\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} %\usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsfig} \usepackage{textcomp} \usepackage{tikz} \usepackage{ifthen} \usetikzlibrary{snakes,arrows,shapes} %%%%%%%TILINGS: \newcommand{\Ar}{\mathrm{Area}} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode \epsfbox{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf Polymetric Brick Wall Patterns\\ \vskip .1in and Two-Dimensional Substitutions } \vskip 1cm \large Michel Dekking\\ 3TU Applied Mathematics Institute\\ and \\ Delft University of Technology\\ Faculty EWI\\ P. O.~Box 5031\\ 2600 GA Delft\\ The Netherlands\\ \href{mailto:f.m.dekking@tudelft.nl}{\tt f.m.dekking@tudelft.nl} \end{center} \vskip .2 in \begin{center} \emph{On the occasion of Jean-Paul Allouche's $60^{th}$ birthday} \end{center} \bigskip \begin{abstract} Polymetric walls are walls built from bricks in more than one size. Architects and builders want to built polymetric walls that satisfy certain structural and aesthetical constraints. In a recent paper by de Jong, Vindu\v{s}ka, Hans and Post these problems are solved by integer programming techniques, which can be very time consuming for patterns consisting of more than 40 bricks. Here we give an extremely fast method, generating patterns of arbitrary size. \end{abstract} % main text \section{Introduction} How can we build a brick wall with different sizes of bricks? This problem was recently discussed by de Jong, Vindu\v{s}ka, Erwin, and Post \cite{JVHP}. %--------------------------------------------------------------------------------------------------- \begin{figure}[h!]% \begin{center} \epsfysize=40mm \epsfbox{polymetsel.eps} \\ % \caption{\footnotesize An example of a polymetric brick wall from www.metselen.net/fotoalbum.html}\label{Fig:1} \end{center} \end{figure} %--------------------------------------------------------------------------------------------------- Let us try to formulate the requirements put forward by architects and builders. These are \begin{itemize} \item[[R1\!\!\!]] The absence of regularity in the patterns, as e.g., periodicity \item[[R2\!\!\!]] Long horizontal joints, to facilitate the building process \item[[R3\!\!\!]] The absence of long vertical joints, for sturdiness and strength \item[[R4\!\!\!]] Pleasing aesthetics, as e.g., not too many large bricks close together \end{itemize} \noindent One can appreciate the fulfilment of all four requirements in Figure~\ref{Fig:1}. In de Jong, Vindu\v{s}ka, Erwin, and Post\cite{JVHP} these four requirements are met by considering the construction of a brick wall pattern as a pallet loading problem, which is then solved with a mixed integer linear program. Here we take a completely different approach, constructing the patterns as pseudo-self-similar tilings generated by 2-D substitutions. Well known properties of such tilings automatically yield the fulfilment of [R1], and also more or less [R4]. Requirement [R2] can very easily be built in, so that [R3] becomes the main challenge. The style of this note will be somewhat informal, for precise definitions and more context see the papers by Solomyak \cite{Solom} and Frank \cite{Priebe-primer}. An early paper connecting tilings and 2D-substitutions is Allouche and Salon \cite{All-Salon}. See Section 14 (and the book cover) of Allouche and Shallit's monograph \cite{Allou} for the subclass of tilings generated by multi-dimensional automatic sequences. Since then the literature on tilings has expanded a lot (cf.