It’s very cool that we can rotate things in three dimensions just by playing with a circular disk. After all, a disk is flat!
What’s also cool is that we can rotate things in four dimensions just by playing with a sphere. It’s not even that difficult to do — but it can be a little difficult to understand.
Just as you can rotate 3D objects by putting a finger down on a disk on a computer touch-screen, then moving the finger around inside the disk, there’s an equivalent action for a spherical region floating in space in front of you.
Namely, you can put your hand inside the sphere, pinch your fingers together to “grab” the sphere, then move your hand around inside the sphere (then unpinch your fingers to “ungrab”).
One way to think about this operation is that the volume inside the sphere is kind of like the area inside a circle, except in one higher dimension. As you move your pinched fingers around inside the sphere, you’re essentially rotating a 4D trackball.
But how do your movements inside that floating sphere translate to rotations in four dimensions? We’ll get to that tomorrow.