summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex')
-rw-r--r--Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex894
1 files changed, 871 insertions, 23 deletions
diff --git a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex
index 8bcd020028c..3016fa2987b 100644
--- a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex
+++ b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex
@@ -1,6 +1,6 @@
-%% $Id: pst-fractal-doc.tex 658 2017-11-26 16:55:41Z herbert $
-\documentclass[11pt,english,BCOR10mm,DIV12,bibliography=totoc,parskip=false,
- smallheadings, headexclude,footexclude,oneside]{pst-doc}
+%% $Id: pst-fractal-doc.tex 677 2017-12-03 09:51:54Z herbert $
+\documentclass[fontsize=11pt,english,BCOR=10mm,DIV=13,toc=bibliography,parskip=false,
+ headings=small, headinclude=false,footinclude=false,oneside]{pst-doc}
\usepackage[utf8]{inputenc}
\usepackage{pst-fractal}
\let\pstFV\fileversion
@@ -8,6 +8,37 @@
\def\PSLenv{\Lenv{pspicture}}
\usepackage{animate}
+
+\def\pSTilingsFibonacci#1#2#3{% #1=n #2=nbre de lignes #3=type 1 ou 2
+ \pstFPDiv\Nbr{#2}{2}%
+ \psset[pst-fractal]{n=#1}%
+ \pstVerb{/n0 #1 def /tabPell [0 1 2 5 12 29 70 169 408] def
+ /Type #3 def
+ Type 2 gt {/Type 1 def} if
+ Type 2 eq {
+% (-1)^2P(n),(P(n+1)
+ /Y1 tabPell n0 1 add get def
+ /X1 -1 n0 exp tabPell n0 get mul def
+ }{
+% (P(n+1),(-1)^2P(n)
+ /X1 tabPell n0 1 add get def
+ /Y1 -1 n0 exp tabPell n0 get mul def
+ } ifelse
+ % le déplacement perpendiculaire
+ /X2 Y1 neg def
+ /Y2 X1 def
+}%
+\pstVerb{/ListColors [0 0 1] def}%
+\multido{\I=-\Nbr+1}{#2}{5 \rput(!X2 \I\space mul Y2 \I\space mul){%
+ \multido{\i=-\Nbr+1}{#2}{%
+%\definecolor[ps]{Couleur}{rgb}{ListColors aload pop}%
+ \definecolor[ps]{Couleur}{cmyk}{ListColors aload pop 0}%
+ \rput(!X1 \i\space mul Y1 \i\space mul){\psFibonacciPolyominoes[fillcolor=Couleur]}%
+ \pstVerb{/ListColors [ListColors aload pop 3 1 roll] def}%
+}}}}%
+
+
+
%\usepackage{auto-pst-pdf}
\addbibresource{\jobname.bib}
@@ -24,7 +55,8 @@
\tableofcontents
-\clearpage
+\vspace{3cm}
+
\begin{abstract}
\noindent
@@ -48,16 +80,18 @@ resolution. Run the examples as single documents to see how it will be in
high quality.
-\section{Sierpinski triangle}
+\section{Sierpinski triangle and curve}
The triangle must be given by three mandatory arguments. Depending to the kind of
arguments it is one of the two possible versions:
\begin{BDef}
\Lcs{psSier}\OptArgs\coord0\coord1\coord2\\
-\Lcs{psSier}\OptArgs\coord0\Largb{Base}\Largb{Recursion}
+\Lcs{psSier}\OptArgs\coord0\Largb{Base}\Largb{Recursion}\\
+\Lcs{psSier}\OptArgs
\end{BDef}
+\subsection{Triangle}
In difference to \Lcs{psfractal} it doesn't reserve any space, this is the
reason why it should be part of a \PSLenv{} environment.
@@ -76,8 +110,7 @@ reason why it should be part of a \PSLenv{} environment.
