## Rotating Objects Through the 4th dimension

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• 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
Message 1 of 2 , Oct 28, 2006
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

---------------------------------------------------------------------
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Homepage: http://www.shlomifish.org/

Chuck Norris wrote a complete Perl 6 implementation in a day but then
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• 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
Message 2 of 2 , Mar 2, 2007
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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