~\cite{Priebe-primer}). \section{Prouhet-Thue-Morse patterns} The Prouhet-Thue-Morse sequence $0,1,1,0,1,0,0,1,\ldots$ is the \emph{one}-dimensional sequence generated by the substitution $0\rightarrow 0\,1,\; 1\rightarrow 1\,0$. It is folklore that in the ``same way" one generates a 2D-sequence (see, e.g., \cite{Photonic}) by the 2D-substitution: $$ \begin{tabular}{ccccccccc} $ $ & $ $ & $1$ & $0$\; &$ $ & $ $ & $ $ & $0$ & $1$ \\ $0$ & $\mapsto$ & $0$ & $1$, &$ $ & $1$ & $\mapsto$ & $1$ & $0$. \end{tabular} $$ Usually this sequence is then represented in the plane by white and black squares. When we take $2\times1$ and $3\times1$ bricks this corresponds to the following substitution rules: \begin{center} \begin{tikzpicture}[scale=.2,rounded corners=0] \def\bricktwoone(#1){\path (#1) coordinate (P0); \draw [fill=red!50!yellow, draw=gray, ultra thick] (P0) rectangle +(2,1)} \def\brickthreeone(#1){\path (#1) coordinate (P0); \draw [fill=red!10!brown, draw=gray, ultra thick] (P0) rectangle +(3,1)} \def\sigma(#1,#2){\path (#2) coordinate (P0); \path (P0)++(0,0) coordinate (P210); \path (P0)++(2,0) coordinate (P211); \path (P0)++(0,1) coordinate (P212); \path (P0)++(3,1) coordinate (P213);%%einde brick21 \ifthenelse{#1=21}{\bricktwoone(P210);\brickthreeone(P211); \brickthreeone(P212);\bricktwoone(P213)}{}; \path (P0)++(0,0) coordinate (P310); \path (P0)++(3,0) coordinate (P311); \path (P0)++(0,1) coordinate (P312); \path (P0)++(2,1) coordinate (P313); %%einde brick31 \ifthenelse{#1=31}{\brickthreeone(P310);\bricktwoone(P311);\bricktwoone(P312);\brickthreeone(P313)}{}; }; %%end\sigma \path (0,0) coordinate (B21); \path (6,0) coordinate (SB21); \path (20,0) coordinate (B31); \path (26,0) coordinate (SB31); \bricktwoone(B21); \sigma(21,SB21); \node at (4,0.4) {$\mapsto$}; \brickthreeone(B31); \sigma(31,SB31); \node at (24.7,0.4) {$\mapsto$}; \end{tikzpicture} \end{center} The third iteration of this substitution starting from the $2\times1$ brick $B_{21}$ yields the following pattern: \begin{figure}[h!]% \begin{center} \epsfysize=1.5cm\epsfbox{Morse-bricks-real-2D-order-3.eps} \end{center} \end{figure} Obviously requirement [R3] will be violated, since we obtain vertical segments crossing the whole pattern. To cope with this, we slightly change the previous substitution to the substitution $\tau$ given by \begin{center} \begin{tikzpicture}[scale=.2,rounded corners=0] \def\bricktwoone(#1){\path (#1) coordinate (P0); \draw [fill=red!50!yellow, draw=gray, ultra thick] (P0) rectangle +(2,1)} \def\brickthreeone(#1){\path (#1) coordinate (P0); \draw [fill=red!10!brown, draw=gray, ultra thick] (P0) rectangle +(3,1)} \def\sigma(#1,#2){\path (#2) coordinate (P0); \path (P0)++(0,0) coordinate (P210); \path (P0)++(2,0) coordinate (P211); \path (P0)++(1,1) coordinate (P212); \path (P0)++(4,1) coordinate (P213);%%einde brick21 \ifthenelse{#1=21}{\bricktwoone(P210);\brickthreeone(P211); \brickthreeone(P212);\bricktwoone(P213)}{}; \path (P0)++(0,0) coordinate (P310); \path (P0)++(3,0) coordinate (P311); \path (P0)++(1,1) coordinate (P312); \path (P0)++(3,1) coordinate (P313); %%einde brick31 \ifthenelse{#1=31}{\brickthreeone(P310);\bricktwoone(P311);\bricktwoone(P312);\brickthreeone(P313)}{}; }; %%end\sigma \path (0,0) coordinate (B21); \path (6,0) coordinate (SB21); \path (20,0) coordinate (B31); \path (26,0) coordinate (SB31); \bricktwoone(B21); \sigma(21,SB21); \node at (4,0.4) {$\mapsto$}; \brickthreeone(B31); \sigma(31,SB31); \node at (24.7,0.4) {$\mapsto$}; \end{tikzpicture} \end{center} \noindent Now a fourth order iteration of $\tau$ starting from the $2\times1$ brick yields the pattern in Figure~\ref{Fig:P-T-M-order-4}. \begin{figure}[h!]% \begin{center} \epsfysize=4cm\epsfbox{Morse-bricks-real-2D-simple-coord-order-4.eps}\\ % \caption{\footnotesize Skewed 2D-Prouhet-Thue-Morse: $\tau^4(B_{21})$.