How do you visualize 4D space?

As in four spatial dimensions?

A dot to a line (0D to 1D) requires that you extend the shape by a unit length in a direction.

A line to a square (1D to 2D) requires that you extend the shape by a unit length in a direction perpendicular to the line.

A square to a cube (2D to 3D) requires that you extend the shape by a unit length in a direction perpendicular to all of the current lines.

A cube to a hypercube (3D to 4D) requires that you extend the shape by a unit length in a direction perpendicular to all of the current lines again.

Now this last step is a problem, because your brain runs out of directions that are perpendicular, because we live in a 3D world. This is the major hurdle to overcome.

Even a 3d model of a hypercube is confusing, because it's projected into 3d space. If you draw a cube on a piece of paper, you have to draw some of the angles as less than or greater than 90 degrees, even though you know they are 90 degrees in reality.

Here is a picture of a hypercube projected onto a 2d space:
http://upload.wikimedia.org/wikipedia/commons/0/0f/Hypercube.png

You can see the cube at the center - and the lines coming out from it are, in 4D space, at 90 degrees to the other 3 lines at the vertexes - and so are the surfaces. Yes - it doesn't help me very much either. It's a very unnatural thing for a human to do - most people just rely on the math and don't worry about the visualisation.