In class, we showed that the equilibrium shape of a two$-$dimensional crystal is

determined by the condition min$({\displaystyle \underset{s}{\overset{\phantom{}}{}}}dl),$ where the integral is taken over the

crystal perimeter $s.$ For a faceted shape, this condition becomes min$({\displaystyle \underset{i}{\overset{?}{}}}{}_i{l}_i)$

where the subscript i represents a given facet on the perimeter. If a rectangular

crystal is bounded by sides of length $l_1$ and $l_2,$ prove by differentiation that the

equilibrium shape is given by the Gibbs$-$Curie$-$Wulff theorem,

$\frac{{}_2}{{}_1}=\frac{l_1}{l_2}$

You can assume that ${}_1$ and ${}_2$ are constants. Note that the area of

two$-$dimensional crystal, $l_1{l}_2,$ should also be taken as a constant. This equation

indicates that relative surface energies are related to facet lengths.