Everyone has probably tried it, but nobody will admit to it. And many people have pulled a muscle while trying to do it. I don't necessarily condone it, but if you're going to try and visualize a hypercube, you might as well be safe.
What is a Hypercube?
Of course, *you* wouldn't know anything about hypercubes, would you? As a way of maintaining this little polite fiction, I'll explain what they are. Start with two line segments. A square can be made by connecting the end point of each segment with the corresponding endpoint of the other segment. Likewise, a cube is made by connecting the corners (vertices) of each square to the corresponding corners of the other sqare.
Notice that any face (square) of this cube is the same as any other. You can rotate it or turn it inside out and it will still look the same.
You can connect one cube to another in the same way, connecting the vertices of one cube to the corresponding ones of another square. Now it becomes a tesseract. A tesseract exists in 4 dimensions, and the lines connecting one cube to another are actually perpendicular to every line in the original cubes. It can also be rotated in four dimensions, and much like a cube has 6 square faces, there are 8 cubic faces in the tesseract. Trying to picture all of them will probably hurt.
The square, cube, and tesseract are instances of a hypercube or n-cube. A square is a 2-cube or 2-dimensional hypercube, a cube is a 3-cube, and a tesseract is a 4-cube. This process can be continued infinitely, possibly making them really hard to draw.
How to Picture a Hypercube
A way to deal with this, if you're not interested in rotating the n-cube or turning it inside out, is to just not draw all the edges. Just draw one edge from each cube to each other. Once you get to a 6-cube and want to connect it to another 6-cube to make a 7-cube, just draw one edge connecting the centers of the two 6-cubes. That line represents 64 connections. The idea is, instead of breaking your head trying to picture an additional dimension, you can just re-use one of the ones you can already picture, provided you don't need to put too much strain on it.
