Folding the Square-Diagonal Lattice

TL;DR

This paper investigates the thermodynamics and combinatorics of folding a square lattice with diagonals, providing analytical and numerical estimates of folding entropy and exploring higher-dimensional and alternative lattice models.

Contribution

It introduces a novel approach by mapping the folding problem onto multiple models, including a 28-vertex, 4-color model, and extends the analysis to higher dimensions and other lattices.

Findings

Folding entropy per face is approximately 0.2299.

Successful mapping onto dense loop, spin, and vertex models.

Extension of folding analysis to higher dimensions and different lattices.

Abstract

We study the problem of "phantom" folding of the two-dimensional square lattice, in which the edges and diagonals of each face can be folded. The non-vanishing thermodynamic folding entropy per face $s \simeq .2299(1)$ is estimated both analytically and numerically, by successively mapping the model onto a dense loop model, a spin model and a new 28 Vertex, 4-color model. Higher dimensional generalizations are investigated, as well as other foldable lattices.