Content uploaded by Orly Shenker
Author content
All content in this area was uploaded by Orly Shenker on Jan 11, 2018
Content may be subject to copyright.
0039-3681(94)00046-8
Fractal Geometry is not the
Geometry of Nature
Orly R. Shenker*
Abstract-In recent years the magnificent world of fractals has been revealed. Some
of the fractal images resemble natural forms so closely that Benoit Mandelbrot’s
hypothesis, that the fractal geometry is the geometry of natural objects, has been
accepted by scientists and non-scientists alike. The present paper critically examines
Mandelbrot’s hypothesis. It first analyzes the concept of a fractal. The analysis reveals
that fractals are endless geometrical processes, and not geometrical forms. A comparison
between fractals and irrational numbers shows that the former are ontologically and
epistemologically even more problematic than the latter. Therefore, it is argued, a proper
understanding of the concept of fractal is inconsistent with ascribing a fractal structure
to natural objects. Moreover, it is shown that, empirically, the so-called fractal images
disconfirm Mandelbrot’s hypothesis. It is conceded that the fractal geometry can be used
as a useful rough approximation, but this fact has no bearing on the physical theory of
natural forms.
1. Mandelbrot’s Hypothesis Concerning the Fractal Geometry of Nature
In recent years the magnificent world of fractals has been revealed. Some of the fractal
images closely resemble natural forms: flora, fauna and landscapes.’ The resemblance
is so great, that it seems it might be possible to explain the origin or causes of such
forms in terms of fractals. Indeed, Benoit Mandelbrot called his book, in which
fractals resembling nature were first presented, The Fractal Geometry of Nature.
Mandelbrot advocates the hypothesis that numerous natural forms are fractals and,
therefore, are to be described and analyzed by the fractal geometry. This geometry,
he argues, is the appropriate one for this matter, and not the traditional non-fractal
geometries, Euclidean and others.* Mandelbrot’s hypothesis has caught on among
scientists, including those specializing in the subject, and thence also among
*Program in the History and Philosophy of Science, The Hebrew University of Jerusalem, Mount
Scopus, Jerusalem, 91905, Israel.
Received 25 October 1993; in revised form 10 July 1994.
‘Numerous examples are found in the literature. See for example Benoit Mandelbrot (1983) The Fracfal
Geometry of Nature (New York: W.H. Freeman) and H. 0. Peitgen and D. Saupe, eds (1988), The Science
of Fracfal Images (New York: Springer).
‘Mandelbrot, in The Fractal Geometry of Nature, calls these geometries ‘Euclid’. This name may be
misleading, as it also refers to non-Euclidean geometries.
Pergamon
Stud. Hist. Phil. Sri., Vol. 25, No. 6, 967-981, 1994.
Copyright 0 1995 Elsevier pp. Science Ltd
Printed in Great Britain. All rights reserved
0039-3681/94 $7.00 + 0.00
967
968 Studies in History and Philosophy of Science
non-scientists3 The prevailing view of the matter, as expressed by Jurgens, Peitgen
and Saupe is, that
Fractals are much more than a mathematical curiosity. They offer an extremely compact
method for describing objects and formations . Fractal geometry seems to describe
natural shapes and forms more gracefully and succinctly than does Euclidean geometry.4
However, in spite of the great visual impact of fractal images, Mandelbrot’s
hypothesis is far from being a satisfactory scientific theory. In fact, with regard to
natural forms, the famous images are all it has to offer at present. That this is the
present-day situation is well known. The interesting question is, whether this
hypothesis has the potential to become a scientific theory. Many think it does; the
present paper challenges this view. Leo P. Kadanoff expressed the problematic state
of the present day ‘science of fractals’, when he noted that
. further progress in this field depends upon establishing a more substantial theoretical
base, in which geometrical form is deduced from the mechanisms that produce it. Lacking
such a base, one cannot define very sharply what types of questions might have interesting
answers Without that underpinning much of the work on fractals seems somewhat
superficial and even slightly pointless. It is easy, too easy, to perform computer simulations
upon all kinds of models and to compare the results with each other and with real world
outcomes. But without organizing principles, the field tends to decay into a zoology of
interesting specimens and facile classifications. Despite the beauty and elegance of the
phenomenological observations upon which the field is based, the physics of fractals is, in
