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Rotating Objects Through the 4th dimension

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  • Shlomi Fish
    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
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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
      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@...
      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.
    • Omer Zak
      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
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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 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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