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Chapter 8
Fractal properties of plants
What is a fractal? In his 1982 book, Mandelbrot defines it as a set with
Hausdorff-Besicovitch dimension
D
H
strictly exceeding the topological
dimension
D
T
[95, page 15]. In this sense, none of the figures presented
in this book are fractals, since they all consist of a finite number of
primitives (lines or polygons), and
D
H
=
D
T
. However, the situation
changes dramatically if the term “fractal” is used in a broader sense [95,
page 39]:
Strictly speaking, the triangle, the Star of David, and the
finite Koch teragons are of dimension 1. However, both
intuitively and from the pragmatic point of view of the sim-
plicity and naturalness of the corrective terms required, it is
reasonable to consider an advanced Koch teragon as being
closer to a curve of dimension log 4/log 3 than to a curve
of dimension 1.
Thus, a finite curve can be considered an approximate rendering
of an infinite fractal as long as the interesting properties of both are
closely related. In the case of plant models, this distinctive feature is
self-similarity.
The use of approximate figures to illustrate abstract concepts has a
long tradition in geometry. After all, even the primitives of Euclidean
geometry — a point and a line — cannot be
drawn
exactly. An in-
teresting question, however, concerns the relationship between fractals
and real biological structures. The latter consist of a finite number of
cells, thus are not fractals in the strict sense of the word. To consider
real plants as approximations of “perfect” fractal structures would be
acceptable only if we assumed Plato’s view of the supremacy of ideas
over their mundane realization. A viable approach is the opposite one,
to consider fractals as abstract descriptions of the real structures. At
first sight, this concept may seem strange. What can be gained by
reducing an irregular contour of a compound leaf to an even more ir-
regular fractal? Would it not be simpler to characterize the leaf using
Fractals vs.
finite curves
Fractals vs.
plants
Complexity of
fractals
176
Chapter 8. Fractal properties of plants
Previous
viewpoints
Fractals in
botany
a smooth curve? The key to the answer lies in the meaning of the term
“simple.” A smooth curve may seem intuitively simpler than a fractal,
but as a matter of fact, the reverse is often true [95, page 41]. Accord-
ing to Kolmogorov [80], the complexity of an object can be measured
by the length of the shortest algorithm that generates it. In this sense,
many fractals are particularly simple objects.
The above discussion of the relationship between fractals and plants
did not emerge in a vacuum. Mandelbrot [95] gives examples of the re-
cursive branching structures of trees and flowers, analyzes their
Hausdorff-Besicovitch dimension and writes inconclusively “trees may
be called fractals in part.” Smith [136] recognizes similarities between
algorithms yielding Koch curves and branching plant-like structures,
but does not qualify plant models as fractals. These structures are pro-
duced in a finite number of steps and consist of a finite number of line
segments, while the “notion of fractal is defined only in the limit.” Op-
penheimer [105] uses the term “fractal” more freely, exchanging it with
self-similarity, and comments: “The geometric notion of self-similarity
became a paradigm for structure in the natural world. Nowhere is this
principle more evident than in the world of botany.” The approach pre-
sented in this chapter, which considers fractals as simplified abstract
representations of real plant structures, seems to reconcile these previ-
ous opinions.
But why are we concerned with this problem at all? Does the no-
tion of fractals provide any real assistance in the analysis and modeling
of real botanical structures? On the conceptual level, the distinctive
feature of the fractal approach to plant analysis is the emphasis on
self-similarity. It offers a key to the understanding of complex-looking,
compound structures, and suggests the recursive developmental mech-
anisms through which these structures could have been created. The
reference to similarities in living structures plays a role analogous to the
reference to symmetry in physics, where a strong link between conser-
vation laws and the invariance under various symmetry operations can
be observed. Weyl [159, page 145] advocates the search for symmetry
as a cognitive tool:
Whenever you have to deal with a structure-endowed entity
Σ, try to determine its group of automorphisms, the group
of those element-wise transformations which leave all struc-
tural relations undisturbed. You can expect to gain a deep
insight into the constitution of Σ in this way.
The relationship between symmetry and self-similarity is discussed
in Section 8.1. Technically, the recognition of self-similar features of
plant structures makes it possible to render them using algorithms de-
veloped for fractals as discussed in Section 8.2.
8.1. Symmetry and self-similarity
177
Figure 8.1: The Sierpi´ski gasket is closed with respect to transformations
n
T
1
,
T
2
and
T
3
(a), but it is not closed with respect to the set including the
inverse transformations (b).
8.1
Symmetry and self-similarity
The notion of symmetry is generally defined as the invariance of a con-
figuration of elements under a group of automorphic transformations.
