Subject: Rotating Objects Through the 4th dimension

From: Shlomi Fish <[email-masked]>
Date: Sat, 28 Oct 2006 22:02:54 +0200

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      [email-masked]
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.

---

From: Omer Zak <[email-masked]>
Date: Fri, 02 Mar 2007 18:01:42 +0200

I am not mathematician, but I would like to offer two thoughts about the
subject.

1. Mathematical description:

You can describe flipping a 2-D object in a 3-D space as a sequence of
the following transformations:

The original object is a set of points {(xi,yi)}.
In 3-D space, the set is trivially transformed into {(xi,yi,zi==0)}.
Now, when we rotate the object by 90 degrees, this can be described as
transformation from (xi,yi,zi) into (zi,yi,-xi).  Applying again the
same rotation, we get now:  (-xi,yi,-zi).  Remember that zi==0 for the
2-D objects which we consider.

Now, for 4-D objects, define a rotation by 90 degrees and apply it twice
(I am not sure it is possible for any arbitrary combination of axes).

2. Chemical description:

Molecules of optically active materials (sugar, amino acids, and other
biologically relevant materials) are assymetric in the way you described
the geometrical object to be rotated.

Thus, in an hypothetical 4-D world, the concept of optical activity
loses its meaning.

                                     --- Omer

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