many ways, a subject waiting to be bom.5
But Kadanoff does not express doubt regarding the very plausibility of
Mandelbrot’s hypothesis. He too seems to believe it has a potential for becoming what
he calls ‘the physics of fractals’.6 The present paper critically examines both
Mandelbrot’s hypothesis that nature has a fractal geometry, and the belief expressed
by Kadanoff that there is a physics of fractals waiting to be born. Let me be more
precise. Mandelbrot’s hypothesis is that fractals are to be found among the spatial
forms of natural objects, as well as among temporal properties of natural systems (for
example, frequencies) and spatio-temporal dispositions (i.e. initial conditions of
chaotic systems). I wish to argue that the hypothesis is basically wrong in its spatial
part. Mandelbrot may have a point regarding the temporal parts of his hypothesis, but
3For an example of scientists’ view of the matter see the sayings of R. F. Voss and M. F. Barnsley in
Peitgen and Saupe, The Science
o
f
Fractal Images p. 21 and pp. 219-220. An instance of a non-scientific
use of Mandelbrot’s hypothesis is found in Larry Short (1991) ‘The Aesthetic Value of Fractal Images’,
British Journal of Aestherics 31, 342. Short sees nature as trivially having a fractal geometry. In this he
follows Mandelbrot’s opinion in Benoit Mandelbrot (1981) ‘Scalebound and Scaling Shapes: a Useful
Distinction in the Visual Arts and in the Natural Sciences’, Leonardo 14, 45.
4H. Jurgens, H. 0. Peitgen and D. Saupe (1990) ‘The Language of Fractals’, Scienti~5c American 263,
40.
‘Leo P. Kadanoff, (1986) ‘Fractals: Where’s the Physics. T’, Physics To&y, February, 6, quoted from
p. 7.
6Mandelbrot himself, however, saw in Kadanoffs words a heresy. See B. Mandelbrot (1986)
‘Multifractals and Fractals’, Physics Today, September, 11-12.
Fractal Geometry is not the Geometry of Nature 969
this I leave for a separate discussion. The theory regarding chaotic systems makes
use of fractals having a temporal element, as they are embedded in the phase space.
The success of this scientific domain has had the unfortunate side-effect of causing
many to believe that Mandelbrot’s hypothesis concerning the spatial properties of
natural objects is as meaningful and as successful. This mistake I wish to correct
presently. (Hereafter, when referring to Mandelbrot’s hypothesis I will mean its
spatial part.)
The criticism regarding the spatial part of Mandelbrot’s hypothesis rests on four
main arguments. First, I wish to propose that fractals are geometrically abnormal, in
the sense that they are better viewed as geometrical processes, not geometrical objects.
In this respect, they are ontologically and epistemologically even more problematic
than irrational numbers. Therefore, the claim that spatial forms of natural objects are
fractals is inherently contradictory. Second, Mandelbrot’s hypothesis is empirically
mistaken. It is based on the so-called fractal images. But some of these are generated
by functions whose very classification as fractals is mistaken. Others are unnatural
because they are embedded in essentially non-isotropic abstract spaces. As to the rest,
their success in imitating nature can be better explained without resorting to fractal
geometry, and therefore has nothing to do with fractals. Third, fractal geometry can
be used as a useful approximation of natural spatial forms in some cases. But, if it
turns out that a system and its approximation are dominated by different scientific
principles, the latter’s nature should not be taken as an indication of the former’s
nature. Finally, as Kadanoff pointed out, Mandelbrot’s hypothesis regarding spatial
forms is a mere collection of abstract functions which are not even serious candidates
for being part of a science, since they lack interpretation. Such an interpretation cannot
be found within known science, as the hypothesis is inconsistent with the atomistic
paradigm. Alternative, radically non-atomistic interpretations have not been offered,
even in the most general terms.
Let us discuss these four arguments, starting with an examination of the concept
of a fractal.
2. Fractal
s
A
s
Geometrical Proce
ss
e
s
The name ‘fractal’, coined by Mandelbrot, denotes a geometrical kind, not unlike
‘line’. The concept of a fractal cannot be defined in terms of basic concepts such as
point, line, etc., whether their interpretation is Euclidean or otherwise, and in this
sense the fractal is a geometrical type in its own right, having the same status as the
line. Fractal geometry has been developed to deal with these unique geometrical
entities7 The set of fractals can be divided into several families. as can the set of lines.