\end{pspicture} }
\end{LTXexample}
-
-\section{Sierpinski curve}
+\subsection{Curve}
There are four special optional arguments for the Siepinski curve:
\begin{itemize}
@@ -115,9 +148,6 @@ There are four special optional arguments for the Siepinski curve:
\end{LTXexample}
-
-\iffalse
-
\begin{animateinline}[controls,% palindrome,
begin={\begin{pspicture}(-4,-4)(4,4)},
end={\end{pspicture}}]{5}% 5 image/s
@@ -141,7 +171,7 @@ There are four special optional arguments for the Siepinski curve:
\end{animateinline}
\end{verbatim}
-\fi
+
\section{Julia and Mandelbrot sets}
@@ -214,7 +244,7 @@ $(x_0;y_0)$ is the starting value.
\begin{LTXexample}[pos=l]
-\psfractal
+\psfractal \qquad
\psfractal[type=Mandel]
\end{LTXexample}
@@ -222,7 +252,7 @@ $(x_0;y_0)$ is the starting value.
The color for the convergent part is set by \Lkeyword{baseColor}.
\begin{LTXexample}
-\psfractal[xWidth=4cm,yWidth=4cm,dIter=30](-2,-2)(2,2)
+\psfractal[xWidth=4cm,yWidth=4cm,dIter=30](-2,-2)(2,2) \qquad
\psfractal[xWidth=4cm,yWidth=4cm,baseColor=yellow,dIter=30](-2,-2)(2,2)
\end{LTXexample}
@@ -235,12 +265,16 @@ The color for the convergent part is set by \Lkeyword{baseColor}.
\psfractal[type=Mandel,xWidth=12.8cm,yWidth=10.8cm,dIter=5](-2.5,-1.3)(0.7,1.3)
\end{LTXexample}
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\subsection{\texttt{cx} and \texttt{cy}}\xLkeyword{cx}\xLkeyword{cy}
Define the starting value for the complex constant number $C$.
\begin{LTXexample}
\psset{xWidth=5cm,yWidth=5cm}
-\psfractal[dIter=2](-2,-2)(2,2)
+\psfractal[dIter=2](-2,-2)(2,2) \qquad
\psfractal[dIter=2,cx=-1.3,cy=0](-2,-2)(2,2)
\end{LTXexample}
@@ -252,17 +286,21 @@ the value of \Lps{iter} added by 400.
\begin{LTXexample}
\psset{xWidth=5cm,yWidth=5cm}
-\psfractal[dIter=30](-2,-2)(2,2)
+\psfractal[dIter=30](-2,-2)(2,2) \qquad
\psfractal[dIter=10,cx=-1.3,cy=0](-2,-2)(2,2)
\end{LTXexample}
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\subsection{\texttt{maxIter}}
\Lkeyword{maxIter} is the number of the maximum iteration until it leaves the loop.
It is predefined by 255, but internally multiplied by \Lkeyword{dIter}.
\begin{LTXexample}
\psset{xWidth=5cm,yWidth=5cm}
-\psfractal[maxIter=50,dIter=3](-2,-2)(2,2)
+\psfractal[maxIter=50,dIter=3](-2,-2)(2,2) \qquad
\psfractal[maxIter=30,cx=-1.3,cy=0](-2,-2)(2,2)
\end{LTXexample}
@@ -274,17 +312,20 @@ value, it is preset by 100.
\begin{LTXexample}
\psset{xWidth=5cm,yWidth=5cm}
-\psfractal[maxRadius=30,dIter=10](-2,-2)(2,2)
+\psfractal[maxRadius=30,dIter=10](-2,-2)(2,2) \qquad
\psfractal[maxRadius=30,dIter=30,cx=-1.3,cy=0](-2,-2)(2,2)
\end{LTXexample}
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\subsection{\texttt{plotpoints}}\xLkeyword{plotpoints}
This option is only valid for the Sierpinski triangle and preset by 2000.
\begin{LTXexample}
\begin{pspicture}(5,5)
\psSier(0,0)(2.5,5)(5,0)
-\end{pspicture}
+\end{pspicture} \quad
\begin{pspicture}(5,5)
\psSier[plotpoints=10000](0,0)(2.5,5)(5,0)
\end{pspicture}
@@ -332,6 +373,10 @@ is assumed.
\end{pspicture}}
\end{LTXexample}
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\subsection{\texttt{c}}\xLkeyword{c}
This is the length of one element in the unit pt.
@@ -349,6 +394,9 @@ This is the length of one element in the unit pt.
\end{pspicture}}
\end{LTXexample}
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\subsection{\texttt{maxIter}}\xLkeyword{maxIter}
This is the number for the iterations.
