  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.
