Rotating Objects Through the 4th dimension

Hi all!

This is a small mathematical diversion I've thought of introducing here for a 
long time. I once read a book of mathematics that made the following 
proposition: if we take the following two triangles:

              ___   ___
          ___/  |   |  \___
      ___/      |   |      \___
  ___/          |   |          \___
 /              |   |              \
*----------------   ----------------*

then in a two-dimensional world they'll not be considered congruent 
(or "Hofefim" in Hebrew) because they cannot be rotated on the plane to 
match. In order for them to match one has to rotate them through the third 
dimension which is perpendicular to the entire plane.

Now I've been thinking, since our objects are three dimensional, what would 
happen if we rotated them through a fourth dimension and back?

Take those two objects for example:

http://www.shlomifish.org/Files/files/images/Computer/Math/

(flip-thru-4th-dim-*.png).

One of them is a cone that has an orthogonal cross-shaped extension on its 
middle side, and an orthogonal cylindrical extension 90 degrees 
counter-clockwise. The other has the cylinder 90 degrees clockwise.

Now, in a three dimensional space, these shapes cannot be considered 
equivalent. But can we rotate one through a 4th dimension to form the other 
one?

I hope to pick the brain of some of this list's mathematicians.

Regards,

	Shlomi Fish

---------------------------------------------------------------------
Shlomi Fish      shlomif-ik1l9ssToec+JF/[email protected]
Homepage:        http://www.shlomifish.org/

Chuck Norris wrote a complete Perl 6 implementation in a day but then
destroyed all evidence with his bare hands, so no one will know his secrets.


 
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