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Re: [hackers-il] Rotating Objects Through the 4th dimension

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  • 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 1 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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