Imagining things in more than 3 dimensions How do mathematicians imagine things while dealing with more than three dimensions in linear algebra?Is there a way to imagine such things or is it not possible?
 A: How do you picture a cube (or a three-dimensional sphere, or any three-dimensional object) on a 2-dimensional piece of paper ? You draw its projection.
Well, it's the same with higher-dimensional objects.
A: I once nearly managed to properly visualize a hypercube in 4 dimensions - but it hurt, and I stopped.  I've had a little luck imaginarily moving my hands over a four dimensional object; the 3d image in your head would be that you're moving your hand through a normal projection of (say) a hypercube, but you can imagine feeling corners along the appropriate planes.  Another trick is that you can assign colors to the 4th axis - in theory, you can assign points 6 dimensions like that, though I've never found it particularly intuitive.  None of these are particularly standardized, but they're things that have helped me a bit.  Beyond that, there's a youtube video I found that gives some ways of dealing with N-dimensional objects: Thinking visually about higher dimensions, by 3Blue1Brown.  In general, dealing with N dimensions is difficult, but gets a somewhat easier with practice.  If you're doing math, specifically (like linear algebra), thankfully the notations mean you don't usually have to imagine them visually, or we wouldn't get very far.  While projections do help, be careful, because sometimes they can lead you astray - some rules that hold in 2 and 3 dimensions don't for higher dimensions.  (For instance, I was surprised to discover that things don't rotate around axes; they rotate around planes, and can rotate around multiple perpendicular planes at once.  This is difficult to infer from a projection alone.)
