Is There a Shape with More than 2^{\aleph_0} Sides in the Euclidean Plane?

[cragar]

Do you think it is possible to have a shape in the euclidean plane that has more
than $2^{\aleph_0}$ sides. maybe some crazy fractal.
and if not in the plane would we maybe have to go to infinite dimensional space.

 

[micromass]

Define "side".

Anyway, we have that $\{(x,y)~\vert~x,y\in \mathbb{R}\}$ has cardinality $2^{\aleph_0}$. So your figure would have to repeat its "sides" infinitely often.

 

[jbriggs444]

Define "side".

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

That sounds like you need a space whose dimensionality is larger than the cardinality of R.

 

[cragar]

thanks for the responses. defining a side with an ordered pair seems like a good idea
you 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.