glass - very expensive

[Quiet Laughs] I have brought with me

this one here, so if you wanna have a play with it

there's one down the front here

it's called the 'Klein-Bottle', it's my all-time favourite shape, it's absoluty amazing

actually i got more then one model

I've got another one here

which is based on a Conical Flask.

Alright and so - all these are so good actually because - alright

- is this sink plumed in?

We're about to find out!

[Laughs]

It's the RI I'm doin' experiments

If you pour water into ... the bottle, you can see it goes in fine

look at that- right? and then if you try and get it back out again

that's a lot more difficult

alright so, I just really I'm doing this over a national treasure right?

this is Faraday's desk and I'm there going oh look at the water right?

and so.. the world's first electrical motor was demonstrated right here

I made some hearts

Ah right

so this is the least side of what's important though right?

actually next time you go to a party

bring drinks in one of these alright

it's so good because you of a bit like oh cheers thanks

What?!

[laugh]

but how did it get in there?

right, so.. oh you can get it back out again

if you get it into the handle, it will go around the handle

and then out oh! look!

[laugh]

all over the floor

OK if you would help me drag the desk back on top of this right? and no

so this right, if you want to play with this

take it over there

ah I'll leave it there

you're welcomed to come and have a look and play out with that

it's absolutely brilliant

although as always people say

right OK, so the Klein Bottle

it's a wonderful 4D - well it's a 3D shadow of a 4D shape

people would say what is the point?

what is the use of it?

well for a start it makes for a mighty fine hat

[laugh]

you can knit the 3D shadow of the 4D

because so, I'm so excited

so when I realised that this could happen

I went to my mum, right, because my mum can knit

I said mom

you have got to knit me the 3D shadow of the 4D twisted doughnut

so I can wear it as a hat

[laugh]

and she was like..

"What have I raised?"

and in the mid of the conversation where I was talking

perfectly reasonable maths

but she was speaking fluent knitting

but eventually we worked it out and we made one

I've got it here

Right I've got a picture of my hat right there.

So you can see...

Oh, if you wanna see what it looks like when I'm wearing it

It looks like that

So...

[laugh]

What happens, is the hat becomes a tube

The tube then goes through the side of the hat

And then if I reach in there and I pull the tube through

The tube folds back on itself

Right?

This is a neverending tube of knitting

It's a doughnut of knitting that goes through itself

And it keeps your head warm, it's so good!

If you want...

That is just creeping me out now...

If you want the knitting pattern

Seriously, send me an e-mail! I'll send you a copy!

It's so good!

So you may have noticed that this hat is stripey

The first one of these my mom made for me

Which I called "the prototype"

Which she called "the perfectly good gift"

It didn't have any stripes, it was all the same color

I said, "Mum, could you make me another one of these,

that has stripes?"

Well yeah, I can do that, I just change the color of the wool as I go!

I said "Oh, good, could the stripes be different thicknesses, could I have thick ones and thin ones?"

She said "Yeah, I just do a different number of rows"

I said "Ah, good, good..."

What if I gave you a long list of numbers,

Could you make each stripe the thickness of the next number on the list?

And so this, Ladies and Gentlemen, these are the digits of Pi!

And these are on a 4D-hat!

This is officially the world's nerdiest hat!

If you would like to meet the hat afterwards,

I will leave it down here

You can come and take a photo of you wearing the hat

That is absolutely fine

If you can convince a loved one to knit it for you

Sorry parents!

They come in a variety of sizes, send me an e-mail!

Now...

Very quickly, before we wrap up, we'll have a little bit of time for questions

The very last thing I wanna show you

Before we stop for questions and then you get to go back into the real world

Is what if, when you go home

You wanna have your own 4D-shape to play with

And you haven't got the time or effort to knit it

Well, you can go online and play with a 4D-cube

And the reason you can do that, is because online there is a 4D-Rubik's cube

So, here's our standard 3D-cube

Pöh

Instead of trying to get 2D-stickers onto the 2D-surface of a 3D-cube

Here you have to get the 3D-stickers onto the 3D-cell of a 4D-cube

And to twist it...

If you click on one of the 3D-stickers, if it's the center of a rotatable face

It will then set it rotating

And it comes back after two twists!

That's why people think USB-cables are 4D-objects

Because you gotta plug it in, doesn't fit, turn it over, plug it in, doesn't fit

Turn it over, Hey!

Proof

The problem is to solve it you gotta move it around in 3D

Because what you are doing is grabbing and moving the 3D-shadow around in front of you

But you can still move it 4 different ways

So you got the standard 3 different ways you can drag it around

Turns out in the internet for the fourth Dimension you hold down Control, doesn't work on many websites

And then it will rotate in the fourth direction

So the outside pops into the inside

And there's always one face you can't see at any point in time

because you're in the wrong dimension

Oh, and one final problem

It looks like you're just moving around the 3D-shadow of the 4D-Rubik's cube

In fact though you're doing it on a computer monitor

And computer mice only move in two dimensions up and down, left and right

So what you're actually doing is you're manipulating the 2D-projection

Of a 3D-projection of a 4D-Rubik's cube

And if that sounds too easy there is of course a 5D-Rubik's cube

[laugh and amazement]

At which point I've gone probably too far

So I'm gonna finish there, thank you all very much

[Applause]