Visualising extra dimensions What is the :


*

*most useful

*prettiest


way to visualise extra dimensions in shapes and charts?
 A: Whenever I think about $S^{2n-1}$, I draw $n$ complex planes and a unit disk in each one.  $S^{2n-1}$ is the set of ordered $n$-tuples of points, one from each disk, so that the sum of the squares of their absolute values is one; you can rotate each "coordinate" in a circle in its own plane independently of the others, but if you move it "outwards" or "inwards," the others have to change in magnitude as well.  From there, it's only a small leap to $\mathbb{CP}^n$.
For complex functions, which live in 4 dimensions, one idea is to graph either the real part or the magnitude as a function in 3-space, and then use color to represent the imaginary part or the argument.  Or you could just do several different graphs, like Wolfram MathWorld does.  This breaks down for things sitting in 4-space that aren't graphs of functions, because you could wind up with two points sharing x-, y-, and z-coordinates but with different "colors."
I believe people who study knotted surfaces like to use "movies," each "frame" of which is a 2-d cross section of the surface.  The advantage of this is that the Reidemeister moves for knots generalize to "movie moves" for knotted surfaces.  I don't know that much about this, though.
Finally, my algebraic topology professor told me that he just thinks of CW-complexes in terms of their cell structure.  I asked him to draw $\mathbb{CP}^n$ and he just wrote $1,0,1,0,\dotsc$ in an upwards column, each number representing the number of cells in that dimension.  Obviously, you'd need to take attaching maps into account as well in the general case.  But this lets you just read off homology, etc.  Thinking like this is really why fields like algebraic topology exist: it's easier to visualize the invariants than the actual objects.  I've yet to talk to a geometer about this, though, and it would be interesting to hear how you think of, say, 3-manifolds when it's no longer "up to homotopy equivalence."
So the answer to your question is: depends on what you want to visualize!
A: The most useful way to think of the $n$-cube is to imagine the set of configurations of $n$-points in the interval. For a more explicit recipe, wiggle your index finger; its set of configurations describes an interval. Wiggle two fingers: now you have a square. Wiggle $4$-fingers and pretend that the motions are independent (or ask an Indian percussionists to demonstrate); the set of configurations describes a $4$-cube. The configuration space method is useful if you want to decompose a (hyper)cube into simplicies. 
A beautiful rendition of an $n$-cube is obtained by projecting the unit coordinate vectors to the $n$-th roots of unity. As for "how to" see the MO link above.
A: Suppose you want to visualize an $8$-dimensional space. One possible approach(in fact the only one I know) is to first visualize an $n$-dimensional space in the form of $\mathbb R^n$, and then let $n = 8$.
A: If you want to visualize four dimensional space time, then imagine three dimensional space except each point has a clock; this is just three space where we associate a number with each point.  When the fourth dimension is curved, then time passes at different rates for these clocks.
