Visualizing the Fourth Dimension Using Color
rdrop.com
rdrop.com
A brief guide for the film that gives an overview: http://www.dimensions-math.org/Dim_tour_E.htm#guide
Watch online (Deutsch, English, French, Spanish, Italian, Japanese, Russian and Arabic subs available): http://www.dimensions-math.org/Dim_regarder_E.htm
(Includes images)
It does take practise to be able to visualise 4d objects, and I agree that 2d projection example isn't helpful. I find it interesting to think of the 3d -> 2d problem that some people have, and extend it to 4d.
A child would recognize a cube since we naturally see 2d projections of 3d objects as it is. Likewise a 2 dimensional being could recognize a 1d projection of a 2d object (looking at the 2d object on it's side.) But would a 3d object projected to 1 dimension make any sense?
However, one thing does strike you that no matter how many senses you incorporate, you will always navigate a particular 3D projection.
EDIT: Cached page: http://webcache.googleusercontent.com/search?q=cache:http://...
How do you mean? I can think of visualizations that do not.
Something almost analogous is if you imagine two circles: one smaller and one larger in 2D. Let's say the smaller circle is inside the larger. It's impossible to move the smaller one outside smoothly without intersecting the larger.
In 3D of course this is trivial, just move the smaller circle up away from the 2D plane, then over the larger one and down on the plane again.
To tie it in a bit tighter with OPs article, I would just add that if we could only perceive two dimensions, the inner circle would disappear as we "move the circle up and away from the 2d plane".
It would only reappear to us once we moved it back down into our plane of perception, safely outside of the other circle.
The relevant bit from OPs article that this is analogous to is:
> "What would we see if you watched this happen in real life? Since we can't see anything outside our 3D slice of 4D space, from our perspective the moving (green) loop would disappear, to reappear later in the unknotted position."