@@ -392,6 +440,9 @@ is assumed. The default \Lkeyword{scale} is set to 10.
\end{LTXexample}
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\section{Koch flake}
\begin{BDef}
@@ -423,6 +474,8 @@ Optional arguments are \Lkeyword{scale}, \Lkeyword{maxIter} (iteration depth) an
for the first rotation angle.
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\section{Apollonius circles}
\begin{BDef}
@@ -461,12 +514,12 @@ and the depth by \Lkeyword{maxIter}. Valid optional arguments are
\medskip
\begin{center}
-\begin{tabular}{@{}>{\ttfamily}lll@{}}
+\begin{tabular}{@{}>{\ttfamily}lll@{}}\hline
\emph{Name} & \emph{Meaning} & \emph{default}\\\hline
\Lkeyword{xWidth} & first base width & 1cm\\
\Lkeyword{minWidth} & last base width & 1pt\\
\Lkeyword{c} & factor for unbalanced trees (0<c<1) & 0.5\\
-\Lkeyword{Color} & colored tree & fasle
+\Lkeyword{Color} & colored tree & false\\\hline
\end{tabular}
\end{center}
@@ -539,14 +592,809 @@ and the depth by \Lkeyword{maxIter}. Valid optional arguments are
\end{pspicture*}
\end{LTXexample}
+
+\section{Fibonacci fractals}
+There are seven different commands which are all defined by Manuel Luque (for more informations see
+\url{http://pstricks.blogspot.de}):
+
+\begin{BDef}
+\Lcs{psFibonacciWord}\OptArgs\Largr{\CAny}\\
+\Lcs{psFibonacci}\OptArgs\\
+\Lcs{psNewFibonacci}\OptArgs\\
+\Lcs{psiFibonacci}\OptArgs\\
+\Lcs{pskFibonacci}\OptArgs\Largr{\CAny}\\
+\Lcs{psBiperiodicFibonacci}\OptArgs\Largr{\CAny}\\
+\Lcs{psFibonacciPolyominoes}\OptArgs\Largr{\CAny}
+\end{BDef}
+
+
+\begin{itemize}
+ \item \Lcs{psFibonacciWord} A Fibonacci word after n iterations
+ \item \Lcs{psFibonacci} Draw the fractal curve of a Fibonacci word
+ \item \Lcs{psNewFibonacci} Draw a bunch of curves obtained from the ``Dense Fibonacci Word''
+ (DFW) by substitutions.
+ \item \Lcs{psiFibonacci} In the article \cite{ramirez} a new family of curves in a row is called
+ ``\texttt{i-Fibonacci Word Fractal}''.
+ \item \Lcs{pskFibonacci} study the
+ following k-Fibonacci and the curves associated with words in the article ``\textit{On the
+ k-Fibonacci
+ words}\footnote{\ url{http://www.acta.sapientia.ro/acta-info/C5-2/info52-4.pdf}}'', this
+ command allows to represent these curves.
+ \item \Lcs{psBiperiodicFibonacci} it is still José L. Ramírez and Gustavo N. Rubiano who in the
+ article ``\textit{Biperiodic Fibonacci Word and Its Fractal Curve}''%
+ \footnote{\url{https://www.researchgate.net/publication/276406650_Biperiodic_Fibonacci_word_and_its_fractal_curve}}
+ extend the notion of Fibonacci sequence with 2 parameters (a, b). This command draws the
+ associated fractal curves.
+ \item \Lcs{psFibonacciPolyominoes} this command draws a Fibonacci tile, also called a Fibonacci
+ flake and allows you to pave the plane in two ways, following the rules established by A.
+ Blondin-Massé, S. Labbé, S. Brlek and M. Mendès-France in their article ``\textit{Fibonacci
+ snowflakes}\footnote{\url{www.slabbe.org/Publications/2011-fibo-snowflakes.pdf}}''.
+\end{itemize}
+
+The valid optional arguments with its default values:
+
+
+\begin{enumerate}
+ \item \texttt{[n=10]} : number of iterations;
+ \item \texttt{[k=5]} : k-Fibonacci series;
+ \item \texttt{[a=5,b=5]} : Biperiodic-Fibonacci series;
+ \item \texttt{[angle=90]} : turn right (-) or left (+) an angle of this value (see examples in
+ the article of José L. Ramírez et Gustavo N. Rubiano.
