Let's imagine a 4 dimensional cube

I was recently thinking about a fun old trick that I used to enjoy showing to students that would allow them to imagine a 4 dimensional object. Today we are going to imagine a hypercube. The key to doing this, is to imagine explaining to a 2 dimensional person what a cube is, if we can figure out how to describe a cube using only 2 dimensional objects, we can then use analogy to imagine a 4 dimensional hypercube with 3 dimensional objects.

So, imagine there is a guy (let's call him Edwin) who lives in a plane, Edwin is a 2 dimensional person and has no concept of a third dimension. We are going to describe a cube by building it out of squares. The first thing you might think to try is the following picture. (Please forgive my poor paint skills)

This doesn't work though, because we only see a cube in this 2D picture because we have seen a 3D cube before. To Edwin this is just a jumbled mess. Think about this as a purely 2D object, I see a little square in the middle, a triangle in the top left and bottom right, and 4 irregular 4 sided polygons. This is just no good at all. We need to do this without overlapping any lines.

So, we start with a single square, and if Edwin walks off any of the 4 sides of the square, he ends up on another square. At this point Edwin probably is picturing something like this.

A pretty good start for sure, we can now try to explain to Edwin that the 4 outer squares are actually pushed into the 3rd dimension. When he crosses those lines he is making a 90 degree turn into a new direction he's never thought of before. Furthermore, the 4 outer squares are connected in ways not shown in this picture. For example, if he goes up, then turns left, when he gets to the edge of the square he will step onto the square to the left. There are basically 2 ways to draw this, we can use colors to show edge identifications or we can distort the picture, here we can see both of those options.

The picture on the left shows the flatness of the squares perfectly, but we have to imagine "jumping" from one square to the next when we hit a colored line. The picture on the right shows the proper spatial relationship between all 5 squares, but the 4 outer squares are distorted since we have pushed them into 2D from their natural 3D space. This picture on the right is the one we want to focus on, it is more useful to us.

Now, we have 5 of the six walls of this cube, we simply have to imagine a 6th square whose 4 edges connect to the 4 outer edges of this picture and we have a cube with 6 walls. We simply have to "fill in" the interior of the cube with three dimensional stuff and we have ourselves a cube. This last bit will be pretty impossible for Edwin to understand, but hopefully he will understand the 2 dimensional walls and how they all fit together.

We are ready to move on to the hypercube, but I want to mention one aside with this picture on the right. Imagine if you have a cube and you put one side right up to your eyeball. This is what you will see, the opposite side is a big square right in the middle of your vision, and the other 4 squares are distorted around the side.

Ok, now we are going redo this work, but we will start with 3D and work into 4D. Imagine you start with a cube, on each face of the cube, we attach another cube, we wind up with the following picture (stolen from here)

Now, if you go into any of the "outer" cube and make a 90 degree turn you should "jump" into one of the other neighboring cubes. Again, we distort the picture (stolen from here)

We are finally ready to imagine the hypercube. Think back to the cube built from squares, you start in the middle square and walk into one of the other squares. When you cross and edge from one square to another, you are really making a 90 degree turn out of the page. The same thing happen here. You are in the middle of the cube in the middle, you can fly into one of the other cubes. When you hit the face that the 2 cubes share, you are really making a 90 degree turn into the 4th spatial dimension. So we have the little cube in the middle of this picture. The 6 cubes adjacent to it that are distorted in this picture, and the final cube that is connected to those 6 that isn't pictured. Those 8 cubes are the boundary to the 4 dimensional hypercube.

Hopefully this made some kind of sense. It is obviously better to do it person, and with physical models instead of pictures and a good bit of hand waving. Just remember, the key is to think about what is going on with Edwin and extrapolate that to yourself.