Dr. Michael Starbird, professor of mathematics at The University of Texas, Austin, shows how to bolster creativity by using analogy to build new concepts.
See and share lecture notes, practice tests, and teaching materials.Get access now
Dr. Michael Starbird: That sounded pretty good. Well, it’s great to be here. Thank you so much for coming. I wanted to talk to you a little bit today about the fourth dimension. I’m hoping that you don’t have much of a sense of what that is. Any idea? Fourth dimension? The fourth dimension is an idea. One of the things that I like to do most is to teach people how to come up with new ideas. To me, that’s the real goal of education. You want to teach people to be creative, to think of new things. I happen to believe that being able to come up with new ideas is not magic, but there are actual steps that you can take that lead to new ideas, ways to solve problems that you didn’t imagine you could solve before, and that these are rather methodical and mundane strategies that we can use. The exploration of, and development of, the concept of the fourth dimension illustrates some of these very well. So I think you’ll enjoy thinking about the fourth dimension.
So let’s start by…. Maybe I should just ask for some of you to ask some questions about the fourth dimension. What would you like to know about the fourth dimension? Let’s just start with that. The way I do it, I don’t ask for questions when I’m teaching. I just tell people to come up with questions. It’s not a question of asking whether you have any questions. You have to come up with questions, so talk to the person next to you. Come up with a question about the fourth dimension, and then I’m just going to point to random people and have you speak. So there’s no getting out of it. You understand this? You understand? So right now, and you just have, like, 30 seconds, go ahead, talk to …
Starbird: OK, so let’s go ahead. Let me just say, everybody’s thought of a question, so I’ll start. Mabel, do you want to tell me a question? You’re Mabel, right?
Starbird: OK, Mabel, go ahead. Tell me a question.
Mabel: What is the fourth dimension?
Starbird: What is the fourth dimension? That’s a very basic question. What is the fourth dimension? That’s a great question. Let me ask for another question. What’s your name?
Starbird: Cecilia. So, Cecilia, what was your question?
Cecilia: Can you measure it?
Starbird: Can you measure it? OK. Can you measure it? Are you …
Starbird: Jesse. I would’ve guessed Jesse. So, Jesse?
Jesse: Does it exist?
Starbird: Does it exist? Does it exactly exist in the physical realm? Yeah, that’s a great question. So let’s see, what was your name? I can’t …
Starbird: Genevieve. So, Genevieve, what do you …
Genevieve: [Inaudible] time?
Starbird: Isn’t time the fourth dimension? Yeah. So, this is a good question. Let’s see, how about over there? Does somebody want to say what’s your name and what’s a question? You have a question? What’s your name?
Audience member: My name’s [inaudible].
Starbird: Yeah, how’s it applicable to real life? OK. OK. These are all good questions. Any other question that you want to ask? Tara, you have a question? Sarah?
Sarah: Is [inaudible]?
Starbird: Discover…. It will be. It will be after today. Or nightmares … things that happen at night. OK, OK. Yes, Alex?
Alex: How [inaudible]?
Starbird: Yeah, how it was discovered, or invented? Yeah, yeah. Yes, and what’s your name?
Katie: Is there a movie or TV show that’s gotten it right?
Starbird: I’ll answer that one right away. There’s a book from 1872 or something by Abbott, a guy by the name of Abbott, who wrote a book called Flatland. Have you heard of Flatland? Anybody heard of Flatland? Flatland, it’s a story about a two-dimensional world, where all the creatures are two-dimensional. It’s really an allegory of the class structure of England back in the 1800s. It does get it right. There’s some students who were actually in a class of mine, produced a movie where they made a cartoon movie of this two-dimensional world, and they got Martin Sheen and Kristen Bell and Michael York to be the voices of these characters. It’s very cute. It’s called Flatland: The Movie. And so you can watch it. It really is very, very cool. And it does get it more or less right. Yeah, so this is right.
OK. And in fact, that actually is a good segue into…. So, you asked what is the fourth dimension. One thing about the fourth dimension is that you have several words there, fourth and dimension. And if you think about dimension, the first thing that you might ask is, well, what is dimension in itself? And what dimensions are we familiar with? If I said I’m going to talk about the third dimension, you’d sort of have a sense that we’re talking about, like, this room, right? And it’s sort of three-dimensional. We view it as sort of three-dimensional because if you go to a spot in the room, a fly is flying up in the room, then you’d need three…. You have sort of three directions you can go: front and back, side to side, up and down, and three numbers would tell you the location of that fly. It goes so much forward, so much to the right, so much up, and that’s where that fly is. So, there are three degrees of freedom for three-dimensional space.
