We're not all Quantum physicists like J so here's a really cool clip going through all the dimensions.

http://www.tenthdimension.com/medialinks.php

Make sure you can listen to it with some attention, or you'll lose track in a second.

I lost myself somewhere during the 5th dimension.

On Quarks! On Leptons! On Dancer! On Prancer!

I hope to have your forgiveness, J, for that I possibly play with the words but "picturing something" does not always have something to do with drawing it (this sentence can fall as well, if we take that writing a letter or number is kind of drawing too). But to the point: if I tell

in an euclidean space, the x is family of points that suffice:
"the distance between x and fixed point O is fixed real number r"

then I just defined a circle with center O and radius r. Note that I didn't tell in which dimension it all happens so if dimension is 2, it is just simple circle in euclidean plane (R^2). If dimension is 3, then it is a circle in our euclidean three-dimensional space (R^3) - that circle has name sphere; If dimension is n, then it is circle in euclidean R^n.

[spoiler]If you do not use euclidean metric, it may happen that a circle actually looks like a square or even point-grid.[/spoiler]

So although I cannot draw a circle in R^n, I know that it is actually an with given specifications (see the definition above). It is easy to understand at that point.

Now why would I need to do do calculations in such spaces, where i.e. ten parameters change (in most) independently, but need to be taken together as that is the way they impact their environment? It could be just a simple control table of ten generators.. You may need a function (eleventh dimension) that calculates what the power all ten generators generate, but you do not need to draw eleven-dimensional picture for that - it is easy enough to follow the whole situation with eleven 2D pics.