[Tex/LaTex] Square fractal generation in tikz

fractalstikz-decorationstikz-pgf

I'm trying to recreate this exact diagram using tikz. So far I have been attempting to use nested decorations in tikz since I used these for other fractal constructions. However, these constructions were all either decorations pre-defined by tikz like the Koch curve or ones I found solutions to on stack exchange, like the Sierpinski triangle.

I've been looking a lot into defining my own decorations but it seems like quite a complicated process for a tikz newbie, and haven't found any examples too similar to what I'm trying to do. I know it will also be possible using lindemayer systems, but only understand how to use them for line constructions.

If it's any help, to my mind it seems the simplest way to do this would be to set the square as the initial shape with origin at the bottom left, then do scale by 1/4 for the bottom left square, scale by 1/4 then translate upwards for the top left square etc. and then set the new shape to replace the initial, ready for the next iteration.

$The fractal in question$

Any help would be much appreciated!

Best Answer

Here's a way with a Lindenmayer system. For orders above 5, compile with LuaLaTeX.

% \RequirePackage{luatex85} % Only for LuaLaTeX and standalone class
\documentclass[varwidth,border=5]{standalone}
\usepackage{tikz}
\usetikzlibrary{lindenmayersystems}
\pgfdeclarelindenmayersystem{square fractal}{%
  \symbol{S}{\pgflsystemstep=0.5\pgflsystemstep}
  \symbol{A}{\pgftransformshift%
    {\pgfqpoint{0.75\pgflsystemstep}{0.75\pgflsystemstep}}}
  \symbol{R}{\pgftransformrotate{90}}
  \symbol{Q}{%
    \pgfpathrectangle{\pgfqpoint{-0.5\pgflsystemstep}{-0.5\pgflsystemstep}}%
    {\pgfqpoint{\pgflsystemstep}{\pgflsystemstep}}%
  }
  \rule{Q -> [SQ[ASQ][RASQ][RRASQ][RRRASQ]]}
}
\begin{document}
\foreach\i in {0,...,5}{%
\tikz\fill [l-system={square fractal, step=5cm, axiom=Q, order=\i}] 
  lindenmayer system;
\ifodd\i\par\bigskip\leavevmode\fi
}
\end{document}

And here's a way with decorations:

\documentclass[varwidth,border=5]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations}
\pgfdeclaredecoration{square fractal}{start}{
\state{start}[width=0pt,next state=draw]{
  \pgfpathmoveto{\pgfpointdecoratedinputsegmentfirst}
}
\state{draw}[width=\pgfdecoratedinputsegmentlength]{
  \pgfpointdiff{\pgfpointdecoratedinputsegmentfirst}%
    {\pgfpointdecoratedinputsegmentlast}
  \pgfgetlastxy\tmpx\tmpy
  \pgfmathveclen\tmpx\tmpy
  \pgfmathparse{\pgfmathresult/4}%
  \let\tmp=\pgfmathresult
  \pgfpathlineto{\pgfpoint{\tmp}{+0pt}}
  \pgfpathlineto{\pgfpoint{\tmp}{-\tmp}}
  \pgfpathlineto{\pgfpoint{3*\tmp}{-\tmp}}
  \pgfpathlineto{\pgfpoint{3*\tmp}{+0pt}}
  \pgfpathlineto{\pgfpointdecoratedinputsegmentlast}
}
\state{final}{
  \pgfpathclose
}
}
\begin{document}
\tikz[decoration=square fractal]
  \fill (0,0) rectangle (4,4);
\tikz[decoration=square fractal]
  \fill decorate { (0,0) rectangle (4,4) };
\\
\tikz[decoration=square fractal]
  \fill decorate { decorate { (0,0) rectangle (4,4) } };
\tikz[decoration=square fractal]
  \fill decorate { decorate { decorate { (0,0) rectangle (4,4) } } };
\end{document}

