As was pointed out by Ehrenfest back in 1917 [4], neither classical atoms nor planetary orbits can be stable in a space with n > 3, and traditional quantum atoms cannot be stable either [5].

citation

Here n represents the number of spatial dimensions (elsewhere in the paper m represents the number of time dimensions)

On the dimensionality of spacetime (space.mit.edu) Some superstring theories have more than one effective low-energy limit corresponding to classical spacetimes with different dimensionalities. We argue that all but the (3 + 1)-dimensional one might correspond to ‘dead worlds’, devoid of observers, in which case all such ensemble theories would actually predict that we should find ourselves inhabiting a (3 + 1)-dimensional spacetime. With more or less than one time dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers.