More on Lindenmayer Systems

September 24, 2013 Leave a comment Go to comments

We briefly explored Lindenmayer systems (or L-systems) in an old post: Toying with Basic Fractals. We quickly reviewed this method for creation of an approximation to fractals, and displayed an example (the Koch snowflake) based on tikz libraries.

I would like to show a few more examples of beautiful curves generated with this technique, together with their generating axiom, rules and parameters. Feel free to click on each of the images below to download a larger version.

Note that any coding language with plotting capabilities should be able to tackle this project. I used once again tikz for \text{\LaTeX}, but this time with the tikzlibrary lindenmayersystems.

name  : Dragon Curve
axiom : X
order : 11
step  : 5pt
angle : 90
rules :
        X -> X+YF+
        Y -> -FX-Y
name  : Gosper Space-filling Curve
axiom : XF
order : 5
step  : 2pt
angle : 60
rules :
	XF -> XF+YF++YF-XF--XFXF-YF+
	YF -> -XF+YFYF++YF+XF--XF-YF
name  : Quadric Koch Island
axiom : F+F+F+F
order : 4
step  : 1pt
angle : 90
rules :
        F -> F+F-F-FF+F+F-F
name  : Sierpinski Arrowhead
axiom : F
order : 8
step  : 3.5pt
angle : 60
rules :
        G -> F+G+F
        F -> G-F-G
name  : ?
axiom : F+F+F+F
order : 4
step  : 2pt
angle : 90
rules :
        F -> FF+F+F+F+F+F-F
name  : ?
axiom : F+F+F+F
order : 4
step  : 3pt
angle : 90
rules :
        F -> FF+F+F+F+FF

Would you like to experiment a little with axioms, rules and parameters, and obtain some new pleasant curves with this method? If the mathematical properties of the fractal that they approximate are interesting enough, I bet you could attach your name to them. Like the astronomer that finds through her telescope a new object in the sky, or the zoologist that discover a new species of spider in the forest.

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  1. green
    March 19, 2014 at 9:30 am

    Hi, what does F stand for? Would you mind providing the LaTeX code of these examples, please?

    • March 20, 2014 at 5:13 pm

      From the tikz manual: “F moves forward a certain distance, drawing a line.”

      Look for example at the code of the Dragon curve:

      %%%%%%%%% start of code %%%%%%%%%

      \pgfdeclarelindenmayersystem{Dragon curve}{
      \rule{X -> X+YF+}
      \rule{Y -> -FX-Y}}

      \begin{tikzpicture}[color=blue]
      \draw [l-system={Dragon curve, axiom=X, order=11, step=5pt, angle=90}] lindenmayer system;
      \end{tikzpicture}

      %%%%%%%%% end of code %%%%%%%%%

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