# Example: Lindenmayer systems

Published 2011-12-18 | Author: Stefan Kottwitz
From Wikipedia: An L-system or Lindenmayer system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms. A L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules which expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems can also be used to generate self-similar fractals such as iterated function systems. L-systems were introduced and developed in 1968 by the Hungarian theoretical biologist and botanist from the University of Utrecht, Aristid Lindenmayer (1925–1989).

TikZ provides the lindenmayersystems library for drawing such fractals. The example uses differently shaded drawing for a nicer visual effect.

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% Lindenmayer systems
% Dec 18, 2011, Stefan Kottwitz
% http://texblog.net
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{lindenmayersystems}
\begin{document}
\pgfdeclarelindenmayersystem{Koch curve}{
\rule{F -> F-F++F-F}}
\pgfdeclarelindenmayersystem{Sierpinski triangle}{
\rule{F -> G-F-G}
\rule{G -> F+G+F}}
\pgfdeclarelindenmayersystem{Fractal plant}{
\rule{X -> F-[[X]+X]+F[+FX]-X}
\rule{F -> FF}}
\pgfdeclarelindenmayersystem{Hilbert curve}{
\rule{L -> +RF-LFL-FR+}
\rule{R -> -LF+RFR+FL-}}

\begin{tabular}{cc}
\begin{tikzpicture}
[l-system={Koch curve, step=2pt, angle=60, axiom=F++F++F, order=4}]
lindenmayer system -- cycle;
\end{tikzpicture}
&
\begin{tikzpicture}
\shadedraw [top color=white, bottom color=blue!80, draw=blue!80!black]
[l-system={Sierpinski triangle, step=2pt, angle=60, axiom=F, order=8}]
lindenmayer system -- cycle;
\end{tikzpicture}
\\
\begin{tikzpicture}
\shadedraw [bottom color=white, top color=red!80, draw=red!80!black]
[l-system={Hilbert curve, axiom=L, order=5, step=8pt, angle=90}]
lindenmayer system;
\end{tikzpicture}
&
\begin{tikzpicture}
\draw [green!50!black, rotate=90]
[l-system={Fractal plant, axiom=X, order=6, step=2pt, angle=25}]
lindenmayer system;
\end{tikzpicture}
\end{tabular}
\end{document}