One of the themes that I really focus on in developing new ideas is to, say, take some complicated idea and think about simpler versions of it or the related analogies in lower dimensions, and see if we can build up from the simple to the more complex. So, let me ask you, if this is three-dimensional space, this room, or it’s actually the abstraction of the concept of this room that goes off indefinitely in every direction, what would two-dimensional space be? Two-dimensional space?
Starbird: The floor or the plane, just a flat plane that goes off. It’s just flat. How about one-dimensional space? A line. A line. One dimensional space is a line, right? It goes off … because you just need one point to determine a point on a line. You just give a number and there it is. That’s a point on the line. So, a line.
Audience member: [Inaudible] ceiling [inaudible] grid.
Starbird: Oh yeah, it does. It does. This is a good analogy for two-dimensional space. We can all look up and see the grid where you have two directions, forward and to the side. OK.
One-dimensional space is a line. How about zero-dimensional space?
Audience member: A point.
Starbird: A point, yeah. If you lived in zero-dimensional space, you would never miss a party.
Starbird: Exactly. You’d be home by now because you’d be…. There’s just one point, you know? So everybody’s in the same place. So, zero-dimensional space has a lot of coziness to it.
So what we have is we have these examples: zero-dimensional space, one-dimensional, two-dimensional, three-dimensional, and we’re trying to create a concept of the fourth dimension. One way to do that is to say, “Well, can we see the relationships among the familiar objects that we have [in] zero-, one-, two-, three-dimensional space? Can we see how they relate to each other with the goal of saying, “Can we create something that is outside of our experience, namely, the fourth dimension?” OK?
Let’s think about the relationships among the different dimensions that we already know. For example, a way to get to one-dimensional space from zero-dimensional space, zero-dimensional space being a point, is that you can just put a lot of points together, namely one for every real number, and you get a line. Now, how can you get from a line to two-dimensional space? What do you do?
Audience member: [Inaudible] directions.
Starbird: Right. You change direction. You move. You took a whole bunch of collections of one-dimensional space, namely one for each real number. No two of them touch each other. You have a bunch of them. They spread out, and that gives you the plane. How do you create three-dimensional space? What do you do? Do it with your hands. Do it with your hands. Everybody, hands, hands. OK? Two-dimensional space, what do you do?
Starbird: Right. Right. You have one plane for every real number, and you get three-dimensional space. So, how do you create four-dimensional space? Ready? All together. One, two, three….
Starbird: OK, let’s start again. You seem to have…. You know, you sort of lost the momentum there. I thought you were doing great, and then suddenly we came to a halt. Let’s try again. OK.
So, here is two-dimensional space represented as a collection of one-dimensional spaces. I’ve just drawn five of them here, but there’s really one for every single real number. So, like [inaudible], I didn’t draw it there, but you could draw it there. Three, four, and so on. So there’s one for every real number, and you stack them up, and that gives you a representation of two-dimensional space, right?
Now, three-dimensional space, as you all said, was that you took two-dimensional space and you took one for every real number, and you stacked them up, and you got three-dimensional space. This representation is a stack of two-dimensional spaces to give you three-dimensional space. And by the way, questions had to do with practicality of this concept: What do you use this vision of three-dimensional space for? Who uses this all the time? Anybody? Talk it over. Find…. Where is this used all the time? I’ll give you a hint. Anybody who has bad knees, like I have from my racquet sporting years, has experienced this concept. Talk it over. What is it?
Audience member: MRI.
Starbird: MRI, right, right, right. MRI. Right? An MRI or a CAT scan. That’s exactly the way the doctor looks at your knee, is that you look at slices and then, conceptually, the surgeon looks at that and is able to see the features by putting together in your mind’s eye how those things fit together. This is a real, practical vision.
So, now you told me, how do I go…. This is a way to construct three-dimensional space from two-dimensional space. Tell me how to construct four-dimensional space from three-dimensional space. What do you do?
Starbird: But how do you take three-dimensional space and construct four-dimensional space?
Audience member: Changing the direction?
Starbird: OK. OK. Let me give you a hint.
Audience member: [Inaudible] box. You could add boxes [inaudible].
Starbird: OK. So this is a problem. So wait, the point is that what you need to do—and this whole lecture is going to be about the one word, which is retreat. Whenever we are asked a question about the fourth dimension, what we’re going to do is retreat to the third dimension and the second dimension. We’re going to see how those ideas were built up, and then by analogy, we’re going to do the same thing to get to the fourth dimension.
