I've on many occasions that there are various numbers of 'extra' dimensions above the 4th. However, I've heard that they are 'compacted' or 'folded' tightly and unimaginably small. Now, as I understand, we live in 3 dimensional space coupled with time. So, by extension, those four dimensions should be present all around us and in our universe. Now, if these hypothetical extra dimensions are folded and compacted, obviously smaller than our universe, doesn't that mean that the extra dimensions are 'somewhere' in the universe and contained within it rather than enveloping it?

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    $\begingroup$ Extra dimensions are at each point of the 4-dimensional spacetime. The whole 10- or 11-dimensional spacetime has the shape of ${\mathcal M}^{3+1} \times X$ where $X$ is the 6- or 7-dimensional manifold and the first factor is the 3+1-dimensional spacetime manifold we used to think is "everything". It's a Cartesian product. It may also be a warped product but I don't want to go into these things. $\endgroup$ – Luboš Motl Sep 23 '12 at 5:36
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    $\begingroup$ @Lubos that would be the sort of thing to put in an answer $\endgroup$ – David Z Sep 23 '12 at 7:41

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