I'm not so experienced with problems in N dimensions but i can suggest one thing: do not confuse the geometry with an analitical process.
Also do not confuse when you are talking about mathematical analysis visualizing it with the help of the geometry and viceversa.
The geometry is a really old branch of the math, sometimes someone describe it as a science apart, the thing is that this discipline was born when the human knowledge had to dial with nature very closely and directly, like was during the ancient egipt or in greece.
The geometry was born from the observation of the reality, nothing more and nothing less, and the way to express geometry consist in the use of the mathematical language because it is universal and unambiguous.
The analitic process is born to approach and to try to solve other kinds of problems and has to dial with concepts that are not properly available in nature like the concepts of approximation, N dimensions, the laws that we think are the correct ones to describe the natures, the concept of infinite, the infinitely small and the infinitely big, and so on.
The analitic process is really focused on the human needs and only this, if you should describe the lenght of a piece of wood to make a bridge with your hands, you probably do not need to have a really precise measurement, you probably do not need it at all, but if you have to abstract that bridge into a project you probably have to deal with an approximation of that measure, so you need a value, simply because you have to put a quantity on a piece of paper, and thanks to this you can skip from geometry to an analitical process .
You simply can't imagine more than 3 dimension because N>3 does not belong to the world you are in, geometrically speaking.