Ah. Over the years, artists, writers, and filmmakers have tried to answer that question, creating some dazzling works of Science Fiction in the process. But are the higher dimensions we see in scifi really fiction . Meet maggie, scientist by day and scifi fanatic by night. Maggies watching a 1950s film about scientists who invent a device that catapults them into the Fourth Dimension, where they can walk through walls and read minds. Like the rest of us, maggie lives in a threedimensional world. But she also appears to have an interest in alternate dimensions. In 1940, master scifi author Robert Heinlein wrote the short story and he built a crooked house. In the story, our hero builds his home from an unfolded fourdimensional hypercube. When an earthquake strikes, the house folds up, creating the worlds first fourdimensional house, where occupants can look down a hallway and see their own backs. Dimension is one of the most important and intriguing ideas in mathematics. We move in three dimensions of space, we live in four dimensions of space. And time, but mathematicians and increasingly scientists of all flavors, from statisticians to biologists, are finding that they need to understand and work in worlds of hundreds and even thousands of dimensions, worlds that reach well beyond human sensory experience. When we start asking questions such as when did our universe begin, and how will it end . Many scientists believe the answers will involve 10, 11, 12, or even more dimensions. But what do we really mean by the word dimension . Generally speaking, the dimension of space or a problem is simply the number of numbers that we need to describe the situation. These numbers are called coordinates. The idea of coordinates was developed in 1637 by french mathematician and philosopher rene cartesius descartes. Descartes, who never got up before 11 a. M. , as the legend goes, was lying in bed watching a fly crawl across the ceiling, and it hit him the basis for what we now call the cartesian coordinate system. What descartes very well could have realized by watching that fly wander across the ceiling is that the flys position on the ceiling could be described by exactly two numbers what we might call the x and y coordinates. Now, i have a feeling descartes would have loved one of my favorite toys, the etch a sketch. Its all about coordinates. With one of these knobs, i can control the lines that go back and forth, the horizontal direction. And with the other, i can control the up and down, the vertical dimension. We can make some pretty pictures with this toy, but our freedom of movement is limited to just these two independent dimensions, or parameters, we might call them. Now of course, we dont move in a world of two dimensions like the etch a sketch. Were able to move not just side to side and back and forth, but up and down as well, so that we need three numbers to describe a point in our physical world. So suppose we had a threedimensional etch a sketch. Well, lets not suppose. I actually have one. Come on, lets see it. Heres my threedimensional etch a sketch. Now, with three knobs, i can draw vertically. Horizontally, and with depth. This is so cool. I now have three independent parameters so that i have freedom of motion in three dimensions, just like in our everyday world. But in the world of mathematics, we deal with dimension in some rather unique ways. In fact, when mathematicians talk about dimension, were talking more than spatial dimension. Were talking about any set of parameters that help define everything from the markets to the Health Benefits of a particular muffin to the latest model car or even the perfect date. Our friend maggie is using a computer dating web site, circa 2007. It starts by asking a series of 250 questions, grouped into 30 categories. By the time maggies done, she will have established values for 30 different parameters to define herself, as well as her dream date. With these coordinates, you can now say were in a 30dimensional world. And in this world, each point represents a human being. And if maggie can find someone whose coordinates are near hers, shes got a date. When were talking about sets of parameters, it seems we can work in as many dimensions as we want. But whats so special about three dimensions . Because we live in a 3d world, our brains think in 3d, but can we actually see things in more than three dimensions . What would a fourdimensional object look like . Well, threedimensional etch a sketches but multidimensional personalities. Can we see those kinds of things . Were here with greg leibon, Research Mathematician at dartmouth college, and, greg, can i see four dimensions without hurting myself . Youre going to have to go through a little bit of pain to see into higher dimensions, but the pain comes in the form of analogies and metaphors. So we would probably start understanding four dimensions by thinking a little about what we know in lower dimensions. For example, we could start with a cube. Okay, great, great. Weve got our cube. Weve got our cube here. And theres a few things worth observing first. That looks like a threedimensional cube to us, but of course this is on a twodimensional screen. This is some sort of projection. Thats true, thats true. Its a total cheat. But our brains are, fortunately in this case, very hardwired to interpret this as a threedimensional object. Four dimensions is much trickier because, for example, here is a picture of a fourdimensional cube that well just look at for a second just to get a sense. It looks very cluttered. Its the same sort of projection where now weve gone from a cube in four dimensions, we put it on our screen down here in two dimensions, and it looks very cluttered. What it would be like as a twodimensional being to experience a threedimensional object, because what we are is essentially, in the way we think spatially, threedimensional beings, and were contemplating a fourthdimensional object. Now, this is not a new idea. I mean, heres flatland. This is a very famous example of this turned into a beautiful story. So this is edwin abbotts flatland, a book first printed in 1884, and its the story of beings that live in two dimensions on the plane, if you like, with no thickness and how they might experience threedimensional life, which is analogous to our problem as beings living in a threedimensional world trying to experience