‘For details of this geometry, see K. Falconer (1990) Fractal Geometry (New York: Wiley).
970 Studies in History and Philosophy of Science
The famous Mandelbrot set belongs to one of them.* The forms belonging to the
different families of fractals may seem at first sight very different from each other,
but they share important properties, and these make them a single geometrical kind.
As Mandelbrot’s hypothesis relates to all the sorts of fractals, the concept of a fractal
has to be examined in the most general terms.
The definition of a line is hard to grasp, since pure length without width and depth
is impossible to visualize. But one can overcome this difficulty by viewing the concept
of a line as a limit concept. The grasping of fractals is a far harder case. In order to
appreciate the special difficulties fractals pose, it is necessary to turn first to their
definition.
For a reason given below, I choose to define fractals, or see the necessary and
sufficient condition for being a fractal, as objects having infinitely many details within
a finite volume of the embedding space.’ Consequently, fractals have details on
arbitrarily small scales, revealed as they are magnified or approached. Let us call this
trait ‘infinite complexity’.”
While each of the various kinds of fractals is mathematically well defined, there
is not yet a generally accepted definition of the general concept of a fractal. However,
there is general agreement regarding which forms belong to the set of fractals and
which do not, with problematic marginal cases. Therefore one may offer a definition
of fractals, as long as it accords with this general agreement, but one is obliged to
justify the offered definition.” It seems to me desirable that the definition should
establish and emphasize the dichotomy between forms that are and are not analyzable
by fractal geometry, for the following reasons. The application of fractal geometry
to finitely complex forms, i.e., forms having a finite number of details, is worse than
superfluous. Analysis of such forms in terms of non-fractal geometries is more exact
and necessitates the use of fewer approximations. This is true even for complex forms,
as long as their complexity is finite. It is especially significant in the limit of the very
small scales used in fractal geometry’s definitions. I* On the other hand, non-fractal
geometries cannot provide a complete description of infinite complexity, neither
locally nor globally. The analysis of fractals by these geometries is, at best, partial
or approximate. If exactness and full description are required, non-fractal geometries
*For a discussion of the various kinds See Falconer, Fractal Geometry; Peitgen and Saupe, The Science
of Fractal Images; and Mandelbrot, The Fructul Geometry of Nature.
‘The concept of a detail is not primitive, and seems reducible to concepts of non-fractal geometries.
This needs a more rigorous discussion than the present one.
‘“In choosing this definition I generally agree with Mandelbrot. It is equivalent to his definition of a
fractal, as a form having fractal dimensions greater than its topological dimensions; see Mandelbrot, The
Fractal Geometry ofNuture, pp. 15 and 361ff. I find the term ‘infinite complexity’ simply more indicative
of the fractal’s nature. For a presentation and explanation of various kinds of fractal dimensions see
Falconer, Fractal Geometry, pp. 25-61.
“For example, Falconer in Fractal Geometp chooses to define fractal as a Wittgensteinian family
concept; see pp. XIII-XXI.
‘*Especially in the definitions of the various fractal dimensions; see Falconer, Fracfal Geometry, pp.
25-61.
Fractal Geometry is not the Geometry of Nature 971
can only be used to analyze finitely complex forms. For these reasons, I prefer a
definition of fractals that includes all and only the forms not analyzable by non-fractal
geometries. I3 For this purpose, infinite complexity seems to serve well.
Equipped with this definition, we can return to the difficulties posed by the concept
of fractal. Prima facie, if a form has infinitely many details, its very specification is
impossible, either as numerical data or as a geometrical entity presented graphically.
Fortunately, certain kinds of infinite complexity can be generated by finitely long
algorithms, including loops that run infinitely many times, constantly generating
additional numerical data. These data can be presented graphically, with the aid of
computerized graphical equipment.
Practically, of course, only the finite amount of data obtained after a finite time can
be presented. An image obtained this way is not a fractal, but an intermediate stage
on the infinite route to the fractal. This is the case whether the process is viewed as
constructing the fractal or as discovering it. The infinitely complex fractal itself cannot
be obtained, since it is generated by an essentially infinite process. We finite creatures
can only examine images that are intermediate stages of this creation, but these images
are necessarily of finite complexity, and therefore are by the above definition not
fractals.
Fractals seem analogous to irrational numbers in their ontic and epistemic essence.
