It doesn't work in a universe with four dimensions of space, nor five, nor six.

So it's an obvious mistake. Why did they make the mistake?

So, a one dimensional space is just a line.

No,he's a local.

(Laughter)

The fourth being time.

Let me explain it to you more intuitively.

This is a vector field.

( ominous theme playing ) ( horn honking ) ( dramatic theme playing )

We got dirty, and we loved it.

We take this curved 12 dimensional space, and transform it into a flat, four dimensional space.

You come here from nowhere, knowing no one here.

like a piece of paper.

Same situation as before. We'd like to go from here to here.

Watch between the rocks. Under your feet too.

The second heuristic we're using looks at the 3D problem but without obstacles.

OK.

And it will be me. And we'll be back together.

Further, if you re represent each data vector using those eigenvectors, you'll find a 3 dimensional space where original data falls into a varity of different places.

They have some deficiencies.

So the only difference, class, is that our object now moves in a 2 dimensional space, where as in class, it moved in a 1 dimensional space.

Fly your ship in a 4d environment

Shinigami killing people and humans killing people aren't the same thing.

We charge RMB1 for the 2nd hour and RMB2 for the 3rd hour.

So the best way of viewing it is by analogy

And I'm sure of it, you know?

So we use a little trick.

Why is that?

Time is like a river it does not return to its source.

It's interesting.

So, How do I do this?

The only way to define a line or a curve in three dimensions, if I wanted to describe the path of a fly in three dimensions, it has to be a parametric equation.

If one of the binomials is 3x squared minus 2x, what is the other binomial?

So now I'm going to show you a number of projects that we've built, from one dimensional, two dimensional, three dimensional and even four dimensional systems.

You, watching me out there on the Internet, why are you spending your time doing this?

When you study the mathematics of string theory, you find that it doesn't work in a universe that just has three dimensions of space.

Revert timing

In a couple of weeks, it'll be the same. Now you can go.

The orthogonal dimension in this direction carries alomst information, so it suffices, in most cases, to project the data onto this 1 dimensional space.

Even though moving one over here and one over there would give a better solution.

Here's a 2 dimensional space.

For our direction to be a dimension, it has to be at right angles to all other dimensions.

Typically a dozen or so will suffice to make a face completely reconstructable, which means you've just mapped a 2500 dimensional feature space into a, perhaps, 12 dimensional feature space on which we can now learn much, much easier.

Beyond it is another dimension a dimension of sound... a dimension of sight... a dimension of mind.

Are there any shifts we could do oh we could totally shift over that (b).