If extra compact dimensions exist, why are they compactified? I will show that Einstein-Maxwell theory with a positive cosmological constant supports black brane solutions for which the interior takes the form of a non-singular cosmological spacetime times a stabilized compact sphere. I will then argue that de Sitter space in six or more dimensions is unstable to the quantum nucleation of such solutions, including realistic four-dimensional universes. While very simple, these solutions point the way to dynamical connections between vacua of different macroscopic dimensionality. (See arxiv:0904.3115.)