+ \item \texttt{[i=6]} : sets the follow-up nature of generalized Fibonacci;
+ \item \texttt{morphism=(0) (1) (2)} : for the command \Lcs{psNewFibonacci}, we will write
+ in the 3 pairs of parentheses the substitutions to be performed (see section~\ref{dfw})).
+ \item \texttt{[PSfont=Times-Roman]} : PostScript font;
+ \item \texttt{[fontscale=8} : fontscale;
+ \item \texttt{[colorF]} : curve color \verb+n-1+ for construction by juxtaposition;
+ \item \texttt{[juxtaposition=false]} allows the juxtaposition of the \texttt{n} and
+ \texttt{n-1} curves to bring up the \texttt{n+1} curve by simply writing
+ \texttt{[juxtaposition]} in the options.
+ \item \texttt{[DFW=false]} to display the ``\textit{Dense Fibonacci Word}'' (DFW) with
+ \verb+\psFibonacciWord[DFW]+~;
+ \item \texttt{[iFibonacci=false]} to display the word ``\textit{i-Fibonacci}'' with
+ \verb+\psiFibonacciWord[iFibonacci]+, obtained with the \texttt{[i]} parameter after
+ \texttt{[n]} iterations.
+\end{enumerate}
+
+The color and the thickness of the line of the fractal curve \texttt{n} are fixed with the usual
+parameters of PSTricks: \texttt{linewidth} and \texttt{linecolor}. The starting point of the curve
+is in $(0,0)$ and the unit is set by the PSTricks \texttt{unit =} option.
+
+This package does not pretend to exhaust the subject on the continuation of Fibonacci, the word of
+Fibonacci and the various fractals which are inspired by it. The subject is very vast and the
+studies very numerous. For those who discover the subject here are some tracks.
+
+The number 478 of the August 2017 issue of \textit{Pour la Science} contains an article by
+Jean-Paul Delahaye ``\textit {The following of Fibonacci \ldots\ and its consequences}'' whose
+title sums up the content of the article with, as usual, detailed explanations and beautiful
+illustrations.
+
+Concerning all the variations on the fractal curve of the Fibonacci word, Alexis Monnerot-Dumaine's
+article entitled ``\textit{The Fibonacci Word fractal}'' is the
+reference\footnote{\url{https://hal.archives-ouvertes.fr/hal-00367972}}.
+
+The site \url{https://fr.wikipedia.org/wiki/Fractale_du_mot_de_Fibonacci} is also very rich in
+information.
+
+The command \verb+\pSTilingsFibonacci+ allows the tiling of the
+plane with the n order Fibonacci tile.
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-1,-1)(1,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=5]
+\rput(0,-0.45){$F_{5}=$}
+\psFibonacciWord[n=5](0.5,-0.5)
+\end{pspicture}
+\begin{pspicture}[showgrid=false](-1,-1)(2,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=6]
+\rput(0,-0.45){$F_{6}=$}
+\psFibonacciWord[n=6](0.5,-0.5)
+\end{pspicture}
+\begin{pspicture}[showgrid=false](-1,-1)(2,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=7]
+\rput(-0.5,-0.45){$F_{7}=$}
+\psFibonacciWord[n=7](0,-0.5)
+\end{pspicture}
+\begin{pspicture}[showgrid=false](-1,-1)(2,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=8]
+\rput(-0.4,-0.45){$F_{8}=$}
+\psFibonacciWord[n=8](0,-0.5)
+\end{pspicture}
+\begin{pspicture}[showgrid=false](-1,-1)(2,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=9]
+\psFibonacciWord[n=9](-0.5,-0.75)
+\rput(-1,-0.75){$F_{9}=$}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-1,-1)(6,3)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},linewidth=0.05cm,n=10]
+\psFibonacciWord[n=10](-0.5,-0.5)
+\rput(-1,-0.45){$F_{10}=$}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(12,8)
+\psFibonacci[unit=0.02,linecolor={[rgb]{0.5 0 0}},n=23,linewidth=0.015cm]
