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The existence of a fourth space dimension needs to be understood in a sensible manner.  In the S4 picture, the four space dimension is purely electromagnetic.  Electromagnetism pervades a spatial four-dimensional universe and it is gravity that constrains the three dimensions.  Thus our experience of three space dimensions is explained, and simultaneously our metaphysical experience of an ‘unphysical fourth space dimension’ is also explained.  Geometrically it is simple to understand as $S^4(1/h)$ is the total space of what would be a $U(1)$-bundle of electromagnetism as we do have circular fibers for each point on a hypersurface of $S^4(1/h)$ of equal length $1/\hbar$.  But the fibers intersect with each other and so it is not a principal fiber bundle.  But the interpretation of ‘purely electromagnetic’ four space dimensions explains why our usual experience is only of three spatial dimensions.  The LHC style experiments are frankly too crude to decide on the existence of a fourth spatial dimension.  Note also that the hierarchy problem of weakness of gravity has a natural and simple explanation in the S4 picture.