Both can be described as generated by essentially infinite processes. Only their
‘heads’ can be obtained, but these alone are of a different category from the whole,
being finitely complex forms or rational numbers, respectively. The epistemic and
ontic status of the ‘tails’ is a problem at the core of the realism versus anti-realism
debate. One can refer to an irrational number by mentioning its name, such as ‘d2’,
‘rc’ or ‘e’, and in this sense one relates to a certain entity. On the other hand, these
entities are generated by essentially infinite algorithms. There are two kinds of such
infinite algorithms. Some of the irrational numbers are generated by algorithms
having finite descriptions, and containing loops that are repeated endlessly. These
irrationals can be referred to through descriptions of their generating algorithms,
though not directly through specifications of their numerals. Other irrational numbers
are composed of an infinitely long random series of numerals. This subset is the larger
one. By definition, these numbers are generated by algorithms no shorter than their
endless series of numerals. Therefore, they cannot be specified even by descriptions
of their generating algorithms. There is no way whatsoever to refer to such numbers.
For a Mathematical Platonist, these facts present no difficulty, since an irrational
number exists whether it can be specified or not. The inability to specify a number
amounts to a difficulty of reference, at best. For a Constructivist, the inability to
construct an irrational number presents an ontological and epistemological difficulty.
“This definition is not circular, although it may seem so at first sight. Agreed, fractal geometry was
developed for the purpose of analyzing forms that were known beforehand (though not by the name
‘fractals’, coined by Mandelbrot). But, once the geometry is given, its properties can be used for the
definition of its proper subject matter.
972 Studies in History and Philosophy of Science
An inconstructible number does not fully exist. Only its constructible ‘head’ exists,
but the non-complete output of the infinite algorithm is a rational number, not an
irrational one.
The analogy with fractals is illuminating. On the one hand, one can refer to certain
fractals by mentioning their names, such as ‘The Mandelbrot Set’ or ‘The Von Koch
Curve’. On the other hand, the numerical data that describe such entities are generated
by algorithms that are by definition infinite. Here, too, two subsets can be discerned.
One includes fractals generated by algorithms having finite descriptions, such as the
above Mandelbrot and Von Koch fractals. The other includes random fractals, whose
infinite complexity can only be described by specifying each and every detail. By the
same logic used with regard to irrational numbers, this second subset is the larger one.
As in the case of the irrationals, fractals of the first subset can be referred to through
descriptions of their generating algorithms, although they cannot be described
directly. Fractals of the random subset cannot be referred to even in this indirect mode.
It turns out, then, that the ontological and epistemological debates and difficulties
regarding irrational numbers are also relevant for and applicable to fractals.14
Fractals present, however, an additional problem not encountered when discussing
irrational numbers. It is the fact that a fractal is, first and foremost, a geometrical
object. True, geometrical objects can be constructed using numerical algorithms and
data, but they are nevertheless different entities. A geometrical object is embedded
in space, be it real or imaginary, Euclidean or other, and of any number of dimensions.
As such it is thought of as existing all at once. Irrational numbers can be thought of
as existing all at once when presented geometrically and not numerically; in fact, the
irrationality of the square root of two was discovered by the Greeks via its presentation
as the diagonal of a unit square. However, when examining fractals one faces an entity
that is, on the one hand, a geometrical object, and, on the other hand, is generated
by an essentially unending process. There is no way to present it all at once analogous
to the geometrical presentation of the irrational 42. For the Mathematical Platonist,
these facts present no difficulties. Fractals exist all at once in the realm of ideas.
Difficulties are encounted only by a Constructivist. From a Constructivist point of
view, a fractal is a constantly created, forever dynamical geometrical object. It may
be described as a geometrical process, rather than as a geometrical object. Only
intermediate stages on the way to a fractal can be said to exist all at once, but these,
as I have already said, are not fractals.”
These considerations lead to the conclusion that fractals cannot exist at any actual
time and place, other than in a Platonist realm of ideas. In light of the elusive nature
of fractals, it is already hard to understand why one might say that natural objects
14For example, Roger Penrose uses the Mandelbrot set to illustrate his argument for Mathematical
Platonism. See R. Penrose (1989) The Emperor’s New Mind (New York: Oxford University Press), pp.
98-104 and 123-128.
“The advantage of the Platonist with regard to the status of fractals can be used as an argument for
Platonism. However this is nor Penrose’s argument in The Emperor’s New Mind.
Fractal Geometry is not the Geometry of Nature 973
are fractals, other than as a very general approximation. The proper application or
scope of such an approximation will be discussed later.
3. Empirically, Natural Ob
j
ect
s
A
re Not Fractal
s
Indeed, I shall now argue that, as a matter of empirical fact, there is no ground to
say that natural objects have a fractal geometry. Mandelbrot’s hypothesis rests on the
fact that the so-called fractal images prima facie seem to imitate natural forms.
However, close examination reveals that this prima facie resemblance is both
superficial and misleading. Superficial, since in most of the cases the images turn out
to be unnatural. Misleading, since in other cases the success in imitating nature is
better explained without resorting to the fractal geometry, and therefore has nothing
to do with fractals. For this reason I do not refer to these images as ‘fractal images’,
but as ‘so-called fractal images’.
Let us now turn to examine why so-called fractal images seem to imitate nature.
When one says that such a form resembles a natural form, one does not relate to the
infinitely remote output of the fractal’s algorithm, but to the image in front of one’s
eyes. That image is not a fractal, for it is of a finite complexity. It is an intermediate
stage on the way to a fractal. There are several families of fractals, characterized by
the kinds of algorithms used to generate them. The intermediate stages of each such
family seem to imitate a different kind of natural form. The explanation for the
imitations’ success is different for each family, because it relates to the algorithm for
its generation.
3. I. Forms Mistakenly Clussijied As Fructuls
Two kinds of forms, that are not fractals, were for some reason mistakenly classified
as such. The first type is that of the L-systems. These are models of plant development,
and thus are also used for synthesizing realistic images of plants.” A variation of these
systems can be used as an algorithm that generates fractals of the self-affine kind (to
be discussed later). This fact created a confusion even among experts, who somehow
started to see the other, clearly non-fractal products of L-systems, as fractals also.”
The other family of forms is the DLA kind (Diffusion Limited Aggregation). This
method is successfully used to simulate phenomena of aggregation, from electrolysis
to moth growth.” The definition of the forms that this method generates includes a
pixel of a finite size, a geometrical atom. Therefore, their complexity can be increased
only by expansion. But, within a given volume of their embedding space their
‘%ee P. Prusinkiewicz and J. Hanan (1989) Lindenmayer Systems, Fractals and Plants, Lecture Notes
in Biomathematics (New York: Springer).
“See Peitgen and Saupe, The Science of Fractal Images, plates 16~ and 16d after p. 114.
IsFor a model of electrolysis see Falconer, Fractul Geometry, pp. 267-272. Moth growth can be seen
in Peitgen and Saupe, The Science of Fractal Images, p. 38.
974 Studies in History and Philosophy of Science
complexity is essentially finite, contrary to the definition proposed above. Therefore,
they are not fractals.
3.2. Unnatural Fractals That Are Embedded In Abstract Non-isotropic Spaces
Complex iterations forms constitute a third kind of so-called fractal images. To this
family belongs the famous Mandelbrot set. Using this method, forms prima facie
reminiscent of microorganisms were generated.” However, these forms are
embedded in the non-isotropic complex space. Saying that they can be used to explain
life forms amounts to holding the radical belief that, with regard to such forms,
physical space is non-isotropic. However, the assumption that physical space is
isotropic is otherwise overwhelmingly corroborated. Furthermore, close inspection
or a simulated magnification of what superficially seems like organs, reveals
characteristics of Julia sets. The resemblance of the latter to life forms is questionable.
It may be that psychological Gestalt effects in the observer force the intricate details
into the known patterns of organic forms. *‘The Julia and Mandelbrot sets of complex
iterations are also used to generate fantastic landscapes.*’ The non-isotropic complex
space considerations are relevant here, too.
The strange attractors fractals are Poincare sections of the phase space trajectories
of dynamic chaotic systems. Their appearance prima facie resembles mixing fluids.
This impression is strengthened when one notices that real fluids’ images also depict
sections or surfaces. However, no far-reaching physical conclusions can be drawn
from this resemblance, as the attractors are embedded in the non-isotropic phase space
of position and momentum, not in the isotropic physical space. Incidentally, strange
attractors are known as embodying the fractal nature of the initial conditions of chaotic
systems. This belongs to the dynamical or temporal part of Mandelbrot’s hypothesis,
which is not discussed here.
