# Four Dimensional Maths: Things to See and Hear in the Fourth Dimension - with Matt Parker

May 16, 2024unexpected directions! Okay, so you can see my shoelace here, in fact, if I tilt it down for a second... It'll be a little bit less... there we are. Alright, so I'm going to tie my shoe mathematically, and the way this works is: you start with the two laces... With a basic knot... And you just tie them together... And most people makes some kind of It's ridiculous to move the cords around a knot, right? Actually what you can do is once you have tied the base knot... If you simply cross the two cords and they will tie themselves.

Therefore… That's the right amount of awe! You would like to learn? Yeah! Very good, I'll do it again. I'll give you a few seconds. If you have shoes with laces, choose your favorite shoe, undo the laces. If you don't have shoelaces, chances are the person next to you does. And they don't wear at least one shoe. So that everyone has laces within reach. Well, here we go. Are you ready? Yes. Okay, so under the laces, make that base knot like what you can see. Until now I have tied my shoe. Then, if you look closely at the base knot, one strand goes backwards and the other goes forwards.

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## four dimensional maths things to see and hear in the fourth dimension with matt parker...

So take the one going backwards, roll it up and up to move it forward, and hold it that way as it goes down. And then take the one that's already moving forward, roll it back and keep it on the way there. Now, all you have to do is take the two pieces you're holding, put them under the other loop, switch hands, tighten and you have a knot. Is there a delay in that? Yes. Oh, sorry, I'll do it again. If you missed it, quickly look a little to the left, okay, 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'll give you about 30 seconds to try it and then we'll continue with the talk. Well here we go. So I'm going to untie my shoelaces here. We have instructions with pictures and images. So, you start with the back one curled forward - hold it on the way down - The front one curled back - hold it on the way down - and then you cross both under the other loop, exchange hands, pull hard and it's done. The best thing about doing that: Not only will you save seconds of your life every day, but the knot you'll finish is mathematically the same as the standard knot you get from doing it the really long way.

So you're not sacrificing the quality of the knot, but you're doing it much faster, and I can say that mathematically it's the same thing because there's an area of mathematics called knot theory. There are mathematicians called knot theorists (which is the best name there is), but they are theorists. They just look at the knots. And so… There may be some puns in this section. And that's why knot theorists study knots, and unfortunately, right now, humanity's understanding of knots is terrible. Humans just... We're not good with knots at all. And knot theorists have no reliable way to figure out how to untie a knot.

And so, for example, if I show you this knot on the screen here (actually, this is an image from my book), that knot... Oh, and the convention when you draw a knot is whether the rope goes underneath it. , then you just leave a little space. Then, where it passes underneath, it disappears briefly and comes out the other side. So it's not just little pieces of rope that go under each other and the rest. We have no idea how best to untie that knot. No idea. And the way we like to undo knots is: you can pick a piece where the rope goes underneath and you can cut it from the bottom, bring it up and tie it together.

We call this a crossover switch, because you take a bit that crosses below and change it to cross above. And at the moment, the world record for undoing this knot is three crossed switches. So if you change it in three places, it will completely unravel and you will get a loop of rope without any tangles. No one has ever found a way to undo this with two crossed switches. No one has ever managed to prove that this knot cannot be undone with two crossed switches. It is an unknown and open part of mathematics. It is called the ten-eleven (10₁₁) knot.

I put it in my book, because what I want people to do is do it with a string and then try it, and it won't be in this arrangement, because people have tried all the cross switches in this arrangement, but if you do it in that arrangement , and then you pick it up and move it so that different parts cross each other, and then you make two crossed switches and take a photo of yourself pointing at them, because if it works and you can't recreate it No deal, but if it works, if you take a photo pointing at two bits, then you make crossover changes at those two points and it gets undone, and then you email me, mathematical fame and fortune is yours... ...for a very narrow definition of fame and fortune .

So, there are others. This is the easiest one I've found. Oh, I have received one email so far! The book came out in October; Someone sent me and said, "I tied the knot!" I thought, oh my gosh, so I invited a friend. We actually had a little knot party; We got some rope and it turns out that the knot they started with wasn't quite a ten-eleven knot. It was a different knot. We spent the entire afternoon. In the end we solved it. I've left myself very open to string-based trolling, now it's occurred to me. So please...only if you think you have it...oh god...uh...email me and I'll check it out.