So, the third dimension was created by taking stacks of two-dimensional space to create three-dimensional space. How do we construct four-dimensional space? Well, it’s very easy. What we do is we just take a stack of three-dimensional spaces, put them together to create four-dimensional space. What’s the problem with that? You said what the problem was. What’s your name again?
Starbird: Binket. Binket. What did you say was the problem with this?
Binket: It’s three dimensions.
Starbird: Oh yeah, but it’s not because—here’s what…. You see, what I’m doing is, I am creating the concept of the fourth dimension. I’m creating a concept. And the way I’m creating the concept is by analogy. It is the result of what you would get by analogy if it were possible to take all of three-dimensional space and then take another copy right near it, and another copy right near it, that didn’t touch at all, but extended in all directions, and other copy that didn’t intersect, but it just went off in all directions. There’s no reason you can’t, in your mind, imagine that that could happen. Is that a problem? Is that a problem? Pardon?
Binket: Hard to visualize.
Starbird: Hard to visualize. Actually, it turns out that it’s not hard to visualize. It’s not hard to visualize, because all of you visualized it today. Somebody asked, “Isn’t the fourth dimension time?” Right? Well, the answer is that we’re not going to use time as an analog of the fourth dimension, and the reason is that time has a very unfortunate future, which is one direction. Yeah. This is very sad. But it just goes in one direction—very, very sad to me. Anyway, it’s going in one direction, and as far as we can tell. The point is that it doesn’t have the freedom and the…. It’s not the same as the other dimensions. So, I don’t really want to use time as a metaphor for the fourth dimension, but it does tell you how to imagine three-dimensional spaces stacked together that are right next to each other, and don’t intersect. And I’ll tell you how.
Think of this room now. Now, think of this room now. Now. Those are all different. Those are all different, right? You had the entire room now, and then you got the entire room now. We don’t have any problem thinking about that. That is a four-dimensional concept that we use every day. Think about your day. The concept of your day, is it really a four-dimensional entity? At every moment of time, you had the entire world here. There was no problem having the world here now and now. Those were two different things. They were close to each other, but didn’t touch. OK?
So this is the concept of…. So, time is a great metaphor for the fourth dimension, and it tells you how you can conceptualize things that you can’t see. The goal of the fourth dimension and understanding it is not to visualize the fourth dimension as a gestalt. That’s not the goal. The goal is that we’re creating a concept by analogy, and the analogy is to how the third dimension is created from the second dimension, how the first dimension is created from the zeroth dimension, how the second dimension is created from the first dimension. We have those familiar concepts, the relationships between dimension and dimension. What we’re going to do is just extend that concept to say, “I’m going to make it so. I’m going to imagine the concept that we can create by analogy with the way the previous dimensions were created.”
OK, so let’s see now how are we going to explore this idea. Well, first of all, we have a way of representing it. Here it is. Right? These are five levels. I could put more levels in there. I can imagine these levels fitting together in the same way that the two-dimensional planes intersect each other, namely, if you have a particular point, like the origin point. If you have the origin point at level zero, and the origin point at point one and at point two and point three, that they’re close together. Every corresponding point is close to one another, and you can sort of understand by analogy what the fourth dimension is.
Let’s get acquainted with these dimensional representations by starting with something familiar. And we’re always going to do this, retreat back to the familiar, and then learn from it to extend to the fourth dimension. So let’s go ahead and start with this. I have drawn an object in the plane, two-dimensional space, and I’ve just shown you how it impinges on the five level lines that I chose. So, what is the object? You have to imagine extrapolating to the levels, between those levels, and tell me what is the object that those dots represent in the plane. So talk it over for just one, 10 seconds.
Starbird: And you know, you might not be exact. Just get a general idea.
Starbird: OK, so let’s go ahead and say what these are. Let’s see. So, Jesse … but I’ve forgotten your name. What’s your name?
Starbird: Alex, so Alex … Alex and Jesse, what do you think it is? It could be anything, because I didn’t tell you what’s in between, but what does it look like it probably is? Yeah, a diamond or a circle or an oval or something like that. Because if you do it with your fingers, it starts at one point. It gets two wider, two wider, two closer together, one at the bottom. So, it basically is a diamond or a circle. You know, I’m not being too careful about which it is. OK.
Here we go. Now, here’s an object in three-dimensional space. You see? And I’ve shown you how it impinges on levels, on level minus one, level minus a half, zero, a half, and one. You can just extrapolate in between what it would be. What is this object? This is depicting an object in three-dimensional space.
So, you are now a surgeon. You’ve seen this on the MRI. What is this object? Talk it over for a second. Talk it over for a second.
Starbird: OK? Let’s go ahead and see what you have. OK. Let’s see…. Now I’ve forgotten all of your names. What’s your name?
