Folding the Square-Diagonal Lattice

We study the problem of "phantom" folding of the two-dimensional square lattice, in which the edges and diagonals of each face can be folded. The non-vanishing thermodynamic folding entropy per face $s \simeq .2299(1)$ is estimated both analytically and numerically, by successively mapping the model onto a dense loop model, a spin model and a new 28 Vertex, 4-color model. Higher dimensional generalizations are investigated, as well as other foldable lattices.