+\rput(5.5,4){n=23}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\subsection{Fractal curves with juxtaposition}
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(7,7)
+\psFibonacci[unit=0.1,linecolor=blue,n=15,linewidth=0.04cm,juxtaposition]
+\rput(4,2.25){$F_{15}$}
+\rput(5.25,3){$F_{14}$}
+\psline[arrowinset=0.1,arrowsize=0.2]{->}(4.8,2)(3,2)
+\psline[arrowinset=0.1,arrowsize=0.2]{->}(5,2.5)(5,4)
+\end{pspicture}
+\hfill
+\begin{pspicture}[showgrid](0,0)(7,7)
+\psFibonacci[unit=0.1,linecolor=blue,n=16,linewidth=0.04cm]
+\rput(5,2){$F_{16}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(8,5)
+\psFibonacci[unit=0.05,linecolor=blue,n=17,linewidth=0.02cm,juxtaposition]
+\rput(2.5,4){\blue$F_{17}$}
+\rput(6,4){\red$F_{16}$}
+\end{pspicture}
+\hfill
+\begin{pspicture}[showgrid](0,0)(8,5)
+\psFibonacci[unit=0.05,n=18,linewidth=0.02cm]
+\rput(4,4){$F_{18}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(10,7)
+\psFibonacci[unit=0.1,linecolor=blue,n=16,linewidth=0.03cm,juxtaposition]
+\rput(5,2){\blue$F_{16}$}
+\rput(9,2){\red$F_{15}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(10,7)
+\psFibonacci[unit=0.1,n=17,linewidth=0.03cm]
+\rput(5,2){$F_{17}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(13,12)
+\psFibonacci[unit=0.03,linecolor=blue,n=21,linewidth=0.02cm,juxtaposition]
+\rput(1.5,3.5){\blue$F_{21}$}
+\rput(8.5,10.5){\red$F_{20}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(13,12)
+\psFibonacci[unit=0.03,n=22,linewidth=0.025cm]
+\rput(9,4){$F_{22}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(14,6)
+\psFibonacci[unit=0.015,linecolor=blue,n=23,linewidth=0.01cm,juxtaposition]
+\rput(4,2){\blue$F_{23}$}
+\rput(10.5,4){\red$F_{22}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(14,6)
+\psFibonacci[unit=0.015,n=24,linewidth=0.01cm]
+\rput(7,6.5){$F_{24}$}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\subsection{Curves with a big number of iterations}
+With \texttt{n=30} it takes a long time and the number is not readable.
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(14,6)
+\psFibonacci[unit=0.0025,n=30,linewidth=0.001cm]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\subsection{Double color curves}
+
+Superposition of two curves are possible by choosing diffrent color and line thickness.
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-1,-1)(15,12)
+\psgrid[style=gridstyleA]
+\psFibonacci[unit=0.4,linecolor=blue,n=13,linewidth=0.2cm]%
+\psFibonacci[unit=0.4,linecolor=red,n=13,linewidth=0.05cm]%
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-1,-1)(14,6)
+\psFibonacci[unit=0.5,linecolor=red,n=12,linewidth=0.2cm]%
+\psFibonacci[unit=0.5,linecolor=yellow,n=12,linewidth=0.05cm]%
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\section{``Dense Fibonacci Word'' and the command \Lcs{psNewFibonacci}}\label{dfw}
+
+In the chapter ``The dense Fibonacci word: a whole family of curves'', Alexis Monnerot-Dumaine
+wrote:
+
+\begin{quote}
+The odd-even design rule is not easy to manage and we can change to
+a more practical rule. As Jean-Paul Allouche suggested, we can create a word of 3
+letters with \{0; 1; 2\} that can draw the Fibonacci fractal with the simplest drawing rules
+following:
+
+\begin{itemize}
+\item 0, draw a segment in line with the previous one
+\item 1, draw a segment by turning to the right
+\item 2, draw a segment by turning to the left
+\end{itemize}
+\end{quote}
+
+
+By replacing in the Fibonacci word 00$\rightarrow$0, 01$\rightarrow$1 and 10$\rightarrow$2. Alexis Monnerot-Dumaine
+defines the ``Dense Fibonacci Word'' (DFW).