}\label{Fig:P-T-M-order-4} \end{center} \end{figure} These skewed P-T-M patterns nicely satisfy [R3], see the following proposition. \begin{proposition} For any $n$ the vertical segments in the patterns $\tau^n(B_{21})$ and $\tau^n(B_{31})$ have length at most 2. \end{proposition} \begin{proof} By induction on $n$, using the following figure. \begin{center} \epsfysize=3.5cm\epsfbox{Morse-bricks-proof.eps} \end{center} \vspace*{-0.9cm} \end{proof} \section{General patterns} We would like the pattern to be generated by a geometric 2D-substitution, mapping rectangles $i\times j$ (bricks $B_{ij}$) to unions of rectangles (bricks), resulting in a self-similar tiling (see \cite{Priebe-primer} for the basic definitions). However, as already clear from the first P-T-M example above, such self-similar tilings will violate the bounded vertical segments condition [R3]. Thus we should rather consider pseudo-self-similar or pseudo-self-affine tilings as studied by Priebe and Solomyak (\cite{PriebeSolom}). In a self-affine (rectangular) tiling with expansion coefficients $\lambda_1$ and $\lambda_2$, all the bricks (usually called proto-tiles in the literature) are scaled by a factor $\lambda_1$ in the $x$-direction and a factor $\lambda_2$ in the $y$-direction, and then are exactly tiled by translations of the original set of bricks. Note by the way that the P-T-M-substitution is self-similar with two $1\times 1$ bricks (expansions $\lambda_1=2$ and $\lambda_2=2$) , but not with a set consisting of the $2\times 1$ and a $3\times 1$ brick. In a pseudo-self-affine tiling the bricks do not tile exactly the expanded bricks, but have bricks `sticking out' and `sticking in', in a way that still preserves the tiling property. We will consider a subclass of 2D-substitutions which have the desired properties, which we call of PP-type, i.e., which have the periodic parallelogram property. With this we mean that the image of a $i\times 1$ brick is a region as in Figure~\ref{Fig:PP-prop}, left. The black horizontal segments have length $i\lambda_1$, and the two curves on the left and the right are translated copies of each other. For a $i\times j$ brick this is repeated periodically $j$ times, cf. Figure~\ref{Fig:PP-prop}, right. \begin{figure}[h!]% \begin{center} \begin{tikzpicture}[scale=.2,rounded corners=0] \def\ypath(#1){\path (#1) coordinate (P0); \path (P0)+(5,6) coordinate (Q0); \path (P0)+(1.4,2.4) coordinate (D0); \path (Q0)+(-2,-2) coordinate (D1); \draw [blue] (P0)--++(0,1)--++(1,0)--++(0,1); \draw [blue] (Q0)--++(0,-1)--++(-2,0)--++(0,-1); \draw [dashed, blue, thin] (D0)--(D1);} \path (0,0) coordinate (left); \path (10,0) coordinate (right); \path (5,6) coordinate (leftt); \path (15,6) coordinate (rightt); \draw[thick] (left)--(right); \draw[thick] (leftt)--(rightt); \ypath(left); \ypath(right); \path (25,0) coordinate (shift); \path (0,0)++(shift) coordinate (left); \path (10,0)++(shift) coordinate (right); \path (5,6)++(shift) coordinate (leftt); \path (15,6)++(shift) coordinate (rightt); \draw[thick] (left)--(right); \ypath(left); \ypath(right); \path (25,6) coordinate (shift); \path (0,0)++(shift) coordinate (left); \path (10,0)++(shift) coordinate (right); \path (5,6)++(shift) coordinate (lefttt); \path (15,6)++(shift) coordinate (righttt); \draw[thick] (leftt)--(left); \draw[thick] (right)--(rightt); \ypath(left); \ypath(right); \draw[thick] (lefttt)--(righttt); \end{tikzpicture} \end{center} \caption{Left: Image region for a $i\times 1$ brick. Right: Image region for a $i\times 2$ brick.