And hopefully, sooner or later, someone will do it. And mathematicians are working toward a much more systematic way of doing this. Instead of just trying, trying to find a way that, given any knot, you can calculate which crossover switches will undo it. And at the moment, bacteria undo knots better than humans. And that's not good! Because when bacteria reproduce, they unzip their DNA and their DNA becomes very tangled. And what they do is they have little enzymes that go around and do cross-talk. They will cut the DNA from one side, move it around another strand, and rejoin it.

And those enzymes in bacteria are more efficient at undoing knots in DNA than anything humans can do. And right now, biologists are working with mathematicians to try to figure out, first, what bacteria are doing and, second, how we can prevent them from doing it, because if we know what they're doing, we can how to prevent their ability to untangle. And there's a whole future wave of medical ways that we could combat that bacteria if we had a better mathematical understanding of knots, and I think that's absolutely amazing. It is a fantastic and ongoing area of mathematical research.

Hmm. But the following is a little more fun math that you can try at home. So if you get tired of playing with ropes, I'll show you a fantastic activity you can do. Oh! I've brought some other

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for show and tell! Before we get to... Knots, true, are great, but there is a related area of links. And links are when you have more than one loop that intersect each other. And there is a famous set of links called Borromean rings. And they are represented on this beer can. This is a 1950's beer can. I spent years stalking it on eBay.I finally managed to buy one. It's the one in the photo here. You can come take a look at it if you want later. And the reason I really wanted it is because the logo is a mathematical set of rings, they're called Borromean rings. And the way they work, the way those three links interlock, if you break any of those three loops, the other two will separate as well. Therefore, it only holds together as long as all three are intact. You break up anyone, the rest break up. So here you can see that they add purity, body and flavor.

Because if you remove any of them from your beer, the drink simply falls apart. And I thought it's an amazing math logo. And you can do it for more links, you can link

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links. So if you cut any one the other three are separated 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 are also separated. And I had## hear

d rumors about this can; I finally managed to buy one; I was very happy. Recently, I was in New York giving some math talks and I went for a short walk before one of the talks I was giving.And I passed by a billboard advertising the same beer. Ballantine beer. They still make this beer. And I saw it; I took a photo with him. He was so… he 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 anguish. Because they changed the logo! They have removed the crosses from the logo! Okay, so I talked and in the talk I said, "We have to get a bottle of this beer." Then we walked around several bars in New York asking if you have this beer and finally found one that does.

I smuggled the bottle into the country. There you are. So if you want I didn't declare it to customs because I have to put what you brought in an empty beer bottle full of disappointment... and a.... so if you look up because 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. Just then they had taken all the brands, people have tried to say okay Matt, they've just turned it into a Venn diagram. I don't believe it, right, they've ruined all the math, all those links, so I'm very upset.

So if you want to come later, you can take a look at the beer's original and modified math logo. Well then. Move on to an activity you can try yourself at home. Now I'm going to show you my second favorite shape of all time. This is not the number one favorite way; We'll work our way up to that, but my second favorite way of all time, and to do this. Oh, this is not my second favorite way, by the way. This is a rectangle. I mean, don't get me wrong, big fan, but we can do better than a simple rectangle!

What you can do is, if you get a rectangle that is thin enough and you join the two ends together, you will get a loop. The mathematical technical name for this is boring form. You can give it an interesting shape by separating the two ends and turning one of them inside out, so that when you glue them together, you get a loop with a twist. Now many of you - it seems like the kind of people who come to these

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- would have seen one of these before. It was invented by a guy called Mr.Möbius at the end of the 19th century. And he was so excited when he made what we now know as the Möbius loop that he proceeded to name the shape. And, uh, the great thing... Not everything tonight is a fact; I should make it very clear. So the strangest thing about the Möbius loop is that it has all kinds of unusual properties. So some of you may have

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d that it only has one side and it also has only one edge, which is surprising. And there are all kinds of strange things; but the best thing you can do with a Möbius loop is an activity when you have two of them, which are exactly the same length and to get two of the same length, you can see that I have used reasonably wide paper.If I make a single cut in the center and since I originally glued them all together at once, I cut the ribbon, I will end up with two loops of exactly the same length. Okay, so I'm back to where I started, and now I've cut it in half... I... What?! I just left you speechless! Okay, so the Möbius loop can't be cut in half. I actually cut it in half, like I cut it right down the center, there's nothing funny there. If you cut a Möbius loop in half, you will get a single piece, a single loop.

