3-DIMENSIONAL FOLDING OF THE TRIANGULAR LATTICE

We study the folding of the regular triangular lattice in three-dimensional embedding space, a model for the crumpling of polymerised membranes, We consider a discrete model, where folds are either planar or form the angles of a regular octahedron. These ''octahedral'' folding rules correspond simply to a discretisation of the 3d embedding space as a Face Centred Cubic lattice. The model is shown to be equivalent to a 96-vertex model on the triangular lattice. The folding entropy per triangle In q(3d) is evaluated numerically to be q(3d) = 1.43(1). Various exact bounds on q(3d) are derived.