How Much Space Is There?

As a three-dimensional person, I do have the intuition—I know exactly how much space there is, and how it all fits together, without even thinking about it. How would I describe the space, if I had to? Well, maybe I'd start by saying that there's some on the left, some on the right, and so on—that is, that space is divided into parts, one part for each direction. Then I could just explain how the parts fit together. In other words, I'd describe a cube!

So, if you want to understand how much space there is in four dimensions, all you need to do is understand how the faces of a tesseract fit together.

Remember how I pointed out that the forward passage was a tesseract? Well, that wasn't the first tesseract we'd seen—we'd already seen one when we were completely enclosed in a single square of the maze, only the wall right in front of us was the back face of the tesseract. The front face was behind us, out of view.

With that in mind, here are some ways to look all around. In three dimensions, I usually do something like the following sequence of turns.

left × 4, up, down, down, up

The same approach works in four dimensions, …

left × 4, up, down, down, up, in, out, out, in

… but there are faster sequences that use loops.

left × 4, in, up × 4, out

(There are other sequences that are even faster, but I don't think any of them bring you back to your original orientation.)

In practice, though, I don't often look all around, I just set the retina size to a value greater than one and keep track of whether there's a passage behind me.

Now that you've got an idea how much space there is, let me go back and tell you the real answer. In three dimensions, the space around you isn't a cube, it's obviously a sphere, and in four dimensions, it's not a tesseract but a 3-sphere!

So what's a 3-sphere like? Well, that's the whole problem … I can tell you that a 3-sphere is a curved three-dimensional volume, just as a sphere is a curved surface; and I can tell you that it satisfies certain equations; but I still can't tell you what it's like to be inside one. The best I can do is say that it's sort of like being inside a tesseract. It's also like being inside a hexadecachoron, if that's any help.