3.5 L-Systems - Lindenmayer-Systems

Lindenmayer-Systems, are like the IFS very close to naturally looking objects. The biologist Aristid Lindenmayer developed this variant to describe plant-forms. Similarly to the transformation rules of IFS, which makes “n” new elements out of one by a certain rule, the Lindenmayer-System in its first step uses symbols as a set of rules. To illustrate this stage, a set of transformations may for example be given by: "l" is replaced by "l+l--l+l".

That means starting with "l", the first iteration for this example runs as follows "l+l--l+l", the second one "(l+l--l+l) + (l+l--l+l) - - (l+l--l+l) + (l+l--l+l)" and so on. But this does not lead to any picture - e.g. plant-form. Therefore a second step is required, which translates the symbols into drawing rules. "l" for example symbolizes a piece of a straight line forward, "-" an angle of e.g. 60 degrees to the right and "+" an angle of 60 degrees to the left. By that it is possible to generate self-similar mathematical fractals but also plants, bushes and trees - see picture 19.

picture 19: Lindenmayer-Systems; the Koch curve:

rule = “l + l - - l + l” with
l = line 
+ = +60 degrees
- = -60 degrees
pictures after 1 and 4 iterations

picture 19: L-Systems; a bush:

rule = “l [ + l ] l [ - l] l” [Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.62.] with
l = line 
+ = +28.58 degrees
- = -28.58 degrees
[ = start of a branch
] = end of a branch
pictures after 1,2 and 4 iterations

picture 19: some more pictures produced with L-Systems

An example for a bush is given in "Scientific American"[01] with "l" being turned into the expression "l [ + l ] l [ - l] l". There "l" once more symbolizes a piece of a straight line forward, "-" a certain angle to the right and "+" one to the left. The new symbol of "[" defines the start and "]" the end of a new branch - see picture 19. The Lindenmayer-System reduces the high information of plants through short instructions, which leads to one possibility of application, the usage in computer-generated film sequences, where computer power has to be reduced.

Footnotes

[01] Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.62.