Can 4-D Figures be Represented on a 3-D Surface?

[Lewis]

If one can draw (or represent) a 3 dimensional figure on a 2 dimensional surface (ie-a cube on a blackboard), is it possible to "draw" (or represent) a 4 dimensional "figure" on a 3 dimensional "surface"?

 

[Nash]

Yes; however, just as in two-dimensional representations of three-dimensional objects, you can only see one view of the object. Also, because we have no experience with four-dimensional objects and because our eyes are limited to a planar visionscape, it would be much more difficult to intuitively grasp a four-dimensional object's representation than it is that of a three-dimensional one.

 

[robphy]

Just as a cube drawn on a blackboard introduces distortions [e.g., some 90-degree angles are not drawn as such, or true lengths are not drawn], a 4-D figure drawn on a 3-D surface will have distortions.

Have you seen the Necker cube?
http://www.yorku.ca/eye/necker.htm
http://dogfeathers.com/java/necker.html

For some attempts to visualize in 4-D, look for my post in this thread
https://www.physicsforums.com/showthread.php?t=101720

 

[Lewis]

Thanks a lot for the replies, they both make quite a bit of sense.

robphy- I have seen the animated Necker cube before, but had forgotten about it. Thanks for the link, it's really neat.