+From the DFW, we get a whole family of curves by doing, for example, substitutions
+following:
+
+\begin{itemize}
+ \item $\mu_1$ : $1 \longrightarrow 10$ ; $0 \longrightarrow 12$ ; $2 \longrightarrow 02$
+ \item $\mu_2$ : $1 \longrightarrow 010$ ; $0 \longrightarrow 0102$ ; $2 \longrightarrow 002$
+ \item $\mu_3$ : $1 \longrightarrow 02$ ; $0 \longrightarrow 21$ ; $2 \longrightarrow 10$
+ \item $\mu_4$ : $1 \longrightarrow 02$ ; $0 \longrightarrow 00$ ; $2 \longrightarrow 10$
+\end{itemize}
+
+We will find all these families of curves with explanations and references in the article
+Alexis Monnerot-Dumaine. These are just brief explanations for using the commands
+PSTricks to draw these families of curves. In their article "Properties and Gener-
+Fractal Exploring Fractal Curves ``alizations of the Fibonacci'' \cite{ramirez} illustrate this
+family of curves with Mathematica by designating them under the name of
+New-Fibonacci. This name seems to me sensible the PSTricks command will be called \Lcs{psNewFibonacci}.
+
+
+\subsection{``Dense Fibonacci Word''}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-1,-0.2)(10,2)
+\uput[r](-0.5,1){FW=}%
+\psFibonacciWord[n=10,fontscale=12](0.5,0.9)
+\uput[r](-0.75,0){DFW=}%
+\psFibonacciWord[n=10,DFW,fontscale=12](0.5,-0.1)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\subsection{Fractal of ``\textit{Dense Fibonacci Word}''}
+
+The curve can be created with \verb+\psNewFibonacci+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-3,0)(10,11)
+\psNewFibonacci[unit=0.2,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm]
+\rput(4,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-4,0)(6,10)
+\psNewFibonacci[unit=0.2,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm,morphism=() (1) (2)]
+\rput(2,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-1,0)(15,6)
+\psNewFibonacci[unit=0.2,linecolor={[rgb]{0.5 0 0}},n=18,linewidth=0.03cm,morphism=(12) (1) (2)]
+\rput(4,2){n=18}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-7,0)(5,10)
+\psNewFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm,morphism=(102) (2) (1)]
+\rput(-2,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-7,0)(4,10)
+\psNewFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm,morphism=(210) (02) (10)]
+\rput(-2,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-5,0)(4,9)
+\psNewFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm,morphism=(21) (02) (10)]
+\rput(-1,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-10,0)(4,12)
+\psNewFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=17,linewidth=0.03cm,morphism=(210) (020) (10)]
+\rput(-4,4){n=17}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-10,0)(4,11)
+\psNewFibonacci[unit=0.075,linecolor={[rgb]{0.5 0 0}},n=18,linewidth=0.025cm,morphism=(102) (2) (1)]
+\rput(-4,2){n=18}
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\section{The command \Lcs{psiFibonacci}}
+
+\subsection{How it works}
+Briefly, (read the article \cite{ramirez} for more details) more
+``I-Fibonacci Word'' depends on the parameter i and the number of iterations n with the following rules,
+according to the authors' notations:
+
+\begin{itemize}
+ \item $f_0^{[i]}$ =0
+ \item $f_1^{[i]} =0^{i-1}1$ : this notation means that it is necessary to put (i-1) 0 before the 1
+ \item $f_n^{[i]}= f_{n -1}^{[i]} f_{n -2}^{[i]}$ pour $n\geq 2$ et $i \geq 1$.
+\end{itemize}
+
+The construction of the associated fractal curves follows the ``even-odd'' rule as for the fractal
+of the word Fibonacci.
+
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+\subsection{Examples}
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-7,0)(3,9)
+\psiFibonacci[unit=0.2,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=3]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(9,9.5)
+\psiFibonacci[unit=0.2,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=4]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-7,0)(3,9)
+\psiFibonacci[unit=0.15,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=5]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(9,9.5)
+\psiFibonacci[unit=0.15,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=6]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-6,0)(3,8)
+\psiFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=7]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-8,0)(3,12)
+\psiFibonacci[unit=0.1,linecolor={[rgb]{0.5 0 0}},n=10,linewidth=0.025cm,i=11]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](0,0)(6,14)
+\psiFibonacci[unit=0.05,linecolor={[rgb]{0.5 0 0}},n=12,linewidth=0.025cm,i=12]
+\end{pspicture}
+\end{LTXexample}
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{The command \Lcs{pskFibonacci}}
+The 2 mandatory parameters are n and k. The following coordinates are optional but
+put, possibly, to center the curve at the origin of the mark.
