Manifolds in Higher Dimensions

You might be wondering why we spent so much time talking about manifolds.
The manifolds we've seen so far are useful because we can use 2D coordinate systems (which we know very well), to understand surfaces in 3D (which we might not know very well).

Now, armed with the idea of manifolds, we can use manifolds built out of 3D coordinate systems (which we also know very well), to understand spaces in 4D!

By the way, you can make manifolds with as many dimensions in your coordinate systems as you want. Manifolds which use 2D coordinate systems are called 2-manifolds, and manifolds which use 3D coordinate systems are called 3-manifolds, and so on.
The beauty of manifolds is that we can still study higher dimensions without knowing how to visualize them. All we need are Cartesian coordinates - which are just lists of numbers - and we can study things we can't imagine.
Now, using manifolds, let's go study things in four dimensions! Next