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Math Help - Designing a Function...

  1. #1
    TTB
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    Designing a Function...

    Hello!

    I'm trying to construct a functions between 2 subsets of the 2d euclidean plane.

    The sets are the Hawaiian earring. ie - the countable union of circles with centres (1/n,0) and radius 1/n,

    and the countable union of circles of radius n & centres (n,0).

    I can construct one using different n's, but is it possible to construct one using just x & y? eg - perhaps some sort of reciprocal function?

    It seems like it should be the case, but I just can't see an equation! :-(

    Any help would be greatly appreciated. :-)

    Many thanks x
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  2. #2
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    Quote Originally Posted by TTB View Post
    I'm trying to construct a functions between 2 subsets of the 2d euclidean plane.

    The sets are the Hawaiian earring. ie - the countable union of circles with centres (1/n,0) and radius 1/n,

    and the countable union of circles of radius n & centres (n,0).

    I can construct one using different n's, but is it possible to construct one using just x & y? eg - perhaps some sort of reciprocal function?
    You can do this using the technique of geometric inversion. The map T defined by T(x,y) = \Bigl(\tfrac{x}{x^2+y^2}, \tfrac{-y}{x^2+y^2}\Bigr) (inversion in the unit circle) takes circles through the origin to lines, and vice versa. It maps the Hawaiian earring to the family of vertical lines x = n/2, and it maps the other family of circles to the family of vertical lines x=1/(2n).

    Let S be the map defined by S(x,y) = (1/(4x),y), which takes the first family of lines to the second family. Then the composite map TST will do what you want.
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