can be easily described by bi-dimensional words constructed using the two simple substitutions $$\sigma(r) = 
\begin{pmatrix}
  r & b \\ 
  b & r
\end{pmatrix} \quad \mbox{and}\quad \sigma(b) = 
\begin{pmatrix}
 r & b \\ 
 b & r 
\end{pmatrix}.$$

By iterating the above substitutions yields the strictly periodic word $$\sigma(R) = R = 
\begin{pmatrix}
  r & b & r & b \cdots\\
  b & r & b & r \cdots\\
  r & b & r & b \cdots \\
  b & r & b & r \cdots\\
  \vdots & \vdots & \vdots & \vdots ~ \ddots
\end{pmatrix}$$ which describes the continuous wave pattern of Figure \[fig:f0\].

Consider now the wave pattern solutions for $B(x,y)$ obtained for $\omega = 0.18$ on various rectangular grid sizes. For $\Omega = [-30, 30]\times [-15, 15]$, $B$ can be described by a $1 \times 4$ discrete pattern delimited by a box in Figure \[fig:f1\] and coded as $E = {}^t(r, b, r, b)$, where the superscript ${}^t$ denotes matrix transposition. $E$ can be decomposed as $\sigma(e) = {}^{t}(r, b)$ and the initial pattern $E$ is given by the substitution $\sigma$ applied to $E = {}^t(e, e)$ that is: $$\sigma(E) = {}^t\bigl(\sigma(e), \sigma(e)\bigr) = {}^t\bigl(r, b, r, b\bigr).$$

