Three-dimensional Folding of the Triangular Lattice

TL;DR

This paper investigates the three-dimensional folding of a triangular lattice, modeling polymerized membrane crumpling, by mapping it to a vertex model and calculating the folding entropy.

Contribution

It introduces a discrete octahedral folding model for the 3D embedding of the triangular lattice and derives bounds and numerical estimates for the folding entropy.

Findings

Folding entropy per triangle is approximately 1.43.

Model is equivalent to a 96-vertex model on the triangular lattice.

Derived exact bounds on the folding entropy.

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