Re: Rotating Objects Through the 4th dimension

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


On Sat, 2006-10-28 at 22:02 +0200, Shlomi Fish wrote:
> 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.
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