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Question about a shape.

  1. Jan 10, 2013 #1
    Do you think it is possible to have a shape in the euclidean plane that has more
    than [itex] 2^{\aleph_0} [/itex] sides. maybe some crazy fractal.
    and if not in the plane would we maybe have to go to infinite dimensional space.
  2. jcsd
  3. Jan 10, 2013 #2
    Define "side".

    Anyway, we have that [itex]\{(x,y)~\vert~x,y\in \mathbb{R}\}[/itex] has cardinality [itex]2^{\aleph_0}[/itex]. So your figure would have to repeat its "sides" infinitely often.
  4. Jan 10, 2013 #3


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    Well, if a "side" is an ordered pair of vertices then you'd need more than [itex]2^{\aleph_0}[/itex] possible values for each vertex in order to have more than [itex]2^{\aleph0}[/itex] possible ordered pairs of vertices.

    That sounds like you need a space whose dimensionality is larger than the cardinality of R.
  5. Jan 10, 2013 #4
    thanks for the responses. defining a side with an ordered pair seems like a good idea
    ya it seems like you couldn't do it in the x-y plane. it does seem like I
    have R^R choices though for possible shapes.
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