slightly favored over FCC.

Non-round unit cells (kinked paths)
-----------------------------------

=2.0truein

Other upper bounds can be obtained by using different prescriptions for the $A\!-\!B$ surface. For example, one could choose a circular $A\!-\!B$ surface of radius $D(\phi)$ for the cylindrical phase. The inner ($A$) volume may be divided into wedges, and the outer ($B$) volume divided into wedges which each satisfy the volume constraint $\phi$ with a partner $A$ wedge [@XiMiln96b]. For a right triangular wedge which subtends an angle $\theta_0$, points at angle $\theta$ on the $A\!-\!B$ surface map to points $\vec{R}(\theta)$ on the boundary of the Wigner-Seitz cell, $$\vec{R}(\theta) = \vec{R}_1 + s\!\left(\theta\right) 
\left(\vec{R}_2 - \vec{R}_1\right)$$ where $\vec{R}_1=\{0,1\}$, and $\vec{R}_2=\{\tan\theta_0,1\}$. The composition specifies the radius, according to $$\phi\tan\theta_0=D^2(\phi)\theta_0.$$ The mapping which obeys the local composition constraint is $$s(\theta) = {\displaystyle{\theta\over\theta_0} - D(\phi)
{\sin\theta\over\tan\theta_0}\over 1-D(\phi)\cos\theta},$$ where $\theta_0=\pi/3$ and $\pi/4$ for hexagons and squares (see Figure \[fig:wedgecut\]), respectively.

Minimizing over the scale of the structure, eq \[eq:fsum\] yields, after some calculation, the following free energy: $$\begin{aligned}
{f\over f_0} &=& \left\{ {2\phi\over 3}\left[ \varepsilon^{-1}
+ {\varepsilon \phi^2\over (1-\phi)^2 D^4(\phi)} \right.\right.\times\\
&&\left.\left.\int_0^{\theta_0} {d\theta\over\theta_0}
|\vec{R}-\vec{r}|^3\left(
3\left|{d\vec{R}'\over d\theta}\right| + \left|{d\vec{r}'
\over d\theta}\right|\right)\right]\right\}^{1/3},\end{aligned}$$ where $$\begin{aligned}
\vec{r}(\theta) &=& D(\phi) {\sin\theta\choose\cos\theta} \\
\vec{r}'&=& \vec{r}\cdot\left[ \bbox{\delta} - 
{(\vec{R}-\vec{r}) 