We discuss a class of graph embeddings into Minkowski space R^{2,2} = C^{1,1}, called t-surfaces, which arise in the study of the planar dimer model. A t-surface consists of a (perfect) t-embedding together with its associated origami map. Perfect t-embeddings were recently introduced as a key tool for proving that the gradient of the dimer height function converges to that of the Gaussian Free Field in a canonically associated metric, under suitable technical assumptions. After introducing the notion of a t-embedding, we will describe a construction of perfect t-embeddings for regular hexagons of the hexagonal lattice. In which the corresponding t-surfaces converge to space-like maximal surfaces in Minkowski space R^{2,1}. As a consequence, these constructions yield a new proof of convergence of fluctuations of the dimer height function to the Gaussian Free Field in the conformal structure induced by the limiting maximal surface.