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Dimensions Defined without Coordinates?

  1. Jun 6, 2004 #1
    Dimensions Defined without Coordinates???

    Mathematicians, I'm sure, have solidly clear idea of dimension but not to me. So allow me to ask some questions for all who have a clear understanding of the concept of dimension.

    Can we define dimension without the use of so called multidimensional reference frames with their equal number of coordinates? 3 coordinates in Cartesian system. 4 coordinates in Einstein's relativities. 10 and 11 coordinates(?) in superstring and M-theory.

    What is the true relational correspondence between dimension and direction?

    How can direction be defined without a coordinate system or a reference frame?

    What is the physical meaning of angles (plane, solid and abstract phase angle)? Can direction be defined without angles?
     
  2. jcsd
  3. Jun 6, 2004 #2

    matt grime

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    The answer to all your questions is yes - there is nothing to require you to have a phyisical model in defining any of these things.

    Take C^n as affine n space over the complex numbers, or any other algebraically closed field. n, the dimension is the krull dimension - the maximal length of a chain of ideals satifying certain properties.

    Any space with a real inner product may have angles between (position) vectors defined by the angle between a and b is given by (a,b)=(a,a)^{1/2}(b,b)^{1/2}cos{theta}

    and all these are coordinate free objects.
     
  4. Jun 6, 2004 #3
    Many thanks for this great enlightenment!
     
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