Question about a shape.

  • Thread starter cragar
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  • #1
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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.
 

Answers and Replies

  • #2
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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.
 
  • #3
jbriggs444
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Define "side".

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.
 
  • #4
2,544
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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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