Welcome to Calculus. I'm Professor Ghrist. We're about to begin Lecture 34 on Volume and Dimension. We've computed the areas of two dimensional shapes, and the volumes of three dimensional shapes. What do you think comes next? The fourth dimension? The fifth dimension? In this lesson, we're going to do them all at once. We begin our lesson with that timeless question, what is the fourth dimension?Now, that sounds kind of spooky. We're going to jack it up a level and answer the question, what is the nth dimension? But we're going to start at the beginning. Let's say n equals 1. What is that? Well, that's simply the real number line that you all know and love. We could coordinatize it by some variable, x. When we move up to the second dimension, well now, we have two variables to state where we are in the coordinate plane. Call it x and y. Three dimensions is a little bit harder to draw, but no less difficult to understand. We have x, y, and z. But now when we get to the fourth dimension, it's a little difficult to draw pictures. We can, however, simply add another coordinate. You might be tempted to use t for time, or some other variable name. But why don't we just use subscripts x1, x2, x3, x4. The reason for this being that when we move to the nth dimension, well, for large values of n, we don't have enough letters, but we do have enough subscripts. Now, drawing pictures in the nth dimension is hard, therefore, we're going to proceed by exploring through volumes and shapes. Three simple shapes, in particular. Cubes, Simplexes which are analogs of triangles or pyramids, and balls. Now, what do we mean when we say volume in high deminsions? Now lets take a moment to think about that. And as dimension of a object goes from zero to n, we know what three dimensional volume means. That's simply a volume. But we also know what two dimensional volume means. That is area. What is one dimensional volume? Well that is really length. Okay, you've got those three down. What is zero dimensional volume? Well a zero dimensional object is simply a collection of points. How many points? That is the volume. Zero dimensional volume is counting. And now with these in place We can move to n-dimensional volume, which we will call hyper-volume if we're feeling epic, or we might just simply call it n-volume, that will work. Let's build our inutition for n-dimensional volume by looking at an n-dimensional cube. This is going to be a cube, where all the sides have unit length. So in dimension, 3, it's the familiar object. What is a 2 dimensional cube? It's simply a square with side lengths 1. What is a 1 dimensional cube? Well, it's got to be some interval. An interval of length one. What's a zero dimensional cube? It's a zero dimensional set whose value or count is equal to one. It's just a simple point, and from that we can extrapolate to higher dimensional cubes. But at this point, pictures fail and we need to translate two equations. Therefore, we'll define the unit n dimensional cube as those points in n dimensional space whose coordinates x sub i satisfy the inequalities x sub i. Bigger than or equal to zero, and less than or equal to one for all i. Now how do you visualize that? Well it's hard to do with your eyes, but you can do it with your hands. If you think of each coordinate as an independent parameter, then it is remarkably similar to what happens when you slide an equalizer or slider bars up and down perhaps you've played with something like this on a sound system, each of those slider bars is like a coordinate. In the n dimensional cube, where n is the number of slider bars. Each can go up or down independent of the other, until you hit the boundary where it has to stop. Now with that in mind, let's take a look at volumes. Is we consider what happens in each dimension. How do we get from 1 cube to the next? At each stage what we're doing is taking the lower dimensional cube and then crossing it with an interval, making a one parameter family. Of such objects. Now, in every case, the volume is 1. We're used to that in area - length times width - or in 3D volume - length times width times height. In each case, it's 1 times 1 times 1. In fact, this is really the basis for how you should think but n dimensional volume. The surface area, well that's a little bit harder to wrap your brain around, but we want to look at the boundary of the cube, and say how much n minus one dimensional volume is there. So if we take a one-dimensional interval, and say what's the zero-dimensional volume of the boundary? That is, how many points on the boundary? Well, simply two. Surface area for a two-dimensional object is what we used to call perimeter. And in the case of a square, the perimeter is four. There are four unit edges. Of course the surface area for a three-dimensional cube is what? Well you look at each of the six faces, and compute its unit area. Adding together gives six. Now in general, one could argue correctly that the surface area of the n dimensional cube is, in fact, Two n, the number of boundary faces that you have. And that pattern works all the way down, even to dimension zero. The diagonal of a cube is the distance between opposite corners. We know from Pythagoras what that is for a two-dimensional cube. That's square root of two. What about for a three-dimensional cube? Well, we would have to apply the pathagreon theorum twice to obtain the square root of three as the length of this long diagonal between opposite corners. If we continue inductively, we can show that the diagonal of the indimensional cube has length square root of n. That's a little crazy because for large values of n, you can have a very, very small unit cube such that the opposite corners are very very far apart. That's a little strange. But this pattern continues down even to dimensions one and zero. Lastly, if we count the number of corners in a dimensional cube we see 4 squared. There's four. For cube, there is eight. And in general, it's not hard to show that there are 2^n corners. Let's move on to a different shape, one that requires some calclus to understand. This is the simplex, this is an in-dimensional generalization of a triangle or a pyramid. The unit simplex is defined algebraically as a subset of the unit cube that satisfies an additional constraint, this being that the sum of the coordinates is less than or equal to 1. But what does that mean in terms of our slider bar analogy? This means that you can take any of the individual bars and slide it all the way up to 1. However you can't do this independently. If you want to move the other slider bars up you have to do so in a way but the sum of the values does not excede the treshold of 1. That means that this is a highly constrained set. It's not a large subset of the n-dimensional cube. It feels much small. We expect to see that reflected in the volume. Let's see how that works. First, lets explore a few properties and then we'll compute the [UNKNOWN] volume. The number of corners of an