Nicole: My name? Nicole.
Starbird: Nicole and …
Starbird: Cecilia. OK. So, Nicole and Cecilia, do you have a guess what this is?
Cecilia: Well, that your possibilities [inaudible] third dimension [inaudible] no sense, what it actually is, could go on infinitely. It could be an egg shape. Could be uneven.
Starbird: So pick one.
Cecilia: I’m going to say it goes on infinitely.
Starbird: But, OK…. Well, I want you to do it with your hands. OK? This is level one. Put your hands up. Level one. OK? What is on level one? A point. OK, that’s good. Use one … OK. Now, what is on level a half?
Starbird: Do it with two hands. OK? What’s at level zero. Two points. Wider. What’s at level minus one … one half. What’s at minus one? So what did your fingers draw?
Cecilia: An oval.
Starbird: Oval. That’s right. That’s what it is. It’s an oval. It didn’t change to something like a round, sphere, or anything like that. It is just an oval because you did it with your fingers. One point, two points, two points, two points, one point. Right? OK?
Now, by analogy, what is this object in four-dimensional space? Ready? One, two, three.
Starbird: Oval, yeah. Look, we saw, it didn’t change its intrinsic character going from one dimension to two dimensions. It was an oval. Going from two dimensions to three dimensions, it was still just an oval. But in three-dimensional space. Now it’s an oval in four-dimensional space, right? Why? By analogy. We’re constructing a concept. We’re constructing a concept by analogy. Let’s do another one.
Here’s an object in three-dimensional space. OK? Can you see what it is? First of all, talk it over, and I want to see you talk it over with your hands, with your hands. Right? You have now … and here’s what you should be doing. You have these levels. You have these levels. Jessica? Is it Jessica?
Starbird: Katie. Katie, I meant. It’s not like this, it’s like this. Level, right? I mean, at each level you know what’s happening at each level. OK? And now it’s putting it together. So, ready? On the count of three … on the count of three, does everybody have an answer? Everybody have an answer?
Starbird: OK, ready, on the count of three. One, two, three …
Starbird: OK, I didn’t hear. I didn’t hear. Let me just ask somebody. What’s your name?
Starbird: Robert. So, Robert, did you have your team there? Nathan? Pardon? A sphere, OK, yes, that sounds good. Why? Let’s do it with your fingers. You have a point. Then what do you have at this level? Circle. Then you have a circle here, and you have a circle here, and then you have a circle here, right?
OK, now you’re not quite right though, because you failed to remember where I come from, the University of Texas. Exactly. Of course, it’s a football, Richard. Exactly—what are you thinking? It’s a football.
Robert: It’s a football.
Starbird: Duh. Of course it’s a football. Thank you, Richard. OK. Sweet potato.
Audience member: Sweet potato. That’s like a football.
Starbird: It’s somewhat like a football, but University of Texas: football. It’s funny, you don’t go root for the sweet potatoes. It doesn’t fill the stadium. Have a bunch of sweet potatoes fighting it out. Now, here’s an object in four-dimensional space. What is this object? On the count of three, one, two, three.
Audience member: Football.
Starbird: Football. It’s a football. It’s the same idea. There’s no … it does not itself change. It’s still a point, a circle, a circle, a circle, a point. It’s still a football.
OK, now here we go. Now, one of things I always like to do is to prepare people for various walks in life. You don’t know what students are going to become. And even yourselves, you may change jobs. Sometimes people have a certain constructive direction in life, but some people choose to take different paths. For example, maybe they want to become a thief. But I want to serve everybody, so I want to help out even the people who want to become thieves. So, let’s suppose you have a…. In fact, let’s do it here. Suppose that you have … this is a safe here. Do you see this? This is an iron safe. You can see it. This is the boundary of the safe. It’s just solid, solid iron, four inches thick of iron all the way around there. That’s a safe. And in it is something of really great value, something that you really want.
Do you see what’s in there? It’s a math book. It’s a math book. You really want this. So you want to steal it, but you can’t get into it because it’s a safe. It’s closed. So, what do you do? But it’s just at the zero level of four-dimensional space, and suppose you’re a four-dimensional creature, how could you steal that book from the safe without opening the safe? It’s a challenge. So what do you do? What do you do when you’re thinking about this? Talk it over for just five seconds. Talk it over.
Starbird: One, two, three, four … OK, there’s only one correct one-word answer. There’s only one correct, one-word answer, and it is one, two, three …
Audience member: Football.