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\pskFibonacci[unit=0.02,linecolor={[rgb]{0 0 0.5}},linewidth=0.02cm,n=6,k=5](-2.3,-3.2)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\pskFibonacci[unit=0.4,linecolor={[rgb]{0 0 0.5}},n=3,k=5](2.8,-3.6)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\pskFibonacci[unit=0.025,linecolor={[rgb]{0 0 0.5}},linewidth=0.02cm,n=6,k=6](3,0.5)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor={[rgb]{0 0 0.5}},griddots=10]
+\pskFibonacci[unit=0.2,linecolor={[rgb]{0 0 0.5}},n=4,k=6,angle=60](-2,0)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-4,-4)(2,1)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\pskFibonacci[unit=0.1,linecolor={[rgb]{0 0 0.5}},n=4,k=4,angle=60](0,0)
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-8,-8)(8,8)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor={[rgb]{0 0 0.5}},griddots=10]
+\pskFibonacci[unit=0.02,linecolor={[rgb]{0.5 0 0}},linewidth=0.02cm,n=6,k=7](6,-4)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{The command \Lcs{psBiperiodicFibonacci}}
+The 3 mandatory parameters are n, a and b. As for the previous command, the coordinates
+following are optional but allow, eventually, to center the curve at the origin of the
+mark.
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\psBiperiodicFibonacci[unit=0.2,linecolor={[rgb]{0 0.5 0}},linewidth=0.1cm,n=5,a=6,b=6,angle=60](0,2.1)
+\psBiperiodicFibonacci[unit=0.2,linecolor=white,n=5,a=6,b=6,angle=60](0,2.1)
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\psBiperiodicFibonacci[unit=0.5,linecolor={[rgb]{0 0.5 0}},n=5,a=3,b=4,angle=120](-1.5,3.5)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-4,-4)(4,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\psBiperiodicFibonacci[unit=0.2,linecolor=black,linewidth=0.1cm,n=7,a=2,b=6,angle=72](2.62,2)
+\psBiperiodicFibonacci[unit=0.2,linecolor=yellow,n=7,a=2,b=6,angle=72](2.62,2)
+\end{pspicture}
+\end{LTXexample}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid=false](-5,-4)(5,4)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\psBiperiodicFibonacci[unit=0.1,linecolor=red,linewidth=0.1cm,n=10,a=2,b=5](3.5,-1.5)
+\psBiperiodicFibonacci[unit=0.1,linecolor=yellow,n=10,a=2,b=5](3.5,-1.5)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-5,-5)(5,5)
+\psgrid[gridlabels=0pt,subgriddiv=0,gridcolor=red,griddots=10]
+\psBiperiodicFibonacci[unit=0.15,linecolor={[rgb]{0 0.5 0}},n=9,a=2,b=5](3.15,-1.35)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-5,-5)(5,5)
+\psBiperiodicFibonacci[unit=0.8,linecolor=black,linewidth=0.1cm,,n=8,a=2,b=3,angle=120](-1,1)
+\psBiperiodicFibonacci[unit=0.8,linecolor=white,n=8,a=2,b=3,angle=120](-1,1)
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\section{The command \Lcs{psFibonacciPolyominoes}}
+
+
+The only mandatory parameter is the order of the tile: n. Coordinates are optional, but
+they will be used for paving the plan.
+
+
+
+\subsection{The order 0, 1, 2, 3, and 4}
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\psset{linecolor={[rgb]{0 0.5 0}},fillstyle=solid,fillcolor=red}
+\begin{pspicture}[showgrid](0,-1)(1,4)
+\psFibonacciPolyominoes[n=0,unit=0.5]
+\end{pspicture}
+\quad
+\begin{pspicture}[showgrid](-2,-1)(1,4)
+\psFibonacciPolyominoes[n=1,unit=0.5]
+\end{pspicture}
+\quad
+\begin{pspicture}[showgrid](-3,-1)(2,4)
+\psFibonacciPolyominoes[unit=0.5,n=2]
+\end{pspicture}
+\quad
+\begin{pspicture}[showgrid](-3,-1)(2,4)
+\psFibonacciPolyominoes[unit=0.2cm,n=3]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}[showgrid](-7,-2)(4,10)
+\psFibonacciPolyominoes[unit=0.2,n=4,fillcolor=cyan,linecolor=blue,fillstyle=solid]
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\iffalse
+
+
+\subsection{Paving the plan with Fibonacci tiles}
+This tiling is done with the \Lcs{pSTilingsFibonacci}\Largb{n}\Largb{N}\Largb{T} command which has 3 arguments: the
+first is the order of the tile (n = 0, 1, 2 etc.), the second the number of tiles per side of the square,
+preferably it must be odd and T = 1 or 2 defines the type of paving. For tiles to be
+colored, activate the \texttt{fillstyle=solid} option in a pre-command.