n-dimensional simplex is much less than that of a cube. The n-dimensional simplex has n plus 1 corners. What is the volume? Well, we know, for a single simplex, it's just a point. The number of points is one. We know for a one dimensional simplex since its the same as a one dimensional cube. We just get a length of 1. Now, a triangle as we all know, gives us area one-half. When we look at a three dimensional simplex, it's a cone over that triangle. We know the volume of a cone is going to be 1 3rd, the height 1 times the the area of the base, 1 half. Now, we start to see a little bit of a pattern here. What if I told you that the 4 dimensional simplex had 4 dimensional volume equal to 1 24th. That's true. And knowing that you would be convinced of the pattern, namely that the volume of the n dimensional simplex v sub n must be one over n factorial. Now that's a good guess, let's see if we can show it. Our strategy for computing volumes of the n dimensional simplexes is the same as that Of a cone. We're going to slice in a direction parallel to the base. And what we're going to see is that when we slice an n+1 dimensional simplex, what we'll get in an n-dimensional simplex whose size is rescaled. By a factor of x in each coordinate, where x is the distance to the top on the simplex. So for a one dimensional simplex the appropriate volume element of the slice that's nothing more than dx. In a two dimensional simplex, the appropriate area element is what? It's simply x d x. In the three dimensional case, well, we've done this before. This is going to be 1/2 x times x d x and in general, the difficult step. Is to argue that the volume element, for the n plus one simplex, is the volume of the base and simplex v sub n, times x to the n. Since we're re scaling each coordinate. By a factor of x. But once we have that, and then multiplying by the thickness dx, we can compute this n plus one dimensional volume as the integral of the volume form. That is the integral of v sub n times x to the n d x. Integrating as x goes from 0 to 1. This is a trivial integral since v sub n is a constant. Routine x to the n plus 1 over n plus 1, evaluated from 0 to 1. That gives us v sub n over n plus 1 and so we can write down all of these volumes by induction and argue that v sub n is in fact 1 over n factorial. That's a nice application of simple integration. Let's move on to an n-dimensional ball of radius 1. These are a little difficult to draw. In 2D this is simply a disk of radius 1. In 1D it's a disk of radius 1, well it's really an interval of length 2, and in 1D it is again a simple point. Higher dimensional balls are not so easy to draw. Now how do we define it rigorously? The unit ball is defined as those set of points with coordinates x sub I between negative one and positive one satisfying the additional constraint that the sum of the squares of the coordinates is also less than or equal to 1. This is what we're used to in 2D. When we say x squared plus y squared, less than or equal to 1, this is simply the generalization of that. Now, in terms of a slider bar analogy. Now, all of the individual bars can go from negative 1 to 1. Each can go to the very top, or the very bottom. But, in between well, you have some freedom to move the individual sliders up and down. But you can't move them all past a certain point where the sum of the squares is less than or equal to one. Nevertheless, it feels like there's a lot of room inside of there to move around, How do we compute the volume? Well again, for a radius 1 ball in dimension n, what is volume going to be? In dimension 0, there is the single point, volume 1. In dimension 1, this interval has length 2. In dimension 2, well, we know the formula, pi r squared. In this case, r equals 1. In dimension 3, volume is Four thirds pie. Moving up to dimension n. Well what are we going to do here, lets call that volume of the unit ball v sub n. And to determine what that is lets consider what happens when the radius is not one, but r in this case. The length of the one-dimensional is 2 times r. The area of the two-dimensional ball is pi r squared. Volume, 4 3rds pi r cubed. In general, having a ball of radius r and dimension n is going to give you the volume. Of the uniball times r to the nth power. That's going to be helpful for us, as we'll see. The surface area is what? Well in the one-dimensional case it's two. In the two-dimensional case we're looking at the circumference. That's 2 pi r. In the three-dimensional case, the surface of the ball is 4 pi r squared. Do you see a pattern? Yes. It's related to the derivative. In fact, it's going to be in the n-dimensional case, n times V sub n times r to the n minus 1. You'll be able to prove that result in multi-variable calculus. What's the diameter? Well in all cases it's equal to 2 times r or in the unit case is 2. Now, let's see if you can figure out what this n dimensional volume of the uni-ball these are then is. Well we're not going to able to prove it in here, we will prove it in the bonus lesson. It's wise to say there's is some work one can show that the volume, the n-dimensional ball of radius one is, when n is an even number, let's say two times k, then the volume is pi to the k over k factorial. When n is odd, that is 2k plus 1, then the volume is pie to the k k factorial 2 to the n over n factorial. that's going to complicated we will show you how to get this in bonus material. For now, the question I want you consider is, what happens to the volume of the unit-ball as the dimension increases? Well, let's see. N, and thus k, are getting bigger and bigger and bigger. But there's a factorial in the denominator. What happens then? This means that the volume does not get bigger as the dimension increases. In fact, the volume goes to 0 as dimension increases, and it goes to 0 rapidly since factorials beat powers. This is caused for some alarm or some puzzlement. What does this mean? Well, let's think in terms of the difference between a ball and cube in dimension n. Let's say we have them fight. Who wins? Well, in low dimensions, the ball of radius 1 definiately has more volume, or area, than the cube of side length 1. This is true in 2D. It's even true in 3D, but it is not true in all dimensions because of those corners in the cube. Those corners eventually stick out from the ball even when the two are concentric. And all of the volume inside the n-dimensional cube lives in those corners. That's why cubes beat balls. This lesson was neither short nor simple. It may take a little time for things to sink in. Don't worry. You're not going to be asked any questions about hyper volumes of balls. On the final exam for this course. And in our next lesson, we're going to return to the more familiar low-dimensional world. But step back for a moment. Think about what you've done. We have, with rational thought and calculus, measured objects that you cannot see. Smell, taste, touch, or experience with your senses. That's not a bad day's work.