Starbird: Retreat. Katie had it right. Retreat. Retreat. Because what you need to do is to say, “I’m learning something by analogy. I don’t know about four-dimensional space, so what I better do is go by analogy to one dimension lower and see how to do it, and then by analogy I’ll be able to deal with the fourth dimension.” So, by analogy I go back to what the analogous situation is in three-dimensional space as a stack of two-dimensional spaces. In other words, I’m looking at the familiar world, three-dimensional space, in a somewhat unfamiliar way. And it’s looking at the familiar world in an unfamiliar way that opens our eyes to the possibility of this new world of the fourth dimension.
OK. So, what is the analogy of having a safe, a solid safe, at the zero level of three-dimensional space, but not having it exist anywhere else? The analogy is to have a safe in two-dimensional space. So, let’s think about, well, what is a safe in two-dimensional space? Well, it’s something that would protect something inside it here. It would protect this from two-dimensional creatures who were outside here. So this is a two-dimensional creature, two-dimensional creature, and this two-dimensional creature can’t get in because the two-dimensional creature is confined to this plane. The two-dimensional creature doesn’t see up or down. There is no up or down. It’s confined to the plane. So that little band there completely obstructs the ability for this creature to get into that safe. That is a completely impregnable safe. Do you understand that? If you were a two-dimensional creature? Right.
But suppose you’re a three-dimensional creature and you wanted to steal that book from that “safe.” Would you have any trouble doing that? No. What would you do?
Starbird: Exactly. Exactly. You would just come down from the top. There’s nothing in the way. There’s nothing at all in the way. You just pick it straight up. You pick it straight up, move it over, you could move it back down. So, if a three-dimensional creature is trying to steal this book from a two-dimensional safe, there is no obstruction to it.
Now, by analogy. Let’s look at the same situation in four dimensions. You have at the zero-dimensional level of four-dimensional space, you’ve got this book inside the solid safe. How do you get the book out? How do you get it out?
Starbird: You’re now a four-dimensional creature. You’re a four-dimensional creature, and you’re trying, at the zero level, the zero three-dimensional space level, you have an iron safe with a book inside. And you’re a four-dimensional creature, and you’re trying to steal the book. How would you do it? Talk it over.
Starbird: OK, Mary, how do you do it?
Mary: [Inaudible] inside.
Starbird: OK, Mary … Mary, what do you do when you don’t know what to do? OK, we’ll try it again. How did you get the book out of this when it was a three-dimensional thing? You had levels, and at the zero level here, you had a safe, which is like where my fingers are. And then there was something inside. Would you have any difficulty getting that out as a three-dimensional creature? Why not? Correct. You would reach in and go straight up. You wouldn’t go to the side. You would go straight up. You change the third coordinate of the book and then you’d go straight up. There’s no obstruction.
Now, we have, at the zero-dimensional level, we have this iron safe, we have a book inside, maybe right at the origin, what would you do? Yes, you can. No, no, no. It’s not a question of visualizing. That’s the point. What we’re doing is creating a concept by analogy. The goal is understanding the analogy, not having a gestalt. You will never have a gestalt of the fourth dimension. That’s not what you’re aiming for. You’re aiming for following the analogy. OK?
So, what did you do to take this little book out of this two-dimensional safe? And did what with it? Right. So, what do you do with this book that’s in this safe here? And do what? What … wait. What word did you use to take this out? What word will you use here to take this out? Up. Right? So where do you go? Where does this book go? Yeah. Correct. You take this book and put it up here. Right? There is no obstruction going straight up. There’s no obstruction of going straight up. The book disappears from here, and it becomes up here. Right? Because there’s no obstruction to it because the only coordinates that are changing are the fourth coordinate.
Think of the origin point. The origin point at this level, at this level, at this level, they touch each other, but they don’t touch any other point. The safe is all around, at not the origin point. So the origin point, there’s no obstruction to going straight up on that line, just like if you take where my thumbs are and go straight up, they form a vertical line. So there’s no obstruction to just going straight up, and the apparent obstacle is just not there. Does that make sense?
Starbird: It’s an enclosed safe, but it’s just in the zero level, the three-dimensional zero level of four-dimensional space, so that when we move to a higher level of four-dimensional space, there’s nothing there, no obstruction.
By the way, if we, in real life, had a solid three-dimensional space, and you thought of time as the fourth dimension, then the safe would exist here, here, here. It would exist all the way up, you see. What this picture is not of a safe in time. That’s not what it is. It just exists at this one level.