And that just scares me. But it gets worse... If you take a longer rectangle to start with, this time when you start with your zero turn cylinder, instead of making one turn, if you make one, then two, then three, you get what's called a Three-turn Mobius loop. And if it looks familiar to you it's because it's the recycling logo, there you have it. You know, the next time you walk down the street and see the recycling logo, you can say "Hey! That's a three-turn Möbius loop!" That only works if there is someone next to you.

Otherwise you would look crazy. Anyway, that's it, a three-turn Möbius loop, I'm going to do the exact same thing again and try to cut it right down the center. In your own minds, without mentioning it, I want you to see if you can guess once I cut a three-turn Mobius loop in half, how many pieces of paper will I end up with? Will it still be a loop? Will I receive two? Or will I receive three? In your own minds decide... Well, turns out I understand... give it a second... here we go. Is it one, is it two, is it three?

Are two? It turns out that it is one, it is a single loop but now it has a knot tied. Right, so I started with a paper bow without knots, I cut it down the center and it got tangled, at no point did I undo the bow. So somehow a knot got there, oh and for any knot fans out there, there's a shamrock. And then (some of you will be here!), I can't get rid of it, it's like a trivial knot: it's right there. In fact, you need a crossover switch to get rid of it. And inIn fact, this is how knots are formed in DNA.

Because DNA is a very, very twisted and very long strand, when it reproduces it opens in half, just like I did, you cut it in half and when you cut a twisted loop in half it ties into 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 tied together, cutting a tangled thing in half makes it incredibly knotted. And that's where the knots come from when DNA reproduces, and you have to make cross changes to get rid of them.

But the fact that cutting a bow results in a knot really bothers me. But it gets worse... I skipped our friend's zero turn, because he has a pretty boring shape, if he cut it in half right in the center, he would only get two zero turns. It's not that exciting. I'm going to make a slight change, I have another cylinder here. I'm going to join both at right angles. I'm going to cut them both in half while they're glued together. Oh, and if any of you are going to try this after returning home (and many of you are!), make sure you put plenty of tape on both sides because if you don't stick it close enough, it will fall apart. while you cut them. there we go, of course, so again in your own mind without saying it without ruining it for anyone else, see if you can guess how many pieces of paper I'm going to get.

And, if you think you have a good idea of that... Guess what shape or shapes those pieces of paper will be. Going through the first, we get that... Going through the second... And I end up with... A square! Yeah! Yeah! So good! So good. Most of the audience down here... and all of you are my favorites. Anyone who is late or other people who now seem surprised are sitting here at the top. And occasionally people have shown up, particularly people who work in IR because They say what this mathematical comedy is about. So they've been coming.

It didn't happen, I was really hoping someone would walk in RIGHT AWAY. Because all they'd see would be me saying, "Hey guys... A SQUARE!" And you all say, "A square?! Oh my God, give it a hand for..." Wait until you see a triangle! My God! There's some advanced math tonight! Anyway! I think it's amazing that two loops cut in half make a square. In fact, you can reverse that to see why. If I put these two ends together again like this. If I identify those two edges, I get a strip of paper with a loop on each end.

And if I then put those two loops back together so they line up. Those are my two original 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. Two Möbius loops glued together at right angles! I'm going to cut! This is the last one, there are no more down there! I'm going to cut... without speaking... both, in half, at the same time. And then I want you to try and guess, in your own minds... without saying it out loud... how many... how many pieces. of paper and what shape or shapes will they have?

Well, I've cut all the first one and I have... that. Well, here we go. It'll take me a second to untangle them when I'm done, by the way, I'm cutting the first one. the second And I end with Two hearts that cross each other! Too late! Okay, so I held this up and you said "nah..." And then there was silence and one person went to wait a minute... We applauded the square... If we don't applaud this, we'll be officially dead inside Although it's already too much late. That's absolutely brilliant, right? You know what I got my girlfriend for Valentine's Day last year!