}\label{Fig:PP-prop} \end{figure} A simple example of a PP-type substitution with $\lambda_1=\lambda_2=2$ and the three brick alphabet $\mathcal{A}=\{B_{11},B_{21},B_{22}\}$ is given by \begin{center} \begin{tikzpicture}[scale=.2,rounded corners=0] \def\brickoneone(#1){\path (#1) coordinate (P0); \draw [fill=red!40!yellow, draw=gray, ultra thick] (P0) rectangle +(1,1)} \def\bricktwoone(#1){\path (#1) coordinate (P0); \draw [fill=red!60!yellow, draw=gray, ultra thick] (P0) rectangle +(2,1)} \def\bricktwotwo(#1){\path (#1) coordinate (P0); \draw [fill=red!10!brown, draw=gray, ultra thick] (P0) rectangle +(2,2)} \def\sigma(#1,#2){\path (#2) coordinate (P0); \path (P0)++(-1,0) coordinate (P110); \path (P0)++(0,1) coordinate (P111); \ifthenelse{#1=11}{\bricktwoone(P110);\bricktwoone(P111)}{}; \path (P0)++(-1,0) coordinate (P210); \path (P0)++(1,0) coordinate (P211); \path (P0)++(0,1) coordinate (P212); \path (P0)++(3,1) coordinate (P213); \ifthenelse{#1=21}{\bricktwoone(P210);\bricktwotwo(P211);\brickoneone(P212);\brickoneone(P213)}{}; \path (P0)++(-1,0) coordinate (P220); \path (P0)++(1,0) coordinate (P221); \path (P0)++(0,1) coordinate (P222); \path (P0)++(3,1) coordinate (P223); \path (P0)++(-1,2) coordinate (P224); \path (P0)++(1,2) coordinate (P225); \path (P0)++(2,2) coordinate (P226); \path (P0)++(0,3) coordinate (P227); \path (P0)++(2,3) coordinate (P228); \ifthenelse{#1=22}{\bricktwoone(P220);\bricktwotwo(P221);\brickoneone(P222);\brickoneone(P223);\bricktwoone(P224);\brickoneone(P225);\brickoneone(P226);% \bricktwoone(P227);\bricktwoone(P228)}{}; }; %%end\sigma \path (0,0) coordinate (B11); \path (6,0) coordinate (SB11); \path (13,0) coordinate (B21); \path (20,0) coordinate (SB21); \path (30,0) coordinate (B22); \path (37,0) coordinate (SB22); \brickoneone(B11); \sigma(11,SB11); \node at (3,0.4) {$\mapsto$}; \bricktwoone(B21); \sigma(21,SB21); \node at (16.7,0.4) {$\mapsto$}; \bricktwotwo(B22); \sigma(22,SB22); \node at (33.7,0.8) {$\mapsto$}; \end{tikzpicture} \end{center} \noindent The patterns $\sigma^3(B_{11}), \sigma^3(B_{21})$ and $\sigma^3(B_{22})$ are \begin{center} \epsfysize=2.5cm\epsfbox{Bricks-path-1by1-2by1-2by2-order-3.eps} \end{center} The pattern $\sigma^4(B_{22})$ is given in Figure~\ref{Fig:max-11}. \begin{figure}[h!]% \begin{center} \epsfysize=6cm\epsfbox{Bricks-path-1by1-2by1-2by2-order-4.eps} \caption{\footnotesize $\sigma^4(B_{22})$.}\label{Fig:max-11} \end{center} \end{figure} \medskip Let $v_{\rm max}(ij,n)$ be the maximal length of a vertical segment in $\sigma^n(B_{ij})$ for $n\ge 1$ and $B_{ij}\in \mathcal{A}$. For the substitution above $v_{\rm max}(11,1)=1, v_{\rm max}(21,1)=2, v_{\rm max}(22,1)=3$, and one can prove that for this substitution $$v_{\rm max}:=\sup\{v_{\rm max}(ij,n): B_{ij}\in \mathcal{A}, \;n\ge 1\}=v_{\rm max}(22,3)=11,$$ but in general it is not an easy task to determine the largest possible length $v_{\rm max}$ of a vertical segment (which, of course, can be infinite). The following approach generates patterns satisfying [R3] in an easy way. We say the pattern $\sigma(B_{ij})$ \emph{has no crossings} if there are no vertical segments consisting of boundaries of the bricks in $\sigma(B_{ij})$ that connect two points on the boundary of the region determined by $\sigma(B_{ij})$. In the substitution above, for example, $\sigma(B_{11})$ and $\sigma(B_{22})$ do not have a crossing, but $\sigma(B_{21})$ does. \begin{proposition}\label{Prop:nocross} Let $\sigma$ be a {\rm{PP}}-type substitution with expansions $\lambda_1$ and $\lambda_2$. Suppose that for all $B_{ij}$ the pattern $\sigma(B_{ij})$ has no crossings. Let $j^*$ be the maximal height of the bricks. Then any vertical segment in any pattern $\sigma^n(B_{ij})$ has length at most $2j^*(\lambda_2-1)$. \end{proposition} \begin{proof} Let $[(x,y),(x,y')]$ be a vertical segment. The points $(x,y)$ and $(x,y')$ lie in some $\sigma(B_{ij})$ and $\sigma(B_{i'\!j'})$. If these are the same, then $|y-y'|] (3,2.5) -- (6,5); \sigma(10,SB10); \draw [->] (3,-0.5) -- (6,-3); \sigma(12,SB12); \bricktwotwo(B2); \draw [->] (25,2.5) -- (28,5); \sigma(20,SB20); \draw [->] (25,-0.5) -- (28,-3); \sigma(22,SB22); \end{tikzpicture} \end{center} Here random means that each expanded brick is replaced by one of the two possibilities with probability $1/2$, independently of the other replacements. At the next levels, this random replacement is repeated, independently of the previous levels, and independently of the other bricks at the same level. \begin{figure}[h!]% \begin{center} \epsfysize=4cm\epsfbox{Bricks-path-2by2-1by2-random-order-4-three.eps}% \caption{\footnotesize Three realizations of $\sigma^4(B_{22})$.}\label{Fig:random-4} \end{center} \end{figure} In this simple example the substitution matrix $ M_{\sigma}$ is non-random, since the two possibilities are just permutations of each other. A simple computation learns that for $n=1,2,\dots$ $$ M_{\sigma}^n=\left( \begin{array}{cc} 2\cdot 4^{n-1} & 4^{n-1} \\ 4^{n} & 2\cdot 4^{n-1} \end{array} \right). $$ Note that different sequences of coin flips will lead to different patterns, so there will be $$2^{\,1+6+ \dots + 6\cdot 4^{n-1}}= 2^{\,2\cdot 4^n-1}$$ different realizations of $\sigma^{n+1}(B_{22})$, since the total number of bricks in $\sigma^{n}(B_{22})$ is equal to the sum of the two entries in the lower row of $ M_{\sigma}^n$. This means that the three realizations in Figure~\ref{Fig:random-4} are picked from a set of $2^{\,2\cdot 4^3-1}=2^{127}\approx 10^{39}$ elements! We next consider a pseudo-self-similar random substitution, where the coin is flipped with a success probability $p$ for some $p\in[0,1]$: \begin{center} \begin{tikzpicture}[scale=.2,rounded corners=0] \def\brickonetwo(#1){\path (#1) coordinate (P0); \draw [fill=blue!30!white, draw=black!40, very thick] (P0) rectangle +(1,2)} \def\bricktwotwo(#1){\path (#1) coordinate (P0); \draw [fill=blue!15!gray, draw=black!40, very thick] (P0) rectangle +(2,2)} \def\sigma(#1,#2){% Deterministisch (!), met 4 mogelijkheden \path (#2) coordinate (P0); \path (P0)++(-1,0) coordinate (P1200); \path (P0)++(-1,0) coordinate (P1201); \path (P0)++(0,0) coordinate (P1230); \path (P0)++(0,2) coordinate (P1210); \path (P0)++(0,2) coordinate (P1211); \path (P0)++(1,2) coordinate (P1220); \ifthenelse{#1=120}{\brickonetwo(P1200);\brickonetwo(P1230);\brickonetwo(P1210); \brickonetwo(P1220)}{}; \ifthenelse{#1=121}{\bricktwotwo(P1201); \bricktwotwo(P1211)} {}; \path (P0)++(-1,0) coordinate (P2200); \path (P0)++(-1,0) coordinate (P2201); \path (P0)++(0,0) coordinate (P2270); \path (P0)++(1,0) coordinate (P2210); \path (P0)++(1,0) coordinate (P2211); \path (P0)++(2,0) coordinate (P2220); \path (P0)++(0,2) coordinate (P2221); \path (P0)++(0,2) coordinate (P2230); \path (P0)++(2,2) coordinate (P2231); \path (P0)++(1,2) coordinate (P2240); \path (P0)++(2,2) coordinate (P2250); \path (P0)++(3,2) coordinate (P2260); \ifthenelse{#1=220}{\brickonetwo(P2200);\brickonetwo(P2270);\brickonetwo(P2210);\brickonetwo(P2220); \brickonetwo(P2230);\brickonetwo(P2240);\brickonetwo(P2250);\brickonetwo(P2260)} {}; \ifthenelse{#1=221}{\bricktwotwo(P2201);\bricktwotwo(P2211);\bricktwotwo(P2221);\bricktwotwo(P2231)} {}; };%end-sigma \path (0,0) coordinate (B1); \path (10,3) coordinate (SB120); \path (10,-5) coordinate (SB121); \path (22,0) coordinate (B2); \path (32,3) coordinate (SB220); \path (32,-5) coordinate (SB221); \brickonetwo(B1); \draw [->] (3,2.5) -- (6,5); \sigma(120,SB120); \draw [->] (3,-0.5) -- (6,-3); \sigma(121,SB121); \node [above] at (3.2,3.75) {\footnotesize $1-p$}; \node [above] at (4.5,-1.75) {\footnotesize $p$}; \bricktwotwo(B2); \draw [->] (25,2.5) -- (28,5); \sigma(220,SB220); \draw [->] (25,-0.5) -- (28,-3); \sigma(221,SB221); \node [above] at (25.2,3.75) {\footnotesize $1-p$}; \node [above] at (26.5,-1.75) {\footnotesize $p$}; \end{tikzpicture} \end{center} \noindent Figure~\ref{perco} shows 2 realizations for $p=1/3$ and $p=2/3$ each. \begin{figure}[h!]