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We begin our lesson with that timeless question, what is the fourth

Â dimension?Now, that sounds kind of spooky.

Â We're going to jack it up a level and answer the question, what is the nth

Â dimension? But we're going to start at the

Â beginning. Let's say n equals 1.

Â What is that? Well, that's simply the real number line

Â that you all know and love. We could coordinatize it by some

Â variable, x. When we move up to the second dimension,

Â well now, we have two variables to state where we are in the coordinate plane.

Â Call it x and y. Three dimensions is a little bit harder

Â to draw, but no less difficult to understand.

Â We have x, y, and z. But now when we get to the fourth

Â dimension, it's a little difficult to draw pictures.

Â We can, however, simply add another coordinate.

Â You might be tempted to use t for time, or some other variable name.

Â But why don't we just use subscripts x1, x2, x3, x4.

Â The reason for this being that when we move to the nth dimension, well, for

Â large values of n, we don't have enough letters, but we do have enough

Â subscripts. Now, drawing pictures in the nth

Â dimension is hard, therefore, we're going to proceed by exploring through volumes

Â and shapes. Three simple shapes, in particular.

Â Cubes, Simplicies which are analogs of triangles or pyramids, and balls.

Â Now, what do we mean when we say volume in high deminsions?

Â Now lets take a moment to think about that.

Â And as dimension of a object goes from zero to n, we know what three dimensional

Â volume means. That's simply a volume.

Â But we also know what two dimensional volume means.

Â That is area. What is one dimensional volume?

Â Well that is really length. Okay, you've got those three down.

Â What is zero dimensional volume? Well a zero dimensional object is simply

Â a collection of points. How many points?

Â That is the volume. Zero dimensional volume is counting.

Â And now with these in place We can move to n-dimensional volume, which we will

Â call hyper-volume if we're feeling epic, or we might just simply call it n-volume,

Â that will work. Let's build our inutition for

Â n-dimensional volume by looking at an n-dimensional cube.

Â This is going to be a cube, where all the sides have unit length.

Â So in dimension, 3, it's the familiar object.

Â What is a 2 dimensional cube? It's simply a square with side lengths 1.

Â What is a 1 dimensional cube? Well, it's got to be some interval.

Â An interval of length one. What's a zero dimensional cube?

Â It's a zero dimensional set whose value or count is equal to one.

Â It's just a simple point, and from that we can extrapolate to higher dimensional

Â cubes. But at this point, pictures fail and we

Â need to translate two equations. Therefore, we'll define the unit n

Â dimensional cube as those points in n dimensional space whose coordinates x sub

Â i satisfy the inequalities x sub i. Bigger than or equal to zero, and less

Â than or equal to one for all i. Now how do you visualize that?

Â Well it's hard to do with your eyes, but you can do it with your hands.

Â If you think of each coordinate as an independent parameter, then it is

Â remarkably similar to what happens when you slide an equalizer or slider bars up

Â and down perhaps you've played with something like this on a sound system,

Â each of those slider bars is like a coordinate.

Â In the n dimensional cube, where n is the number of slider bars.

Â Each can go up or down independent of the other, until you hit the boundary where

Â it has to stop. Now with that in mind, let's take a look

Â at volumes. Is we consider what happens in each

Â dimension. How do we get from 1 cube to the next?

Â At each stage what we're doing is taking the lower dimensional cube and then

Â crossing it with an interval, making a one parameter family.

Â Of such objects. Now, in every case, the volume is 1.

Â We're used to that in area - length times width - or in 3D volume - length times

Â width times height. In each case, it's 1 times 1 times 1.

Â In fact, this is really the basis for how you should think but n dimensional

Â volume. The surface area, well that's a little

Â bit harder to wrap your brain around, but we want to look at the boundary of the

Â cube, and say how much n minus one dimensional volume is there.

Â So if we take a one-dimensional interval, and say what's the zero-dimensional

Â volume of the boundary? That is, how many points on the boundary?

Â Well, simply two. Surface area for a two-dimensional object

Â is what we used to call perimeter. And in the case of a square, the

Â perimeter is four. There are four unit edges.

Â 6:36

Of course the surface area for a three-dimensional cube is what?

Â Well you look at each of the six faces, and compute its unit area.

