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Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$?

Have anyone met any fractal of dimension $\pi$?

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    There's more than one definition of dimension, which are you interested in? (Some are restricted to the natural numbers)2012-09-18
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    Your ordinary every-day dimension is going to be a cardinal number, but then again it might be one of the numerous dimensions Ben M mentions, which I know nothing about.2012-09-18
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    I believe he's talking about Hausdorff Dimension, in which case the Hausdorff Dimension Theorem says such spaces exist.2012-09-18
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    I'm interested in it for every suitable meaning of 'dimension'.2012-09-18
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    @JacobSchlather Completely reedited this comment. I *thought* I remembered what Hausdorff dimension was, but it turns out I forgot a lot :) In any case, $\mathbb{R}^\alpha$ screams vector space to me, and I have no idea how to interpret it as having a strange Hausdorff dimension.2012-09-18
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    @rschwieb, I agree that $\mathbb R^\alpha$ looks like a vector space. I was guessing hausdorff dimension based on his question concerning a fractal of dimension $\pi$. I don't know too much about spaces with fractional Hausdorff dimension. But it seems like there isn't a canonical space of dimension $\pi$ or a way to make sense of $\mathbb R^\pi$. But there are subsets of euclidean space with Hausdorff dimension $\pi$.2012-09-18
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    @JacobSchlather cool :)2012-09-18

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