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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