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Diffstat (limited to 'Master/texmf-dist/doc/generic/pst-fractal')
| -rw-r--r-- | Master/texmf-dist/doc/generic/pst-fractal/Changes | 4 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/pst-fractal/README | 2 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.bib | 7 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.pdf | bin | 7201737 -> 11285757 bytes | |||
| -rw-r--r-- | Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.tex | 894 |
5 files changed, 882 insertions, 25 deletions
diff --git a/Master/texmf-dist/doc/generic/pst-fractal/Changes b/Master/texmf-dist/doc/generic/pst-fractal/Changes index e297cdd6b27..9452a519d83 100644 --- a/Master/texmf-dist/doc/generic/pst-fractal/Changes +++ b/Master/texmf-dist/doc/generic/pst-fractal/Changes @@ -1,4 +1,5 @@ pst-fractal.pro -------- +0.05 2017-12-02 added fibinacci curves 0.04 2017-11-26 add sierpinski curve 0.03 2011-02-10 cometic changes 0.02 2010-02-10 fix bug in tx@fern @@ -6,7 +7,8 @@ pst-fractal.pro -------- pst-fractal.tex -------- -0.08 2017-11-26 - add sierpinski curve +0.09 2017-12-02 - added fibinacci curves +0.08 2017-11-26 - added sierpinski curve 0.07 2013-06-17 - allow level 0 for sierpinski 0.06 2010-02-10 - add another type of a Sierpinski triangle 0.05 2007-09-25 - allow unbalanced trees with option 0<c<1 diff --git a/Master/texmf-dist/doc/generic/pst-fractal/README b/Master/texmf-dist/doc/generic/pst-fractal/README index fe1f4a9f4f6..8685dbfa7ba 100644 --- a/Master/texmf-dist/doc/generic/pst-fractal/README +++ b/Master/texmf-dist/doc/generic/pst-fractal/README @@ -11,7 +11,7 @@ TeX Frequently Asked Questions: PSTricks is PostScript Tricks, the documentation cannot be run with pdftex, use the sequence latex->dvips->ps2pdf. -To gett he documentation of this document you have to run +To get the documentation of this document you have to run pst2pdf pst-fractal-doc --Iext=.jpg %% This program can be redistributed and/or modified under the terms diff --git a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.bib b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.bib index 038d2b36a0a..a1400940514 100644 --- a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.bib +++ b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.bib @@ -123,3 +123,10 @@ year = {2012}, url = {/graphics/pstricks/contrib/pst-tools} } + +@online{ramirez, +title= {Properties and Generalizations of the Fibonacci Word Fractal Exploring Fractal Curves}, +url = {http://www.mathematica-journal.com/2014/02/properties-and-generalizations-of-the-fibonacci-word-fractal/}, +author= {José L. Ramírez and Gustavo N. Rubiano}, +} +
\ No newline at end of file diff --git a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.pdf b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.pdf Binary files differindex c9909824bbb..df9353720e4 100644 --- a/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.pdf +++ b/Master/texmf-dist/doc/generic/pst-fractal/pst-fractal-doc.pdf 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 |