3.3. Images That Resemble Nature Because They Are Finitely Complex Non-fractals
Some of the so-called fractal images do seem to resemble natural forms, and quite
closely. I shall presently argue that these images resemble nature due to certain
characteristics of theirs, characteristics that have nothing to do with fractals or with
the fact that they can be viewed as intermediate stages on the way to fractals. In fact,
they resemble nature just because they areJinitely complex, and therefore just because
they are not fractals. In other words: these images resemble nature, not because they
are stages on the way to fractals, but rather in spite of this fact.
“See C. A. Pickover (1986) ‘Biomorphs: Computer Displays of Biological Forms Generated From
Mathematical Feedback Loops’, Computer Graphics Forum $3 13, and A. K. Dewdney (1989) ‘Computer
recreations’, Scientijic American July, 92.
*“Compare the different scales of the pictures in pages 314 and 315 in Pickover. ‘Biomorphs: Computer
Dis
?P
lays of Biological Forms Generated From Mathematical Feedback Loops’.
See Peitgen and Saupe, The Science of Fracral Images, covers and colored plates after p. 114.
Fractal Geometry is not the Geometry of Nature 975
The first kind of such images are self-affine forms.22 Many plants and crystals are
self-affine (or self-similar) in a finite number of scales. There is no wonder, then, that
images revealing a finite self-affinity resemble these natural forms. However, if one
continues the algorithm that generates self-affinity, wishing to approach the self-affine
fractal by generating self-affinity in numerous scales, the obtained image ceases to
resemble the natural one. The reason seems trivial, and would not have required any
detailed discussion, were it not for the widespread false idea that these images
resemble nature due to their being stages on the way to a fractal, rather than in spire
of this fact. Let us then discuss this matter briefly, by examining an example.
A paradigmatic example of self-affinity is the image of a fem. A small number of
self-affine steps generates a persuasive image of a fem.23 However, as one expects
the self-affinity to end at the scale of the smallest leaves, additional self-affine details
spoil the natural impression. In order to compare the real fern and the fractal in smaller
scales, magnification is needed. A magnification of a fractal is not done with the aid
of a magnifying glass, since such an action would reveal the details of the paper or
monitor on which the fractal is plotted. The substitute is a simulated magnification,
done by plotting a portion of the original image on the whole page. Were the result
a graphical representation of a natural form, the simulated magnification would have
had the effect of a real magnification, of observing a fern with a magnifying glass,
or of approaching the object. But it does not.24 With regard to natural ferns, one can
easily tell the scale that one is observing, with possible errors ranging between the
few scales in which the fern is self-affine. But with regard to the fractal fern this is
not possible, as all scales appear the same. One cannot tell whether one is observing
the scale of one centimeter or of one Angstrom. True, such is the effect of self-affinity.
But, it is not a natural effect. Therefore, the fact that the so-called fractal images
resemble ferns is a result of their$finite complexity, of the fact that they are self-affine
in only a finite number of scales. In other words, they resemble natural forms due
to their not being fractals.
This fact has surely not escaped the eyes of Mandelbrot and the proponents of his
hypothesis. However it seems not to have belittled the role they give to fractal
geometry in describing nature. There lies the mistake. Forms that can be described
by non-fractal geometries at all, are more exactly described by them. Therefore a
physics of such forms should be based on their non-fractal geometrical description.
The proper role of the fractal geometry, in approximating natural forms, will be
discussed later.
The non-natural characteristics of self-affine images, encountered in simulated
magnifications, are met also in images of the statistically self-similar kind of fractals.
‘*For a discussion of self affine fractals, see Falconer, Fracral Geometry, pp. 113-137.
23For example, see Peitgen and Saupe, The Science of Fractal Images, plate 32 after p’. 114.
*?See Peitgen and Saupe, The Science of Fractal Images, p, 239.
976 Studies in History and Philosophy of Science
These images are used to depict landscapes.” In each of the images, be they of
coastlines, mountains, or whole planets, not more than a few scales are effectively
visible. The largest visible scale is statistically ineffective as one can see only a small
portion of it, such as a single mountain. The medium scales, of which only a few are
practical, are effectively and clearly visible, and determine the general appearance
of the image. The yet smaller scales are not visible, and therefore have no effect on
the depicted landscape’s appearance. Now saying that natural landscapes are
statistically self-similar amounts to saying that, in nature, the invisible scales are
statistically self-similar to the visible ones. However, empirically this is not the case.
