TITLE: Visualizing the 4th dimension.
QUESTION [59 upvotes]: In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can at least see their 3-D cross-sections. A Tesseract hypercube would be a good example.
 Can we conclude that a 3-D cube is a shadow of a 4-D tesseract?  
But,  how can a shadow be 3-D? Was the screen used for casting shadow also 3-D; or else, what way is it different from basic physics of shadows we learnt?  

edit: The responses are pretty good for 4th dimensional analysis, but can we generalize this projection idea for n dimensions, i.e. all n dimensional objects will have n-1 dimensional projections?
   This makes me think about higher dimensions discussed in string theory.
  What other areas of Mathematics will be helpful?

REPLY [5 votes]: I view time as 4th dimension. 4th dimension would be 3D object's state in time. Time is only 1 straight line logically, but what if you let time also have vertical and horizontal axis? then that's 5th dimension. then if you have time with z axis that would be 6th dimension. Beyond 6 would be hard to visualize but it would be an extension of the above concepts.