Code

\documentclass[tikz,border=2mm]{standalone}
\usepackage{xifthen}

\newcommand{\mycolorbar}[3]% height,width,colors
{   \begin{tikzpicture}
        \foreach \x [count=\c] in {#3}{ \xdef\numcolo{\c}}
        \pgfmathsetmacro{\pieceheight}{#1/(\numcolo-1)}
        \xdef\lowcolo{}
        \foreach \x [count=\c] in {#3}
        {   \ifthenelse{\c = 1}
            {}
            {   \fill[bottom color=\lowcolo,top color=\x] (0,{(\c-2)*\pieceheight}) rectangle (#2,{(\c-1)*\pieceheight});
            }
            \xdef\lowcolo{\x}
        }
    \end{tikzpicture}
}

\begin{document}

\mycolorbar{10}{1}{red,orange,yellow,green,cyan,blue,violet,red}
\mycolorbar{10}{1}{black,blue,white,red,black,yellow,white,green,black}

\end{document}

Output

enter image description here

Edit 1:

Now with labels. You might have a look at the source of the pgfplots manual, in the section color maps you can find quite a few examples.

Code

\documentclass[tikz,border=2mm]{standalone}
\usepackage{xifthen}

\newcommand{\mycolorbar}[6]% height,width,colors,label min,label max,label step
{   \begin{tikzpicture}
        \foreach \x [count=\c] in {#3}{ \xdef\numcolo{\c}}
        \pgfmathsetmacro{\pieceheight}{#1/(\numcolo-1)}
        \xdef\lowcolo{}
        \foreach \x [count=\c] in {#3}
        {   \ifthenelse{\c = 1}
            {}
            {   \fill[bottom color=\lowcolo,top color=\x] (0,{(\c-2)*\pieceheight}) rectangle (#2,{(\c-1)*\pieceheight});
            }
            \xdef\lowcolo{\x}
        }
        \draw (0,0) rectangle (#2,#1);
        \pgfmathsetmacro{\secondlabel}{#4+#6}
        \pgfmathsetmacro{\lastlabel}{#5+0.01}
        \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
        \foreach \x in {#4,\secondlabel,...,\lastlabel}
        { \draw (#2,{(\x-#4)/(#5-#4)*#1}) -- ++ (0.05,0) node[right] {\pgfmathprintnumber{\x}};
        }
    \end{tikzpicture}
}

\begin{document}

\mycolorbar{10}{1}{red,orange,yellow,green,cyan,blue,violet,magenta,red}{-0.4}{0.4}{0.1}
\mycolorbar{10}{1}{black,blue,white,red,black,yellow,white,green,black}{-13}{11}{2}

\end{document}

Output

enter image description here

With Tikz Tree how can I make two nodes grow up while the rest grow down

Update: Here is a more general solution.

Here is a solution using forest, which is somewhat more flexible than qtree. You can add manual adjustments to the individual nodes by using before drawing tree. You can add or subtract to the x or y coordinates by using y±=<length>,x±=<length>, or set the coordinates with y=<coordinate>,x=<coordinate>.

I was unclear if you also wanted the d and e subtrees to be shifted so they are symmetric with respect to the c node.

\documentclass{article}

\usepackage{forest}

\begin{document}

\begin{forest}
for tree={draw,circle,minimum width=2em,anchor=center}
[c
    [a,before drawing tree={y=1cm,x=-1cm}]
    [b,before drawing tree={y=1cm,x=1cm}]
    [d
        [d1]
        [d2]]
    [e
        [e1]
        [e2]]]
\end{forest}

\end{document}

For a more flexible solution, in case the a and b nodes have nonempty subtrees, you can create two empty children for the c node and set their grow directions. But before computing the position of the empty children, use before computing xy={l=0,s=0} to place the empty nodes at the root. The syntax if level=1{<do this>}{<else do this>} can be placed in the main for tree.

\documentclass{article}

\usepackage{forest}

\begin{document}

\begin{forest}
for tree={minimum width=2.5em, anchor=center, circle,
    if level=1{no edge, before computing xy={l=0,s=0}}{draw}}
[c
[, for tree={grow'=north}
    [a[a1][a2]]
    [b[b1][b2]]]
[, 
    [d
        [d1]
        [d2]]
    [e
        [e1]
        [e2]]]]
\end{forest}

\end{document}

Best Answer

Related Solutions

[Tex/LaTex] Creating a line separated gradient bar in Tikz

Code

Output

Edit 1:

Code

Output

With Tikz Tree how can I make two nodes grow up while the rest grow down

Related Question