Three-dimensional Folding of the Triangular Lattice

Abstract

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 $\ln q_{3d}$ is evaluated numerically to be $q_{3d}=1.43(1)$. Various exact bounds on $q_{3d}$ are derived.

55 pages, uuencoded, uses harvmac (l mode) and epsf, 19+2 figures included

References in corpus (6)

• Renormalization Theory for Interacting Crumpled Manifolds
• Entropy of Folding of the Triangular Lattice
• Renormalization and Hyperscaling for Self-Avoiding Manifold Models
• Folding Transition of the Triangular Lattice
• On the Crumpling Transition in Crystalline Random Surfaces
• Renormalization of Crumpled Manifolds

Cited by in corpus (13)

• Geometrically constrained statistical systems on regular and random lattices: From folding to meanders
• Folding transitions of the triangular lattice with defects
• Folding transition of the triangular lattice in a discrete three--dimensional space
• Folding of the triangular lattice in a discrete three-dimensional space: Density-matrix-renormalization-group study
• Folding of the Triangular Lattice with Quenched Random Bending Rigidity
• Geometrical Folding Transitions of the Triangular Lattice in the Face-Centred Cubic Lattice
• Crumpling transition of the discrete planar folding in the negative-bending-rigidity regime
• Folding the Square-Diagonal Lattice
• Crumpling transition of the triangular lattice without open edges: effect of a modified folding rule
• Exactly solvable flat-foldable quadrilateral origami tilings
• Folding of the triangular lattice in a discrete three-dimensional space: Crumpling transitions in the negative-bending-rigidity regime
• Folding transitions of the square--diagonal two--dimensional lattice
• Folding of the Triangular Lattice in the FCC Lattice with Quenched Random Spontaneous Curvature