Commonly considered transformations are congruences, which can be
obtained by composing rotations, reflections and translations. Could
we extend this list of transformations to similarities, and consider self-
similarity as a special case of symmetry involving scaling operations?
On the surface, this seems possible. For example, Weyl [159, page 68]
suggests: “In dealing with potentially infinite patterns like band orna-
ments or with infinite groups, the operation under which a pattern is
invariant is not of necessity a congruence but could be a similarity.”
The spiral shapes of the shells
Turritella duplicata
and
Nautilus
are
given as examples. However, all similarities involved have the same
fixed point. The situation changes dramatically when similarities with
different fixed points are considered. For example, the Sierpi´ski gasket
n
is mapped onto itself by a set of three contractions
T
1
,
T
2
and
T
3
(Fig-
ure 8.1a). Each contraction takes the entire figure into one of its three
main components. Thus, if
A
is an arbitrary point of the gasket, and
T
=
T
i
1
T
i
2
. . . T
i
n
is an arbitrary composition of transformations
T
1
,
T
2
and
T
3
, the image
T
(A) will belong to the set
A.
On the other hand,
if the inverses of transformations
T
1
,
T
2
and
T
3
can also be included
in the composition, one obtains points that do not belong to the set
A
nor its infinite extension (Figure 8.1b). This indicates that the set of
transformations that maps
A
into itself forms a semigroup generated
by
T
1
,
T
2
and
T
3
, but does not form a group. Thus, self-similarity is a
weaker property than symmetry, yet it still provides a valuable insight
into the relationships between the elements of a structure.
178
Chapter 8. Fractal properties of plants
Figure 8.2: The fern leaf from Barnsley’s model [7]
8.2
Plant models and iterated function sys-
tems
IFS definition
Barnsley [7, pages 101–104] presents a model of a fern leaf (Figure 8.2),
generated using an
iterated function system,
or IFS. This raises a ques-
tion regarding the relationship between developmental plant models
expressed using L-systems and plant-like structures captured by IFSes.
This section briefly describes IFSes and introduces a method for con-
structing those which approximate structures generated by a certain
type of parametric L-system. The restrictions of this method are ana-
lyzed, shedding light on the role of IFSes in the modeling of biological
structures.
By definition [74], a planar iterated function system is a finite set
of contractive affine mappings
T
=
{T
1
, T
2
, . . . , T
n
}
which map the
plane
R × R
into itself. The
set defined by
T
is the smallest nonempty
set
A,
closed in the topological sense, such that the image
y
of any
point
x
∈ A
under any of the mappings
T
i
∈ T
also belongs to
A.
It can be shown that such a set always exists and is unique [74] (see
also [118] for an elementary presentation of the proof). Thus, starting
from an arbitrary point
x
∈ A,
one can approximate
A
as a set of
images of
x
under compositions of the transformations from
T
. On
8.2. Plant models and iterated function systems
179
Figure 8.3: A comparison of three attracting methods for the rendering of
a set defined by an IFS: (a) deterministic method using a balanced tree of
depth
n
= 9 with the total number of points
N
1
= 349, 525, (b) deterministic
method using a non-balanced tree with
N
2
= 198, 541 points, (c) stochastic
method with
N
3
=
N
2
points
the other hand, if the starting point
x
does not belong to
A,
the con-
secutive images of
x
gradually approach
A,
since all mappings
T
i
are
contractions. For this reason, the set
A
is called the
attractor
of the
IFS
T
. The methods for rendering it are based on finding the images
T
i
k
(T
i
k−1
(.
. .
(T
i
1
(x))
. . .))
=
xT
i
1
. . . T
i
k−1
T
i
k
, and are termed
attracting
methods.
According to the
deterministic approach
[123], a tree of trans-
formations is constructed, with each node representing a point in
A.
Various strategies, such as breadth-first or depth-first, can be devised to
traverse this tree and produce different intermediate results [60]. If the
transformations in
T
do not have the same scaling factors (Lipschitz
constants), the use of a balanced tree yields a non-uniform distribu-
tion of points in
A.
This effect can be eliminated by constructing a
non-balanced tree, using a proper criterion for stopping the extension
of a branch [60]. An alternative approach for approximating the set
A
is termed the
chaos game
[7] (see also [107, Chapter 5]). In this case,
only one sequence of transformations is constructed, corresponding to
a single path in the potentially infinite tree of transformations. The
transformation applied in each derivation step is selected at random.
In order to achieve a uniform distribution of points in the attractor,
the probability of choosing transformation
T
i
∈ T
is set according to
its Lipschitz constant. Figure 8.3 illustrates the difference between the
stochastic and deterministic methods of rendering the attractor. The
Rendering
methods

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