I suppose I could just tell you what sort of things in four dimensions correspond to yesterday’s post. But that wouldn’t be much fun, would it?
Instead, let’s think about this a bit. Here we had a virtual trackball that looked like a circle. It had some straight lines on it, but when we looked at it from the side, it turns out those lines were actually curves. That’s because those straight lines were, in fact, painted onto the front of a sphere.
So what looked at first like a flat disk was actually an image of the outside surface of a sphere.
What if we bring this whole discussion up to one higher dimension? At first we see something that looks like a sphere floating in space. But in fact it is something more.
The sphere we are looking at is the “front” side of a four dimensional sphere. Anything that is drawn inside our sphere is actually being drawn onto the outside of this four dimensional sphere.
If any of this seems mysterious, just go back to the three dimensional case: Something that looks like a flat disk is actually the front face of a sphere.
We’re just taking the same argument to one higher dimension: Something that looks like a sphere is actually the front “face” of a four dimensional sphere.
That’s a pretty weird idea, so I’m going to give it a day to sink in before going on.