Q:

Where do our 4 macroscopic spacetime dimensions reside in multidimensional models of the universe?

In models such as M-theory with 7 'higher dimensions' plus the 4 macroscopic spacetime dimensions, where do our 4 macroscopic spacetime dimensions reside ordinally? My reason for asking is TV shows such as the 'Fabric of the Cosmos' that alludes to 'lower' dimensions in string theory. I can understand if our 4 are, say the 7th, 8th, 9th and 10th so there are dimensions lower than us but that we cannot readily detect.   

A:

Think of a very thin wire. It is a 3-dimensional object, but for many purposes you can describe it just as a 1-dimensional line or curve. The two remaining dimensions are curled up in a tiny cross section.
In a similar way, the speculations (not the slightest experimental hint exists that it should be so) about our world possibly being higher dimensional rest on the assumption that the remaining dimensions of our world are curled up so that for most purposes, only four of them are observable to us. The diameter of the curled up part is thought to be of the order of the Planck length, a scale far below the best current spatial resolution:
http://en.wikipedia.org/wiki/Planck_length
Note that in string theory, the higher-dimensional universe has no boundary, so the wire is an imperfect lower-dimensional analogue. But the surface of an infinitely long wire has no boundary (though only one extra dimension - it is a 2-dimensional universe embedded in a 3-dimensional space) and is a correct 1+1-dimensional analogue of a 4+6 or 4+7 dimensional universe without boundary and with all but 4 dimensions compactified.
There is no ordering of the dimensions, just as the single dimension of the wire is not related to the x- y- and z-axis labeling three spatial dimensions.

