Can anyone define the fourth dimension geometrically?

I'm not sure what you mean by this. A manifold with a Euclidean metric works the same regardless of the number of dimensions, so if you understand how 2- and 3-dimensional Euclidean spaces work, you understand how 4-dimensional Euclidean spaces work. What else do you need to know?

 

Ok, good.

If you don't understand what a manifold is, then you need to learn that on your own first. Trying to answer your questions at this point would amount to giving you a course in geometry and topology, and that's beyond the scope of PF.

 

No; "generally" meaning it's a rule unless a particular exception is made, and any exceptions to the rule are at the discretion of the moderators.

These are not the same thing at all. A 1-dimensional straight line has fewer dimensions than a chart (at least as long as the chart has 2 or more dimensions). A 4-dimensional space has more dimensions than a 3-dimensional graph.

As I said in my previous post, you need to learn the basics for yourself first. That includes manifolds, per my previous post; it also includes coordinate charts and projections (since in order to represent a space of n dimensions on a graph of fewer than n dimensions, you need to do some kind of projection). The questions you are asking are too general at this point to be answered within the scope of PF. Thread closed.