Question about higher dimensions and what connects them

[Cody Richeson]

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?

 

[andrewkirk]

The additional lines do not go in any direction in which one can point, or that one can draw, as we can only point in three dimensions and draw in two. Drawings that show additional dimensions by diagonal lines are intended only to create intuition and have no mathematical rigour. Personally I find such drawings less than helpful, but if they help others than that's good for them.

The extra lines don't have to be perpendicular. But the only component of the new lines that is of interest is that which is perpendicular to all lines that have already been drawn, because when you remove the perpendicular component the remainder can be decomposed into a set of lines each of which is parallel to a line that has already been drawn. So one generally talks about a new line being perpendicular to the ones already drawn because it is simpler and neater.

 

[Cody Richeson]

You say the additional lines "do not go in any direction in which one can point," but we are constantly embedded in however many dimensions there are. I know we only perceive three, but if the dimensions are all there, all the time, then how come we can't point in the direction of additional lines?

 

[andrewkirk]

Because at every instant in time we are in a three-dimensional spatial submanifold, and we can only point within that submanifold. The fact that that submanifold is embedded within an n-dimensional manifold does not change our submanifold, or where we can point, no matter how big n is.

 

[Cody Richeson]

Is that true, or does the limited perception of humans make it seem as though that were true?