Environmental Engineering Reference
In-Depth Information
The potential of such an L-system can easily be explored by playing around with
parameters: If, for example, the initial delay in the branches is reduced from 6 to 2,
branches will emerge earlier and a more compact form of the structure results
(Fig.
11.4b
).
11.5 Fractals
Patterns like the ferns in Fig.
11.4
show a self-similar structure, i.e. affine transforma-
tions exist which map the whole pattern to some parts of it. Such
self-similarity
occurs frequently in nature, e.g., in crowns of older trees in the form of “reiterations”.
Self-similarity is an indication that there are some structural rules governing the
pattern which can be used to specify it in a very condensed form, like our ferns were
described by the three L-system rules above. Alternative methods to generate self-
similar patterns also exist, e.g. the direct specification of the structure-preserving
affine transformations by matrices - an approach known as “Iterative Function
Systems” (IFS; see Barnsley 1988).
Self-similar structures can be characterized as
fractals
, which means, as geomet-
rical objects which have a “broken” (or non-integer) dimension. “Dimension” in this
context does not refer to the usual algebraic definition of the dimension of a manifold
(as the number of coordinates which is necessary to fix positions in it), but to the
degree to which the object fills the space. For instance, the ferns in Fig.
11.4
are more
space-filling than a straight line (dimension 1) but less than a plane-filling object like
a filled triangle (dimension 2). Hence, their fractal dimension is a broken number
between 1 and 2. Exact definitions of fractal dimension are given in measurement
theory, a branch of mathematical topology (see Edgar 1990). Some well-known
Fig. 11.4 (a) Fern leaf produced by a parametric L-system (see text), (b) variant with reduced
delay parameter for branch emergence (from Kurth 2007)
