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We develop a new approach to building cosmological models, in which small pieces of perturbed Minkowski space are joined together at reflection-symmetric boundaries in order to form a global, dynamical space-time. Each piece of this patchwork universe is described using post-Newtonian gravitational physics, with the large-scale expansion of the Universe being an emergent phenomenon. This approach to cosmology does not require any assumptions about nonlocal averaging processes. Our framework clarifies the relation between the weak-field limit of general relativity, and the cosmological solutions that result from solving Einstein's equations with a set of symmetry assumptions. It also allows the effects of structure formation on the large-scale expansion of the Universe to be investigated without averaging anything. As an explicit example, we use this formalism to investigate the cosmological behavior of a large number of regularly arranged pointlike masses. In this case we find that the large-scale expansion is well modelled by a Friedmann-like equation that contains terms that take the form of dust, radiation, and spatial curvature. The radiation term, while small compared to the dust term, is purely a result of the nonlinearity of Einstein's equations. [ABSTRACT FROM AUTHOR]