File Name : figs1.pdf

Caption : fitted parameters $a,b,c$ and $\phi^{ m  min}$ for varying patch width $w$. the blue squares denote the 6.6.6 tiling, the cyan circles denote the false tiling, and the green triangles denote the 4.8.8 tiling. \f{fig:testfits} shows that the parameter-fitting procedure introduces little additional error.

File Name : figs2.pdf

Caption : tiling free energies for various patch widths. the dark green curves are the original numerical calculations (using the parameters $a,b,d,\phi^{\min}$) and the cyan lines are the semi-analytical solutions using the fitted parameters $a,b,c$ and $\phi^{ m  min}$; see fig.~ ef{fig:fits}. the dark grey and the cyan curves closely match in all cases, indicating that little error is incurred during the parameter-fitting procedure.