More on Lindenmayer Systems

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 , 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.

Apollonian gaskets and circle inversion fractals

Recall the definition of an iterated function system (IFS), and how on occasion, their attractors are fractal sets. What happens when we allow more general functions instead of mere affine maps?

The key to the design of this amazing fractal is in the notion of limit sets of circle inversions.

Toying with basic fractals

I will skip all the mathematical theory behind Fractals (dimensions, measures, etc) to focus directly into the description and implementation of some of the most basic examples. In this post, I will cover the ideas behind three classical techniques:

• Iterated function systems
• Membership tests
• Lindenmayer systems