Â Adding together gives six. Now in general, one could argue correctly

Â that the surface area of the n dimensional cube is, in fact, Two n, the

Â number of boundary faces that you have. And that pattern works all the way down,

Â even to dimension zero. The diagonal of a cube is the distance

Â between opposite corners. We know form pathagarys ??

Â what that is for a two-dimensional cube. That's square root of two.

Â What about for a three-dimensional cube? Well, we would have to apply the

Â pathagreon theorum twice to obtain the square root of three as the length of

Â this long diagonal between opposite corners.

Â If we continue inductively, we can show that the diagonal of the indimensional

Â cube has length square root of n. That's a little crazy because for large

Â values of n, you can have a very, very small unit cube such that the opposite

Â corners are very very far apart. That's a little strange.

Â But this pattern continues down even to dimensions one and zero.

Â Lastly, if we count the number of corners in a dimensional cube we see 4 squared.

Â There's four. For cube, there is eight.

Â And in general, it's not hard to show that there are 2^n corners.

Â 8:22

Let's move on to a different shape, one that requires some calclus to understand.

Â This is the simplex, this is an in-dimensional generalization of a

Â triangle or a pyramid. The unit simplex is defined algebraically

Â as a subset of the unit cube that satisfies an additional constraint, this

Â being that the sum of the coordinates is less than or equal to 1.

Â But what does that mean in terms of our slider bar analogy?

Â This means that you can take any of the individual bars and slide it all the way

Â up to 1. However you can't do this independently.

Â If you want to move the other slider bars up you have to do so in a way but the sum

Â of the values does not excede the treshold of 1.

Â That means that this is a highly constrained set.

Â It's not a large subset of the n-dimensional cube.

Â It feels much small. We expect to see that reflected in the

Â volume. Let's see how that works.

Â First, lets explore a few properties and then we'll compute the [UNKNOWN] volume.

Â The number of corners of an n-dimensional simplex is much less than that of a cube.

Â The n-dimensional simplex has n plus 1 corners.

Â What is the volume? Well, we know, for a single simplex, it's

Â just a point. The number of points is one.

Â We know for a one dimensional simplex since its the same as a one dimensional

Â cube. We just get a length of 1.

Â Now, a triangle as we all know, gives us area one-half.

Â When we look at a three dimensional simplex, it's a cone over that triangle.

Â We know the volume of a cone is going to be 1 3rd, the height 1 times the the area

Â of the base, 1 half. Now, we start to see a little bit of a

Â pattern here. What if I told you that the 4 dimensional

Â simplex had 4 dimensional volume equal to 1 24th.

Â That's true. And knowing that you would be convinced

Â of the pattern, namely that the volume of the n dimensional simplex v sub n must be

Â one over n factorial. Now that's a good guess, let's see if we

Â can show it. Our strategy for computing volumes of the

Â n dimensional simplexes is the same as that Of a cone.

Â We're going to slice in a direction parallel to the base.

Â And what we're going to see is that when we slice and n+1 dimensional simplex,

Â what we'll get in an n-dimensional simplex whose size is rescaled.

Â By a factor of x in each coordinate, where x is the distance to the top on the

Â simplex. So for a one dimensional simplex the

Â appropriate volume element of the slice that's nothing more than dx.

Â In a two dimensional simplex, the appropriate area element is what?

Â It's simply x d x. In the three dimensional case, well,

Â we've done this before. This is going to be 1/2 x times x d x and

Â in general, the difficult step. Is to argue that the volume element, for

Â the n plus one simplex, is the volume of the base and simplex v sub n, times x to

Â the n. Since we're re scaling each coordinate.

Â By a factor of x. But once we have that, and then

Â multiplying by the thickness dx, we can compute this n plus one dimensional

Â volume as the integral of the volume form.

Â That is the integral of v sub n times x to the n d x.

Â Integrating as x goes from 0 to 1. This is a trivial integral since v sub n

Â is a constant. Routine x to the n plus 1 over n plus 1,

Â evaluated from 0 to 1. That gives us v sub n over n plus 1 and

Â so we can write down all of these volumes by induction and argue that v sub n is in

Â fact 1 over n factorial. That's a nice application of simple

Â integration. Let's move on to an n-dimensional ball of

Â radius 1. These are a little difficult to draw.

