  We study the folding of the regular two-dimensional triangular lattice
embedded in the regular three-dimensional Face-Centred Cubic lattice, a
discrete model for the crumpling of membranes. Possible folds are complete
planar folds, folds with the angle of a regular tetrahedron (71 degrees) or
with that of a regular octahedron (109 degrees). We study this model in the
presence of a negative bending rigidity K, which favours the folding process.
We use both a cluster variation method (CVM) approximation and a transfer
matrix approach. The system is shown to undergo two separate geometrical
transitions with increasing |K|: a first discontinuous transition separates a
phase where the triangular lattice is preferentially wrapped around octahedra
from a phase where it is preferentially wrapped around tetrahedra. A second
continuous transition separates this latter phase from a phase of complete
folding of the lattice on top of a single triangle.
