Visualizing higher dimensions

Take a flat wall and poke a line into it perpendicular to the wall. The imprint is a point. Now hit it with a square with the face perpendicular to the wall. The imprint is then a line. If you hit it with a cube, with four faces perpedicular to it, it will leave the imprint of a square. Follow me? Now, the fun part: an object leaving the "imprint" of a cube on that wall is a 4D hypercube (there are other 4D objects that would leave this imprint, just as there are many 3D objects that would leave a square.) Continuing in this manner, a 5D hypercube leaves the "imprint" of a 4D hypercube, and so forth.

Comparing dimensions

We compare higher dimensions by analogy, i.e. 1D is a shadow of 2D is a shadow of 3D and so on... so, if a point has no parts, a line has no breadth or height, and a plane has no height, then by analogy a cube would have no... what? Time? This would be impossible. If it had no time or duration then we could not observe it; we are constantly moving through time and would have "passed" it before it even existed. So what is the fourth dimension? It's another dimension of space just as is the fifth, sixth, etc. (but I don't say ad infinitum because I think that time may be the "infinite" dimension). The fourth dimension is actually well defined by topology (as are about 20 or so other higher dimensions, although the mathematics become pretty much unreadable after four). Eric is absolutely right about the Klein Bottle, it has no self intersections or singularities in 4D (it is thus said to be "embedded" in 4D). It would have the "non-intersecting" characteristics that a torus has in 3D (the torus is said to be embedded in 3D). Following that analogy we would postulate that the torus would then be self intersecting in 2D, and indeed it is. A knotted sphere is another example of a manifold which is self intersecting in 3D but embedded in 4D (imagine a deflated basketball, it looks like a bowl, if you could somehow pull the "outer skin" of the bottom of the bowl through the inner while keeping the inner bowl intact you'd have a knotted sphere, the surface contour of which would look similar to that of a slightly smaller basketball with a rope, equal in length to the ball's circumference, wrapped around the ball).