OK, so this is how to be a thief. By the way, if instead of being a thief, you wanted to be a surgeon … you see? If you had a person here and you looked at that person, you would be able to see…. If you’re a four-dimensional creature, you’d be able to se their brains and their guts and their stomach and their appendix. And there’s be no obstruction to just reaching that appendix and pulling it straight up without any incision. That’s good. That’s good. How are we doing here? I don’t know. I’m getting a little concerned with the way things are going.
OK, let’s move on. I want to ask you a question about a four-dimensional cube, OK? My first question is, what is a four-dimensional cube? How would you construct a four-dimensional cube? By the way, there’s only one correct activity to take, which is … retreat. Right? In other words, what does retreat mean?
Audience member: Go back.
Starbird: Right, you go back and look by analogy about how do you create a three-dimensional cube from a two-dimensional cube. Right. And so, you have to decide, well, what is a two-dimensional cube. Talk it over. Talk it over. Talk it over. How do you create a four-dimensional cube?
Audience member: Super [crosstalk]. Thank you so much.
Starbird: Oh, good to see you. Bye, now.
Starbird: OK, so let’s go ahead. How do you create a four-dimensional cube? Let me ask somebody. I don’t know your names back there.
Starbird: Cody, yes, that’s right. And …
Starbird: John. Cody and John. So, Cody and John, how are you going to create a four-dimensional cube?
Cody: A cube is a square, stacked up.
Starbird: Correct. Correct. Correct. That’s what … so, John and Cody are doing exactly the right thing, which is to say retreat. What is a three-dimensional cube? From the point of view of three dimensions as being a stack of two-dimensional spaces. By that analogy is what’s going to allow you to create the concept of a four-dimensional cube. So, here is what you just said, that a three-dimensional cube consists of taking a square, which is a two-dimensional cube, a square, and stacking them on top of each other, from let’s say, level minus one up to level one. Think of it solid. I haven’t drawn them as solid. But, if each one were solid, you would get a solid three-dimensional cube. Right? Great, great.
So now how are you going to create a four-dimensional cube? John and Cody?
Starbird: That’s absolutely right. There’s no challenge with this. You just take a solid cube at level minus one. You take a solid cube at level a half, and every other number between minus one and plus one, and you would stack them together and that gives you a four-dimensional cube. That’s great. Right? No problem.
Audience member: It seems like [inaudible].
Starbird: Does it seem like just another two-dimensional shape when you took a stack of two-dimensional squares and stacked them on top of each other? Well, OK, if you take a square, and then you take another square, and another square, and another square….
Audience member: It now has [inaudible].
Starbird: It has volume. That’s right. It’s now a three-dimensional, solid figure. Now, if you do the analogous thing, and you take a solid cube at level minus one, and then a solid cube at level higher, level higher, up to level one, you’re getting something that has now four-dimensional volume.
Audience member: [Inaudible] inside each other?
Starbird: No, no. They don’t touch each other.
Starbird: No, no. We have to retreat. Three-dimensional space is created from two-dimensional space by putting a bunch of two-dimensional spaces together. No two of them intersect each other, and yet they can be as close as they want. In other words, corresponding points like my … see the ends of my thumbs, they’re very close to each other. They don’t touch, though they’re corresponding points. Likewise, if I stack three-dimensional spaces to make four-dimensional space, corresponding points can be very close to each other, but the entire space does not touch. It’s a concept. You just accept it as a concept. It’s not…. You don’t want to say, “Oh, I can’t visualize it,” because that’s right. You can’t visualize it. Nobody can visualize it, because it’s a concept. It’s a creation of the mind. This is a concept. But, it’s a coherent concept, and we can ask questions about it and develop a whole theory and interesting features that we all can agree on, even though it’s just a concept.
OK, so here is a four-dimensional cube. That’s great. We have this four-dimensional cube. Now how are we going to explore the four-dimensional cube? I’ll tell you, there are lots of ways to do it. One thing that I think is sort of fun is to, when you’ve developed a concept like a four-dimensional cube, one thing that you can do is you can…. You know, I’m a math guy, so one thing to make nuance in your understanding of the world is to quantify things, to count things, like counting features of your object.
So, let’s count some features. What are features of, like, a regular cube that you might want to count? You know?
Audience member: Size.
Starbird: Size, yeah.
Audience member: Corners.
Starbird: Corners. Yeah. So you can count vertices. Those are the points. You can count edges. Those are the lines that connect points. You can count two-dimensional faces. And you can talk about three-dimensional faces when we go to a four-dimensional cube.
So, let’s think by analogy, and let me just ask you some questions. How many vertices does a four-dimensional cube have? How many vertices are there in a four-dimensional cube? And what are you going to do, on the count of three: one, two, three …
Audience member: Retreat.
Starbird: You’re going to retreat. What does retreat mean?