% \begin{center} \epsfysize=2.8cm\epsfbox{Bricks-path-2by2-1by2-random-simple-2-p-s.eps}% \caption{\footnotesize Two realizations of $\sigma^4(B_{22})$; left: with $p=1/3$, right: with $p=2/3$.}\label{perco} \end{center} \end{figure} The maximal length of a vertical segment now is a random variable $V_{\rm max}(ij,n)$, and trivially $V_{\rm max}(22,n)=2$ for all $n$ when $p=1$, and so $V_{\rm max}=2$ in this case. However, for all $p<1$, we have $V_{\rm max}=\infty$ with probability 1. This can be deduced from the fact that there is a lot of independence between the bricks in $\sigma^n(B_{ij}), ij=12$ and $ij=22$, despite the local dependencies. In general the matrix $ M_{\sigma}$ will be random, and its expectation will give us the mean growth of the number of bricks in the patterns $\sigma^n(B_{ij})$. In the last example the mean $M_{\sigma}$ equals $$ \left( \begin{array}{cc} 4(1-p) & 2p \\ 8(1-p) & 4p \end{array} \right), $$ which has eigenvalues 0 and 4 (independent of $p$). This can be considered as an instance of a version of Proposition \ref{Prop: eigenvalues} for random pseudo self-similar tilings. The theory of multi-type branching processes can be applied to obtain further information on these patterns. The basic limit theorem, for instance, tells us that the number of bricks $B_{12}$ in the pattern $\sigma^n(B_{22})$ divided by $4^n$ converges almost surely to a random variable $W=W_p$, which is positive almost surely. \begin{thebibliography}{4} \bibitem{All-Salon} J.-P.\ Allouche and O.\ Salon, Finite automata, quasicrystals, and Robinson tilings, in I.\ Hargittai, ed., \emph{Finite Automata, Quasicrystals, and Molecules of Fivefold Symmetry}, VCH Publishers, New York, 1990, pp.\ 97--105. \bibitem{Allou} J.-P. Allouche and J. Shallit, \emph{Automatic Sequences, Theory, Applications, Generalizations}, Cambridge University Press, 2003. \bibitem{Priebe-primer} N. P. Frank, A primer of substitution tilings of the Euclidean plane, \emph{Expo. Math.} {\bf 26} (2008), 295--326. \bibitem{JVHP} J.\ de Jong, H.\ Vindu\v{s}ka, H.\ Erwin, and G.\ Post, Polymetrisch metselen, \emph{Stator} {\bf 13} (2012), 9--12 (In Dutch). \bibitem{Photonic} L. Moretti and V. Mocella, The square Thue--Morse tiling for photonic application, \emph{Philosophical Magazine} {\bf 88} (2008), 2275--2284. \bibitem{Pey} J.\ Peyri\`ere, \emph{Frequency of patterns in certain graphs and Penrose tilings}, {\it J. Physique} {\bf47} (1986), C3-41--C3-62. \bibitem{PriebeSolom} N. M.\ Priebe and B. Solomyak, Characterization of planar pseudo-self-similar tilings, \emph{Discrete \& Computational Geometry} {\bf 26} (2001), 289--306. \bibitem{Queff} M.\ Queff\'elec, \emph{Substitution Dynamical Systems -- Spectral Analysis}, Springer, 2nd ed., 2010. \bibitem{Solom} B.\ Solomyak, Dynamics of self-similar tilings, \emph{Ergodic Theory Dynam. Systems} {\bf 17} (1997), 695--738. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 52C23; Secondary 05B45, 52C20. \noindent \emph{Keywords: } pseudo-self-similar tiling, 2D-substitution, automatic sequence. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received May 15 2012; revised version received August 10 2012. Published in {\it Journal of Integer Sequences}, March 2 2013. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}