+
+
+\begin{verbatim}
+\def\pSTilingsFibonacci#1#2#3{% #1=n #2=nbre de lignes #3=type 1 ou 2
+ \pstFPDiv\Nbr{#2}{2}%
+ \psset[pst-fractal]{n=#1}%
+ \pstVerb{/n0 #1 def /tabPell [0 1 2 5 12 29 70 169 408] def
+ /Type #3 def
+ Type 2 gt {/Type 1 def} if
+ Type 2 eq {
+ % (-1)^2P(n),(P(n+1)
+ /Y1 tabPell n0 1 add get def
+ /X1 -1 n0 exp tabPell n0 get mul def
+ }{
+ % (P(n+1),(-1)^2P(n)
+ /X1 tabPell n0 1 add get def
+ /Y1 -1 n0 exp tabPell n0 get mul def
+ } ifelse
+ % le déplacement perpendiculaire
+ /X2 Y1 neg def
+ /Y2 X1 def
+}%
+\pstVerb{/ListColors [0 0 1] def}%
+\multido{\I=-\Nbr+1}{#2}{5
+ \rput(!X2 \I\space mul Y2 \I\space mul){%
+ \multido{\i=-\Nbr+1}{#2}{%
+%\definecolor[ps]{Couleur}{rgb}{ListColors aload pop}%
+ \definecolor[ps]{Couleur}{cmyk}{ListColors aload pop 0}%
+ \rput(!X1 \i\space mul Y1 \i\space mul){\psFibonacciPolyominoes[fillcolor=Couleur]}
+ \pstVerb{/ListColors [ListColors aload pop 3 1 roll] def}%
+}}}}%
+\end{verbatim}
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture*}(-9,-7)(8,9)
+\psset{unit=0.125,linestyle=solid}
+\pSTilingsFibonacci{3}{13}{1}%
+\psset{fillstyle=solid,linecolor=blue}%
+\pSTilingsFibonacci{3}{7}{1}%
+\end{pspicture*}
+\end{LTXexample}
+
+
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture*}(-9,-7)(8,9)
+\psset{unit=0.125,linestyle=solid}
+\pSTilingsFibonacci{3}{13}{2}%
+\psset{fillstyle=solid,linecolor=blue}%
+\pSTilingsFibonacci{3}{7}{2}%
+\end{pspicture*}
+\end{LTXexample}
+
+
+\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\fi
+
+
+\section{The command \Lcs{psFibonacciWord}}
+
+\psset{unit=1cm}
+\begin{LTXexample}
+\begin{pspicture}(-1,0)(10,5)
+\rput(0.15,5){\small$F_{1}=1$}
+\rput(0.15,4.5){\small$F_{2}=0$}
+\multido{\i=3+1,\I=3+1,\n=4.0+-0.5}{8}{%
+ \psFibonacciWord[n=\i](0.5,\n)
+ \rput(0,\n){$F_{\I}=$}}
+\end{pspicture}
+\end{LTXexample}
+
+
+\begin{LTXexample}
+\begin{pspicture}(-1,0)(10,6)
+\psset{n=5}
+\multido{\i=1+1,\I=1+1,\n=3.5+-0.5}{6}{%
+\psFibonacciWord[i=\i,iFibonacci](0.5,\n\space 0.1 sub)
+\rput(0,\n){$F_{5}^{[\i]}=$}
+}
+\end{pspicture}
+\end{LTXexample}
+
+
\section{List of all optional arguments for \texttt{pst-fractal}}
+
\xkvview{family=pst-fractal,columns={key,type,default}}
-\bgroup
\nocite{*}
\printbibliography
-\egroup
\printindex