Environmental Engineering Reference
In-Depth Information
fractals from mathematics are the Mandelbrot set and Julia sets (Peitgen and Richter
1986). Many natural objects can also be characterized, at least approximately, by a
fractal dimension - this is true for some plants, venation patterns, coastlines, clouds,
galaxies, mineral surfaces (cf. Mandelbrot 1977), or movement paths of organisms.
However, fractality is more an analytic notion which is normally not directly suitable
to design procedures for synthetic generation of structures. Hence we will not explore
it deeper here.
11.6 Relational Growth Grammars
Despite their successful use for realistic-looking structural models of plants, L-systems
still have certain limitations (even if some extensions of the original concept are
included). First of all, in interpreted L-systems (with turtle geometry and with
brackets for branching), only two possible relations can exist between simulated
objects: A can be a
direct successor
of B or can be supported by B as a
branch
.
However, many more sorts of relations between objects are possible in reality and
can be worth modelling. Another problem with L-systems is that they are not an
appropriate tool for the creation of truly two-dimensional or even three-dimensional
arrangements, like tessellations in the plane (tilings with a collection of simple
shapes without leaving any space) or cellwork systems (e.g., in tissues). There
are extended formalisms like “map L-systems” and “cellwork L-systems” (see
Prusinkiewicz and Lindenmayer 1990) for this purpose, but their definitions and
usage are rather complicated. The core of this problem is that the classical interpre-
tation of bracketed strings by the turtle can only yield structures with an underlying
topology which is locally one-dimensional and has a structure similar to trees.
Hence,
cycles
and
networks
can be created only if additional tools are allowed.
Another weakness of L-systems is apparent when they are viewed from the per-
spective of software engineering: As a programming language they are poor in
comparison to modern languages; particularly, the
object-oriented programming
(OOP; see Chaps. 4 and 12) style, which is currently the standard approach amongst
professional programmers, is not supported. The fundamental units of parametric
L-systems are only symbols with some numbers attached, no objects in the sense of
OOP. Furthermore, no hierarchy of object classes, where specialized classes inherit
attributes and methods from more general classes, can be defined in classical L-
systems.
Finally, from a biologist's viewpoint, it is a drawback that
genotype
and
pheno-
type
of an organism cannot easily be captured in a model based only on classical L-
systems (although the DNA molecule has basically the structure of a string).
These were reasons to design a new rewriting formalism, “relational growth
grammars” (RGG). An RGG operates on
graphs
instead of strings. By a “graph” we
mean a mathematical structure consisting of nodes and directed arcs (also called
“edges”) connecting some of these nodes. “Trees” (in the abstract sense) are special
graphs, but in general, cyclic substructures, which do not occur in trees, are allowed
