Discrete Folding

TL;DR

This paper investigates folding models of a triangular lattice in discrete space to understand crumpling transitions in membranes, revealing a first-order transition in 3D folding on a face-centered-cubic lattice.

Contribution

It introduces a 96-vertex model for 3D folding and analyzes the phase transition from crumpled to flat states.

Findings

3D folding exhibits a first-order transition.

Model captures crumpling behavior of membranes.

Both planar and 3D folding cases are studied.

Abstract

Models of folding of a triangular lattice embedded in a discrete space are studied as simple models of the crumpling transition of fixed-connectivity membranes. Both the case of planar folding and three-dimensional folding on a face-centered-cubic lattice are treated. The 3d-folding problem corresponds to a 96-vertex model and exhibits a first-order folding transition from a crumpled phase to a completely flat phase as the bending rigidity increases.