It's my understanding that, if we ignore the temporal dimension and just focus on spatial ones, then you get to the third dimension by starting with a point and adding perpendicular lines to them. Once you've done this a couple of times, you get three dimensions. Obviously, to the layman, it appears that there are no more perpendicular lines to add in order to reach higher dimensions. There are, of course, and in crude visual analogies a simple 4-dimensional cube appears to have diagonal lines protruding from the 3D dimensional edges. This is not how it "really" would look, as it's a shadow of the 4D actuality, but can anyone explain what direction these additional perpendicular lines are going in, and why they appear diagonal when downgraded to a shadowed projection? Also, why do the lines have to be straight, and why do they have to be perpendicular?