Two Möbius handles and some scissors! And instructions! Oh, and we got married! Last July! Then, you know, boom! Works! This is the most expensive math test I have ever taken. The wedding ring is an iron-nickel meteorite. It's a very different talk... See you later. What substance! Anyway, that's absolutely brilliant! If you want to make one of these There is a secret you should keep in mind When you make a Mobius loop, you are actually faced with a choice: you can turn it clockwise or counterclockwise and you get a right-handed or left-handed mobius loop They are mirror images between yes They have different shapes For the hearts to work You have to have the right-handed one attached to the left-handed If you join them both, and they are both the same spin, then you don't get two hearts.

You get like... ...a boat. And one thing, I don't know. And they're not even joined together! It's a world of disappointments! Oh, teachers! I just planned your next lesson! This is very good, particularly with sixth graders. Because if you do this with sixth graders, you do the whole lesson like this. If you email me, I will send you the worksheets. And as you go along, at the beginning of the lesson you say "Okay class, make sure you pay close attention and follow all the instructions very carefully." And everyone says "pfft, please!" "Follow instructions?!" And then you don't tell them about the twist And a third of them will accidentally get it right and get the two hearts And the rest will get the pot and the thing And then you walk around saying, "I told you to pay attention!" They will do it again, and they will do exactly what their friend is doing.

And it worked for his friend, but it won't work for him. Then his head will explode. I still miss teaching sometimes... It's absolutely so much fun. But I want to get to my favorite form, which is in the

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th## dimension

. Now I will show you what the#### fourth

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looks like. Let's rush to this. Many people are vaguely familiar with the dimensions of an area of mathematics. maybe you didn't expect it. So, when you are in school, you will have been forced to draw diagrams. You may 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 have only two directions in which you can move. You have up and down, and you have left and right. And that's an x-y axis graph, where you plot things on it. And in fact, you can start plotting a variety of coordinates. I have put all the combinations of zero and one here. And you do this quite a bit. You do it in math because it's fun. You do it in science, because if you have done an experiment and there are two different things. measure So you can plot two data bits simultaneously as data points on an x-y coordinate.

In fact, more people should use two-

### dimensional

graphics than do. Most stick to one-### dimensional

graphics, which I think is ridiculous. For example, football. The main football league. Have a leaderboard with only one dimension. It's just vertical up and down. If you wanted, you could plot all the major teams in two dimensions. And you would get a 2D leaderboard. It would look a little like this. I have drawn the network. wins That's the number, subtract the number of losses Neither of these two points is ridiculous And then in this one, I have 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 simply calculate which team is furthest from the origin. And they win, it's all very simple. I can't believe they haven't implemented this! You can use arguments to resolve arguments. If you're trying to find your favorite team, there you go. I think you've already done more sports than you expected. Do not be disappointed. You can plot great things in two dimensions. In this case, you have made all the combinations of zero and one. So what you've actually done is plotted the coordinates of the corners of the square. We can do the same in three dimensions.

Now we have three different directions we can move in: up, down, left, right And now you're out This is the reality we live in We have 3 directions we can move in, up and down, left and right And out and back. And then combinations of those, so we can cross and exit. Or we could just take a shortcut there. And if you put each combination of zeros and ones when you have 3 coordinates. And in science this happens a lot, you do an experiment with 3 different types of data. You can plot it on a 3D diagram. But if you join all the combinations of zeros and ones, in a 3D graph you get the corners of a cube.

Now you can make this one dimension higher. If you had a 4D plot, and the problem here is that we can't imagine that. Because for a 4D plot you would have left and right, up and down, out and back, and then another direction. Another direction we can't even try to visualize. Which is a shame. But it would be very useful. In science, you would do experiments where you have 4 different things that you are measuring. If you have 4 different types of data and you want to plot them on the same graph, you need a 4D graph. And you can do this, but unfortunately you can't see it easily.

Then you lose that visual link. But there are still some ways to visualize what a 4D cube will look like. And I'm going to try to show you tonight what a 4D cube would look like. And to do that, what's really helpful 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 that was perfectly flat, right? You could take a cube... one way is to take your cube and then deploy it to your network. So that's one way to do it. down one dimension.

You can take a 3D cube and deploy it on your network. In fact, what I have here is a video of a network folding and unfolding into a 3D cube Oh! I'm not trying to patronize you. But are we all happy, that's a video of a 2D network folding into a 3D cube? Well. Wow, mixed, okay. Because that is not it. That's not a video of a 2D network folding into a 3D cube. It's a video of a 2D web shadow folding into a 3D cube. Correct. At the top, I have the 3D cube. Then I can deploy it to your 2D network.

