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I was wondering how many dimensions can be embedded within a 2D space, or more generally within N-dimensions. Is there a formal demonstration?

This question came to me when I read about the Holographic principle, it says is possible to encode a volume inside a black hole into the area that surrounds it. So there is a one to one correspondence between 3D and 2D.

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    I've edited the question for clarification2012-12-30

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The Abelian group $(\mathbb R,+)$ is isomorphic to that of $(\mathbb R^k,+)$, for any finite $k$, and in fact even more.

It follows that a single dimension can encode infinitely many dimensions.