now consider different patterns of the form $$\begin{aligned}
X(\bx) = h_k^2 \left(  e^{i \bk_0 \cdot \bx} + e^{i \bl_0 \cdot \bx}  + A e^{i \bmm_0 \cdot \bx} + c.c. \right), \label{eqPatterns}\end{aligned}$$ where $A$ is the (possibly complex) amplitude of the third Fourier component. This pattern can be seen as two independent rolls ($\bk_0$ and $\bl_0$) plus a nonlinear component ($\bmm_0$). This nonlinear component is given by a quadratic coupling [@Kim1979], $$\begin{aligned}
X(\bmm_0)= {A} \, X(\bk_0) X(\bl_0) \label{eqXNLpatterns},\end{aligned}$$ where $A$ is a coupling coefficient. If one thinks of $X(\bmm_0)$ as being due to such a coupling, the nonlinear fraction of the variance is $$\begin{aligned}
\fNL &= \frac{|X(\bmm_0)|^2}{|X(\bk_0)|^2+|X(\bl_0)|^2+|X(\bmm_0)|^2} = \frac{|{A}|^2}{2 + |{A}|^2},\end{aligned}$$ which is given in the panels in the bottom row in Figure \[figPatterns\].

A hexagonal pattern is given by $A=1$; see bottom left panel in Figure \[figPatterns\]. Equation  with $A=1$ is equivalent to the horizontal dependence in Equation (249) in Section 16c of @Chandrasekhar1961. Hexagonal patterns have the advantage that they can be used to model a convective pattern that is characterized by a central upflow region surrounded by a network of downflows [@Chandrasekhar1961; @Toomre1977; @Busse2003], similar to supergranulation. In addition, hexagonal planforms are not symmetric with respect to a 