Starbird: Right. And what question will you ask about a three-dimensional cube?
Audience member: How many layers?
Starbird: No. Because the question that we want to ask is how many vertices does a four-dimensional cube have, so what’s the analogous question?
Starbird: How many vertices does a three-dimensional…. In other words, we want to keep the analogy as close as we can. We want it to ask the same question, but about one dimension lower so that we can develop our strength. OK? Very good.
So, sir, are you all right with that? OK. How are you going to figure out how many vertices there are in a four-dimensional cube? There’s only one right word. One, two, three.
Audience member: Retreat.
Starbird: Sarah? Retreat. Right. Very…. She was getting reluctant. I’m going to have trouble with Sarah. OK. OK. Here we go. I’ve drawn the two of them next to each other so we can look at them side by side.
My first question to you is you want to count the number of vertices in a three-dimensional cube. Now, if I actually had a box here, a cube, it’s rather easy to count the vertices on a box, right? How many are there, by the way? How many vertices?
Starbird: Eight. There are eight. But instead of … the concept of retreating is to look at a familiar object, namely the three-dimensional cube, in the rather unfamiliar way, namely as a stack of two-dimensional spaces. And if I can understand the correct answer in that vision of the cube, then I can apply my strategy by analogy to the fourth dimension. So, let’s go ahead and do that. Where are the eight vertices of this depiction of a three-dimensional cube? Where are they? Talk it over. Where … This, you see, is a three-dimensional cube, like an MRI. Here … you see? This is a three-dimensional cube, an MRI of a three-dimensional cube. So my question is: Where are the vertices in that depiction of a three-dimensional cube? Do you see where they are?
So, Jesse and Alex, do you see where they are? OK. So where are the eight vertices? Ready? You tell me where to put them. All four corners at the top. One, two, three, four.
Starbird: All four on the last one. And the other ones are not vertices. Right? Because, as you’re sweeping up, they’re on the middle of edges. They’re not vertices. OK. How many vertices are there in a four-dimensional cube? Sarah and Tara, I’m going to ask the two of you, so talk it over. How many vertices are there in a four-dimensional cube? And I want you to tell me where they are in this picture. Ready? Talk it over. Talk it over.
Starbird: Richard, turn around and help Sarah and Tara in case they….
Richard: Was I right?
Starbird: I don’t know. I didn’t hear you say, but just talk it over. Talk it over.
Starbird: OK, so how many are there? Tara and Sarah, are you ready? Are you ready, Sarah and Tara?
Sarah: We were thinking … retreat.
Starbird: Yeah. Which is over here.
Sarah: Four on the bottom, or eight on the bottom….
Starbird: That’s exactly right. That’s exactly right. You have these eight here, and then these eight here. Right. For a total of … because by analogy, the vertices just appeared on the bottom, they appeared on the top, because when you were sweeping, you started with the bottom square. You swept up to the top square. The vertices at the bottom were still there. They were still vertices. At the top, they’re still vertices, and nothing in between was. So, by analogy, you have exactly the same thing.
Now, let’s count edges. How many edges are there in a four-dimensional cube? How many edges are there? What are you going to do? Exactly. You’re going to retreat, meaning you’re going to answer that same question about a three-dimensional cube. How many edges does a three-dimensional cube have? But, I want you to do it by looking at the three-dimensional cube in this stacked way, and then ask yourself where does each edge come from. Where does it arise from? OK?
Starbird: OK. My question that I’m asking you is: How many edges are there in a four-dimensional cube, and you’re going to do it by analogy, right? By retreating, by asking how many edges does a three-dimensional cube have, and understanding where they came from. So, Katie, and what’s your name again? Nicole. So, Katie and Nicole, go ahead and tell me how many edges are there in a four-dimensional cube, and what are you going to do first?
Starbird: Exactly. So the first thing that you’re going to do when I ask you a question about the fourth dimension is to ignore the question. Right? Instead, answering and really working on the easier question. This idea of understanding the simpler cases deeply is a strategy that leads to success in everything you do. If you take something, a complicated question, and I don’t care what subject it’s in, and instead of working hard on something that’s too hard, instead say, “Is there an easier but analogous question that can inform me about the harder question, but just working on the simpler question will allow me to use the insights to proceed to the more difficult question later?”
So, go ahead and tell me what, when you retreated down to three-dimensional space, how do you look at this and tell me where the edges are?
Audience member: [Inaudible.]
Starbird: Which is here…. Oh, when I point to this, I’m not pointing … I realize this … OK … which is here….
Audience member: Four on the top.
Starbird: Four on the top.
