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.
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)
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.