All I was showing you before was the bottom part. You can see that there is a light above it and it casts a shadow on that flat surface. But your brain is so used to things being in 3D That when I showed you the bottom part... your brain imagined the top part Oh! Of course, now you're seeing this projected on a wall. But don't think about it too much. Good! And then, just take the essence, okay! Okay, so what you're looking at at the bottom is the shadow of what's happening above. But you can reconstruct the situation above simply by looking at the shadow below.

And the cool thing is, just like 3D objects can cast 2D shadows... 4D objects can cast 3D shadows So, while I can't 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 we have here... This is our 2D network of a 3D cube. What I have next to me is the 3D network of a 4D cube. And just like before, you could see its shadow folding one dimension higher. I'll show you the 3D shadow of the 3D network folding into a 4D cube And it looks a little like this...

It goes back and forth That's not so bad... Now, if you look at the one you're used to If you start to feel a little dizzy ...in a bit of "dimension sickness", right! Look at this side. Because this one is good. This is for sure... And if you think about it When the lid spins on the hub on this one, we're used to it It looks like, in the shadow, it looks like it's stretching And we know it's not stretching We know it's just spinning, but when it spins one dimension higher, his shadow appears as if it is stretching.

So here, that purple cube at the top is not stretching. It's spinning. But when a cube rotates in 4D, it can be seen. as if it were extended in 3D And at the actual moment it is folded It looks like this, here That is the 3D shadow... of a 4D cube... But because it is a shadow, it has been projected in such a way that has "Perspective". "So, that blue cube in the middle is actually the same size as the red cube on the outside, except it's farther away in 4D. And when things are moved further away in 4D, they look smaller...

In 3D , as if I were to show you this... Everyone knows that the blue square is smaller than the red square. Wow... Well, of course, the blue square is smaller than the red square, it's just... it's the same size. It just looks smaller because it's farther away. So you know, red square, blue square, same size. But one looks smaller in its shadow actually, if you put this cube in rotation. , they take turns being the big one. Right! The blue is big... And then the red is big... And it looks like they are going through each other But they are not really going through each other. front and back But if you just look at what's happening on the screen The red and the blue are pretty much in the same place, it looks like they're passing... there And then they take turns getting big and small I can do the same I can take a shadow 3D of a 4D cube and then I can make the 4D cube rotate 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. same size all the time They're just getting closer and farther apart in 4D And it seems like they're passing through each other But they're not passing through each other They're going in front and behind, one higher dimension When they're going in front and behind in 4D from our... you know... pathetic 3D point of view.

They seem to pass through each other. In thisnerdiest hat in the world! If you later want to know the hat, I'll leave it down here. You can come and take a photo with the hat. That's absolutely fine. If you can convince a loved one to knit it for you. Sorry, parents! They come in a variety of sizes, email me! Now... Very quickly, before we finish, we'll have a little time for questions. The last thing I want to show you before we stop to ask questions and then return to the real world is What If? , when you get back home You want to have your own 4D shape to play with And you don't have 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 are a 4D Rubik's cube.

So, here is our standard 3D Pöh cube. Instead of trying to place 2D stickers on the 2D surface of a 3D cube. Here you have to place the 3D stickers on the 3D surface. cell of a 4D cube And to rotate it... If you click on one of the 3D stickers, if it is the center of a rotating face, then it will start to rotate AND it will come back after two turns! This is why people think that USB cables are 4D objects. Because you have to plug it in, it doesn't fit, turn it over, plug it in, it doesn't fit.

Turn it around, hey! Try The problem is that to solve it you have to move it in 3D Because what you are doing is grabbing and moving the 3D shadow in front of you But you can still move it 4 different ways So you have the standard 3 different ways you can drag it It turns out on the Internet for the

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Dimension you hold down Control, it doesn't work on many websites And then it will rotate in the fourth direction Then the outside appears the inside And there's always a face that you can't see at any time because you're in the wrong dimension.Oh, and one final problem. It looks like you are simply moving around the 3D shadow of the 4D Rubik's cube. In fact, even though you're doing it on a computer monitor and the computer mice only move in two dimensions up and down, left and right, then what you're actually doing is manipulating the 2D projection of a 3D projection of a 4D Rubik's Cube And if that sounds too easy, of course there is a 5D Rubik's Cube. At that point I've probably gone too far. So I'll end there, thank you all very much.

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