Armed with the idea of manifolds, we can start to understand 4-dimensional shapes. For example, we saw earlier that a shape built out of two circle coordinates is a torus. What if we add another circle coordinate? What shape do you get by combining three independent circle coordinates?
Let's call this new shape a 3-torus, because it's made out of 3 circle coordinates. To avoid confusing the 3-torus with the bagel-shaped torus you're used to, the one built from 2 circle coordinates, we'll call the bagel shape a 2-torus.
In order to draw a bagel-shaped 2-torus as a smooth surface (without self-intersecting) we needed three dimensions. If we add another independent circle coordinate to a 2-torus, making a 3-torus, we'll be adding another dimension to a torus which already lives in 3 dimensions. Somehow, whatever this 3-torus is, it's going to be four-dimensional. How do we imagine it?

To understand a 3-torus, first let's think about a shape we're already familiar with: the 2-torus. But this time, let's think of a torus as a manifold. Can we make a 2-dimensional coordinate system which covers the entire shape?

Yes - since a 2-torus is a 2-manifold, by definition we can always find 2D coordinate systems which cover a 2-manifold. In fact, a 2-torus is a particularly nice shape because we only need one coordinate system.
Here's how to find it: to give a point some coordinates, look at these two circles, and measure the angle of the point on each circle. To the left is 0°, to the right is 360°, and moving past 360° is the same as starting from 0° again.

Then, combine angle #1 and angle #2 into a 2D coordinate system. This new XY coordinate system covers the entire torus. Dragging left and right will change a point's first angle coordinate, moving the point the long way around the torus, while dragging up and down will change a point's second angle coordinate, rotating the point through the "hole" of the torus.

The exact same technique works with the 3-torus. For each of the three circles we're building a 3-torus out of, we can measure the angle of a point on that circle and use it a coordinate. As long as we remember that an angle of 360° is the same as an angle of 0°, we can use the angles on those three points as our 3D coordinate system! Because the 3-torus is a manifold, instead of twisting our brain trying to imagine four-dimensional shapes, we can use three-dimensional coordinate systems.

Here's our 3D coordinate system. Each point in this 3D coordinate system represents a point on the four-dimensional 3-torus.

Varying each coordinate gives us a different point in the coordinate system. It may seem like the points teleport when they reach the edge of the coordinate system, but that's just because 360°=0°. If you want to be technical, you could say it's because of the transition function we're using for this coordinate system in this manifold. In four dimensions, there's no teleporting, because the faces of this cube are actually connected in 4D space.

Now, armed with this 3D coordinate system, we can move around the 3-torus in whatever way we want. One reason manifolds are so useful is because we can study four-dimensional objects, just using our old friends the 3D coordinate systems. Next