Audience member: Four on the bottom.
Starbird: Here they are. One, two, three, four. Four on the bottom. One, two, three, four …
Audience member: [Inaudible.]
Starbird: And where do the four in between come from?
Audience member: Sides that connect to the bottom.
Starbird: Correct. When you sweep a vertex on the bottom, and you sweep it up, it sweeps up an edge. Right? So, for example, if I take the upper right-hand corner, the upper right-hand corner, the upper right-hand corner, the upper right-hand corner, the upper right-hand corner, I get a vertex. That is a stack of vertices … I mean an edge. I get an edge, because it’s a stack of vertices that create an edge. Right? Every vertex in the base gives rise to an edge in the higher dimension, because it sweeps up to create an edge. Right? And so, therefore, we have four more, namely one for each vertex of the bottom. Each one gives rise … this one, this one…. So there are four vertical edges, and then there were the edges you had at the bottom, the edges you had at the top, and then the ones that arose from vertices. OK?
Now, tell me how many edges are there in a four-dimensional cube?
Audience member: Twelve on the bottom.
Starbird: Twelve on the bottom. Right. I won’t draw them all, but there are 12 down here.
Audience member: There’s 12 on the top.
Starbird: Twelve on the top. Exactly right.
Audience member: And there’s the ones that connect those.
Starbird: And how many are those?
Audience member: Twenty-four.
Starbird: How many of the vertical edges are there?
Audience member: Twelve … eight.
Starbird: Eight. There are eight, because each vertex down here gives rise to a one-dimensional edge. So, four, five … 12 plus 12 …
Audience member: Twenty-four.
Starbird: Plus …
Audience member: Eight.
Starbird: Plus eight.
Audience member: Thirty-two.
Starbird: OK. Yes. Thirty-two edges. OK. Now, I’m going to ask how many two-dimensional faces are there in a four-dimensional cube? How many two-dimensional faces are there? I haven’t met you yet. What’s your name?
Starbird: John, John. And I haven’t met you.
Starbird: OK, so John and Irving, I want you two to tell me this, after one minute, I want you to tell me how many two-dimensional faces are there in a four-dimensional cube. Talk it over. Talk it over.
Starbird: OK, let’s go ahead and start. So John, what are you going to do to work on this question?
Starbird: You’re going to retreat. That’s exactly right. And what question, in retreating, what question will you ask?
John: What’s the formula for the number of faces in a cube?
Starbird: Correct. What are the … and by faces, two-dimensional faces of a three-dimensional cube. And where are they when looked at in this left-hand picture of a cube? OK? And where are they, John?
John: The top square.
Starbird: The top whole square…. Oh, wait a minute. Here, what am I doing? I have an opportunity to use colors. The top square is one.
Starbird: The bottom square. That’s two.
John: Faces [inaudible] around the sides.
Starbird: Right. And so, in fact, in what sense does … when you say a face around the side … for example, this edge, you see this edge? All the way up, exactly. When you slide an edge up like this…. Look at this, if I have an edge and I slide it up, what do I get?
Starbird: A two-dimensional face. Right? A one-dimensional edge gives rise to a two-dimensional face. So, John, how many two-dimensional faces are there in a four-dimensional cube?
Starbird: OK. Six on the bottom.
Starbird: Six on the top.
Starbird: OK, and how many is that? That’s exactly correct. Exactly correct.
Starbird: Twelve more. Twelve that are swept edges. For example, this edge sweeps up to give a two-dimensional face of a four-dimensional cube. So you have six plus six plus 12, for a total of 24. Six plus six plus 12. That’s the way to think of it. The ones at the bottom, the ones at the top, and the ones that are created by sweeping.
So you are now in a position to fill out a chart, such as this chart right here, of vertices, edges, two-dimensional faces, three-dimensional faces, of every dimensional cube. And there’s no reason we need to stop at four dimensions. Once you have a four-dimensional cube, a five-dimensional cube by analogy is that you just create a stack of four-dimensional cubes to create a five-dimensional cube.
This is a great thing. And by the way, you can think about unfolding the boundary of a hypercube. I recommend you do that. Unfolding the boundary of a hypercube by analogy, and that’s a great exercise that I encourage you to do.
So this adventure of the fourth dimension was an example of how to create a concept. It’s a concept that’s coherent in itself, because what we’re taking is an analogy of how we can construct lower dimensions from the yet lower dimensions, and by analogy, we can push it to the fourth dimension or further. I think it’s a great example of the creation of concepts and how you can force yourself to be creative in a particular strategy, among many, of how to create new ideas. So that’s the fourth dimension.
Audience: Thank you.