fourdimensional life. So the twodimensional being is living on this sort of infinitesimally thin surface, right . So they can only move in two dimensions. And somebody enters their world from above, a sphere we can imagine if a sphere, its this very symmetric object, and were going to imagine it in the following sense this sphere which actually lives in three dimensions is going to dip into this twodimensional beings world, and the twodimensional being is going to experience an aspect of this sphere, what we think of mathematically as slices of the sphere, and theyre going to look at the geometry of those slices over time as the sphere moves through their world, and theyre going to use that as a summary statement as to what three dimensions look like. And as the sphere dips through the space, of course, this point instantaneously blows up into a little circle. Because as the sphere dips into the plane, it grows, it gets to some critically large length, which is of course the equator circ of our sphere, and we see it comes back and it shrinks back down to a point. And so what the person has experienced who lives in this plane is a little movie of some circle getting bigger and collapsing back down to a point. And this is their way that theyre going to initially think about the twodimensional sphere that lives in 3d that were used to, is theyre going to think about it as a bunch of slices at different times, and theyre all put together by the continui of into an object. And we see that object here because time we can think of as our third axis, our third dimension here. As it dips through there, that dimension changes. So theres a sense in which we see the movie all at once, but they have no ability except to see one frame. That is exactly the analogy that we want to have in mind. Because now were going to image looking out into our room here. And now were going to have something that starts off as a point. It expands into a sphere, it reaches some critical size, it shrinks back down to a point. And what have we just seen . Well, to us we just saw a movie of a little explosion and contraction of some kind, and were not sure what weve seen. Now, it could well have been that the threedimensional sphere in four dimensions has just dipped into our world to give us a visit, okay . And now we understand that analogy because we can experience exactly what happened in the twodimensional case, and thats the power of the flatland analogy, is that now something that we can witness in our world, we can now understand that via this analogy of what the flatlander experienced. So what you have just experienced is an object that lives naturally in four spatial dimensions. The threedimensional sphere in four spatial dimensions could be thought of as this little movie as you might experience it and as our flatlander did experience it. The problem is we cant step back and see the movie all at once, okay, and we dont do that very naturally. But using this analogy, we can get a grip on what higher dimensions might how they really do look in our lives. Right, yeah, and a way to start to begin to think about them as mathematicians. Thats correct. So now maybe we should go to the more fun object, which are these cubes. So now lets imagine once again were in our flatland world, and now a cube is going to dip into this world. Its nice to look at this trivial direction where it comes in and all of a sudden, instantaneously, we see a square in our twodimensional world. It floats through and its totally staticlooking, it looks like a static square that just suddenly disappears. So now we can get a different view, or we can let our flatlander get a different view of the cube by actually changing the way in which the cube comes into his world. Thats right. Our original way where it came down with the flat parts first gave us this boring movie, and in many ways its sort of a boring look at the cube as a threedimensional object. Its gleaning very little from it. But if we bring it down from another angle, the flatlander learns a lot about the geometry of the shape, so lets come down vertex first. And as we come down kind of at a slight angle here, but we can watch it on the screen, this vertex, when it comes down, first he sees a point, just like in the spherical case, a point. But that point, rather than exploding then into a little circle has now become a little triangle, okay . And this triangle, as we lower it down, every time one of the vertices of our cube in three dimension hits it, an edge is born, and we see this birth process of these edges as our triangle changes into this quadrilateral, which first has this very short side, which gets longer. And then it changes into this pentagontype shape, fivesided object, into this hexagon. And now of course this hexagon has gone back to a pentagon, or a pentagontype object, fivesided, then to this quadrilateral that we see, finally going down to this triangle and a point. So its a lot like the sphere case. Greg, what i like so much about this discussion is that its actually part of things that happen in the real world all the time. For example, when people are trying to image your body, they actually do it by constructing slices of you using some kind you know, like xrays or whatever it is, and they rebuild your body from those slices. And the other way in which its come up is in cubist art, in fact. And picassos idea when he was creating this was actually inspired by discussions of how to see fourdimensional things as threedimensional slices, which is how were actually going to try to see four dimensions, like the flatlanders try to see three. Absolutely. So its easy to see how art and this would interact in the sense that your understanding here is very much so a metaphor in the same way that the art might be used. But lets look at it and see what were talking about. Here were going to be playing that same exact game where we dipped our cube into the flatlanders world, but were going to dip a hypercube into our world and see what we see. The cube that lives in fourdimension space, at least a projection of it. So what are we watching . We are watching a fourdimensional object dipping into our world. Here it starts, we get our vertex. As it dips in, now we get to a simplex, a tetrahedron with four vertices, as it comes in the corner. And now watch it evolve. As it hits these corners, the vertices are chopped off and we