Â In 2D this is simply a disk of radius 1. In 1D it's a disk of radius 1, well it's

Â really an interval of length 2, and in 1D it is again a simple point.

Â Higher dimensional balls are not so easy to draw.

Â Now how do we define it rigorously? The unit ball is defined as those set of

Â points with coordinates x sub I between negative one and positive one satisfying

Â the additional constraint that the sum of the squares of the coordinates is also

Â less than or equal to 1. This is what we're used to in 2D.

Â When we say x squared plus y squared, less than or equal to 1, this is simply

Â the generalization of that. Now, in terms of a slider bar analogy.

Â Now, all of the individual bars can go from negative 1 to 1.

Â Each can go to the very top, or the very bottom.

Â But, in between well, you have some freedom to move the individual sliders up

Â and down. But you can't move them all past a

Â certain point where the sum of the squares is less than or equal to one.

Â Nevertheless, it feels like there's a lot of room inside of there to move around,

Â How do we compute the volume? Well again, for a radius 1 ball in

Â dimension n, what is volume going to be? In dimension 0, there is the single

Â point, volume 1. In dimension 1, this interval has length

Â 2. In dimension 2, well, we know the

Â formula, pi r squared. In this case, r equals 1.

Â In dimension 3, volume is Four thirds pie.

Â Moving up to dimension n. Well what are we going to do here, lets

Â call that volume of the unit ball v sub n.

Â And to determine what that is lets consider what happens when the radius is

Â not one, but r in this case. The length of the one-dimensional is 2

Â times r. The area of the two-dimensional ball is

Â pi r squared. Volume, 4 3rds pi r cubed.

Â In general, having a ball of radius r and dimension n is going to give you the

Â volume. Of the uniball times r to the nth power.

Â That's going to be helpful for us, as we'll see.

Â 15:53

The surface area is what? Well in the one-dimensional case it's

Â two. In the two-dimensional case we're looking

Â at the circumference. That's 2 pi r.

Â In the three-dimensional case, the surface of the ball is 4 pi r squared.

Â Do you see a pattern? Yes.

Â It's related to the derivative. In fact, it's going to be in the

Â n-dimensional case, n times V sub n times r to the n minus 1.

Â You'll be able to prove that result in multi-variable calculus.

Â What's the diameter? Well in all cases it's equal to 2 times r

Â or in the unit case is 2. Now, let's see if you can figure out what

Â this n dimensional volume of the uni-ball these are then is.

Â Well we're not going to able to prove it in here, we will prove it in the bonus

Â lesson. It's wise to say there's is some work one

Â can show that the volume, the n-dimensional ball of radius one is, when

Â n is an even number, let's say two times k, then the volume is pi to the k over k

Â factorial. When n is odd, that is 2k plus 1, then

Â the volume is pie to the k k factorial 2 to the n over n factorial.

Â that's going to complicated we will show you how to get this in bonus material.

Â For now, the question I want you consider is, what happens to the volume of the

Â unit-ball as the dimension increases? Well, let's see.

Â N, and thus k, are getting bigger and bigger and bigger.

Â But there's a factorial in the denominator.

Â What happens then? This means that the volume does not get

Â bigger as the dimension increases. In fact, the volume goes to 0 as

Â dimension increases, and it goes to 0 rapidly since factorials beat powers.

Â This is caused for some alarm or some puzzlement.

Â What does this mean? Well, let's think in terms of the

Â difference between a ball and cube in dimension n.

Â Let's say we have them fight. Who wins?

Â Well, in low dimensions, the ball of radius 1 definiately has more volume, or

Â area, than the cube of side length 1. This is true in 2D.

Â It's even true in 3D, but it is not true in all dimensions because of those

Â corners in the cube. Those corners eventually stick out from

Â the ball even when the two are concentric.

Â And all of the volume inside the n-dimensional cube lives in those

Â corners. That's why cubes beat balls.

Â This lesson was neither short nor simple. It may take a little time for things to

Â sink in. Don't worry.

Â You're not going to be asked any questions about hyper volumes of balls.

Â On the final exam for this course. And in our next lesson, we're going to

Â return to the more familiar low-dimensional world.

Â But step back for a moment. Think about what you've done.

Â We have, with rational thought and calculus, measured objects that you

Â cannot see. Smell, taste, touch, or experience with

Â your senses. That's not a bad day's work.

Â