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Infinite Dimensional Infinitesimal

  1. Feb 9, 2004 #1
    Any smooth connected 1 dimensional manifold is diffeomorphic either to the circle, or to some interval of real numbers.

    Take a line segment of length 1. It is one dimensional.

    A-------B

    Find the midpoint of the line segment and rotate it into 2 dimensions

    A
    |
    |
    |------B


    Each leg is 1/2

    Rotate into 3 dimensions, and each leg is 1/3

    A
    |
    |
    |------C
    |
    |
    B



    rotate into N dimensions and each leg is 1/N

    Continue this process as a limit

    N---->oo

    By the above process, an infinite dimensional universe is a point.
     
  2. jcsd
  3. Feb 9, 2004 #2







    A
    I
    I
    I
    D------I------B
    I
    I
    I
    I
    C

    So this is four-dimenisional?..how do you account for the fact there is only 26 letters in the alphabet?

    The problem is that a Four dimensional space has inner products made from A+B..A Rotation does not constitute a added Dimension?
     
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