get these things that have faces that look like our original quadrilaterals. And the perfect analogy to what was going on for the flatlanders. And now we go through the whole sequence and we see that that shape changed over and over again until were back at our vertex. And we have a beautiful movie that captured this object in four dimensions. This rigid object, which was a perfectly stationary fourdimensional object, has now been experienced by this little film. And in fact a great deal of its geometry can be understood by dwelling on this, just like the flatlander, if they were so inclined, could understand a lot about this cube they were looking at by understanding their movie. And these analogies and metaphors are very important for getting our initial glimpses into Fourth Dimension. And in fact, were looking at probably the one that is the most common way to introduce Fourth Dimensions now, and is, of course, time. And i think maggies also going to use time to help us see four dimensions, so lets go take a look. Great. In 1910, Scientific American held a contest asking readers to describe the Fourth Dimension. The 15 winning entries went into great detail about a fourth spatial dimension. But not one entry mentioned the other Fourth Dimension the dimension of time. Albert einstein changed all that when he put forth his theory of relativity. Einstein said you cant talk about space without dealing with time. Looks like maggie has an appointment at the dating service, which appears to be on the corner of x and y. She needs to go up to the mezzanine level and be there by 1 00. Hi, im here for the speed dating. Im sorry, it started at 1 00. But since maggie has arrived 30 minutes late, shes got the wrong coordinate in the time dimension. And her online dream date just walked out the door with someone who showed up on time. So the Fourth Dimension time . I dont know, maybe. But in fact, as far as a mathematicians concerned, it could sort of be any kind of statistic, right . It could be height, weight, i mean, whatever you like. Yeah. Any parameter, so to speak. Yeah, youre free to have 30 of them, 30 of these parameters. Youre in some big 30dimensional space where your coordinates and parameters which not even you can see. Yes, and in some ways see is kind of the wrong thing, and part of thats because these things might have very different sorts of units. You know, like space, its all in the same unit of measurement, and so you can do all kinds of cohesive things make measurements and do things that are across these dimensions in a sensible way. And there are notions of dimension that have everything to do with that objects living in a place where there really are units that are so were going from a notion of dimension as simply number of parameters to almost a measurement. Maybe we should start with a simple example, okay . Were going to imagine that i have a ruler, and my ruler is a foot long, and you have a yardstick. Lets build a line. Okay. So my line is three times as long as your line. Thats right. So now lets build a square. If i use 20 rulers and you use 20 rulers, ive still got a line three times as long. Thats right. So these things are going to scale perfectly. So now lets suppose that you go ahead and make a square out of your yardstick with one of your units on each side. And we have this square here. And now my question is im going to build the same thing, but in my units i have this footlong ruler. So my square is nine times as big as your square. Thats right, your square is nine times as big, or three squared. The line was three times as big; this is three squared. Now, that notion of dimension, the square coming in, is our next notion of dimension ah, twodimensional square in my thats right. How many more of my objects is it going to take to fill yours . Its a spatial notion. Lets build a cube. Okay. Now, heres your big cube and my little cube, and this is a problem that people that make statues run into all the time. How many of mine are going to be taken to put inside of yours . What is it . Im betting its going to be three cubed. Thats right because my ruler is three times as large and im going in three dimensions now, so its going to be three to the three, so 27 of your cubes are going to be in mine. So we have these 27 cubes filling up that space, and this is a very important thing, the way that objects scale, how its scaling the sides. If i make my ruler three times as big, it takes 27 times as much. Heres a fun one that brings us back to where we were before. Lets take a hypercube; youre going to build a hypercube. Im going to build a hypercube out of a yardstick. Out of a yardstick. So youre some fourdimensional being who has this luxury. Youve built the hypercube and now were kind of used to looking at this picture. There it is. So im going to guess, okay, that im going to need three raised to the fourth power of your hypercubes to fill up my hypercubes. So three to the fourth is 81. So thats my guess. And lets watch. Eightyone of my cubes fill up your hypercube. So this is its all true. I couldnt have seen it, but its all true. And this notion that were using an analogy to understand dimensions is going to be very powerful for our next understanding because weve just gone through some nice friendly dimensional counts. Lets build something a little bit more interesting. Lets build something called a koch snowflake. So what im going to first have you do is take your yardstick and build an equilateral triangle. All the sides are the same. That i can do. That we can do. And now im going to have you take each side of that and split it into three equalsized pieces, which now happen to be the size of my ruler because theres three feet in your yard. Youre going to take that middle piece, and youre going to remove it. And youre going to replace it with the edges of the equilateral triangle that would have that as a base, so now we see this little star thing. Im adding more points. So its more complicated. Nowhere near what you might think of as a snowflake, but now were going to do something in math we call iterate. It means were going to repeat this process. So youre going to take each one of your edges, and youre going to do that again. Youre going to remove the middle piece, and now youre going to do something that mathematicians like to do youre just going to do it forever. That is the mathematicians credo ta