what are L-systems?
deterministic, stochastic, parametric

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Deterministic L systems - simple string rewriting
In 1968 Aristid Lindenmayer, a biologist who worked with yeast and filamentous fungi,  published a description of L-systems (Lindenmayer systems). These were string rewriting systems that could be used to describe the growth of a filamentous cyanobacterium (blue-green alga) called Anabaena. Rewriting is a technique for defining complex objects by successively replacing parts of a simple initial string using a set of rewriting rules or productions. (Prusinkiewicz & Lindenmayer, 1990). However, the application of L-systems took a large leap when the strings generated by the production rules were interpreted using turtle graphics by Smith (1984), so that a simple axiom and production rules could code for a relatively complex plant. This was further developed by Prusinkiewicz, Hanan and Lindenmayer in their extensive work with modelling plants and trees.

Turtle Graphics
The most common interpretation of L-systems is a graphical one using Turtle graphics, where pen movements are represented by symbols such as F (pen down, move forward one step), f (pen up, move forward one step), + (turn left by predefined angle), - (turn right by predefined angle).

Generations
At each generation, when the string is rewritten, the string gets longer and the resulting image (tree, microbe, fractal etc) is more complex.

L-system
/* Koch curve from figure 1.9f pg. 10 */
#define maxgen 4
START : F-F-F-F-
p1 : F -> F-F+F-F-F

Turtle graphics - generations 0, 2, 3
Stochastic and Parametric L-systems
A number of variations to L-systems have been introduced, including Stochastic L-systems, where irregularity is introduced by giving probabilities to the production rules. Another important improvement was the introduction of Parametric L-systems, which include the ability to pass parameters. Examples of these are presented below (no image). Other types of L-systems have also been described.

Stochastic L-system
/* * Fran Soddell * honours thesis fig 24 */#define maxgen 9
#define delta 10
START : Fb
p1: b -> (0.334)[+Fb][-Fb]
-> (0.333)[++Fb][--Fb]
-> (0.333)[+++Fb][---Fb]

 

gen 5

Parametric L-system
/* Koch curve looking like a row of trees Figure 1.37a pg. 48 (L-system on page 47) */
#define maxgen 6
#define width 0
#define delta 86
#define c 1
#define p 0.3
#define q c - p
#define h (p * q) ^ 0.5
START : F(1)
p1 : F(x) -> F(x*p)+F(x*h)--F(x*h)+F(x*q)

gen 4
Uses of L-system
L-systems have been extensively used to model plant growth and are used in computer graphics to generate complex images from simple rules eg fractal objects.

References

  • Prusinkiewicz P & Hanan (1989) Lindenmayer systems, plants and fractals. Lecture Notes in Biomathematics. Springer-Verlag: Berlin.
  • Prusinkiewicz P & Lindenmayer A (1990). The Algorithmic Beauty of Plants. Springer-Verlag: Berlin.
  • Smith A (1984) Plants, fractals and formal languages. Computer Graphics 18 (July) 1-10

Some L-systems Links