Again, the reason seems trivial, but the widespread belief to the contrary necessitates
a brief discussion.
The creator of a statistically self-similar image chooses the right fractal dimension
in order to generate a realistic impression. Actually, the choice relates mainly to the
few medium scales, the only ones that are effective in creating the general impression
of the image.26 Now, imagine that one obtains a realistic landscape, and then simulates
its magnification. Assuming that nature is statistically self-similar, one expects to
experience the effect of approaching the depicted landscape. But this does not happen.
After a simulated enlargement by 107, equivalent to approaching from ten kilometers
in the air to one millimeter off the ground, the landscape looks as if it is seen from
the initial altitude.27 True, again, this is an expected effect of good statistical
self-similarity, but it is not the expected effect of simulated enlargements of natural
landscapes. In reality one can usually quite successfully guess one’s altitude, using
the non-similarity of the different scales.28 The conclusion is, that the landscape
images resemble nature only in a finite number of scales, and therefore just because
they areBnitely statistically self-similar. In other words, they resemble natural forms
just because they are not fractals.
Summing up the explanations for the success in imitating nature, the finitely
self-affine and statistically self-similar forms resemble some scales of natural forms
due to their finite complexity, i.e. due to their not being fractals. Other so-called fractal
images are either embedded in essentially non-isotropic spaces, or do not resemble
a5Examples of such landscapes can be found in Peitgen and Saupe, The Science of Fracrul Images, front
and back covers and colored plates after p. 114, and J. Feder (1988) Fractals (New York: Plenum Press),
colored plates. For a discussion of statistically self similar algorithms that can generate fractals, see
Falconer, Fracfal Geometry, pp. 224-253, and Mandelbrot, The Fractal Geometry of Nature, pp. 12-13
and 200-276.
26For example, in Mandelbrot, The Fractal Geomefry of Nature, colored plate at p. C9, the ‘Moon’ is
of fractal dimension 2.2 and the Earth of 2.5. The effect and the process of choosing the right dimension
can be seen at Peitgen and Saupe, The Science of Fractal Images, colored plates 11-13, and Mandelbrot,
The Fracral Geometry of Nature, pp. 264-267 and plate C14. The effect of varying the random function
is exemplified in Peitgen and Saupe, The Science of Fracral Images, colored plates 8-10.
27See Peitgen and Saupe, The Science of Fructal Images, p. 23.
‘*The photographs in the final section of Mandelbrot, The Fractal Geometry of Nature, confirm this
argument; different scales are statistically non-self-similar in those pictures.
Fractal Geometry is not the Geometry of Nature 977
nature at all, or are not fractals in the first place. Thus, the empirical data disconfirms
Mandelbrot’s hypothesis.
4. The Use of Fractal Geometry As a Rough Approximation in the Analysis of
Natural Forms
Agreed, the fact that no natural form is actually a fractal need not prevent one from
using fractal geometry whenever it turns out to be useful. One might raise the
objection that so-called fractal images, being intermediate stages on the route to
fractals, are approximations of fractals, in the same sense that a drawn line is an
approximation of the ideal one-dimensional line. By stating that the approximation
is qualitatively different from the approximated, I have only repeated an idea well
known in non-fractal geometries. The objection is, why stress this point? Why not
call the approximated fractal ‘fractal’ in the same sense that one calls an approximated
line ‘line’, while remembering the approximation at the back of one’s mind? In order
to answer this objection, let us see how such an approximation can be carried out.
Geometrical forms can be ascribed to physical objects by mapping each point of
the physical space to a point of an abstract space having some suitable and convenient
geometry and a coordinate system. A natural form is a geometrical form in the chosen
abstract space that conforms to a physical object by the said mapping.29 Accordingly,
natural objects can be analyzed using fractal geometry in the following manner. First,
the natural object is to be mapped to the abstract space in which it is ascribed a
geometrical form. The obtained geometrical form is not a fractal, but a finitely
complex form. Infinitely many details are then added to the abstract form in the
abstract space. For example, if the original natural object is finitely self affine, its
self-affinity is to be continued ad in$nitum. It must be remembered that the fractal
thus obtained is not the natural form of the natural object, but an abstract geometrical
form in the abstract space, based on the object and with infinitely many details added.
Fractal geometry can be applied to this fractal. The consequences of such an
application, such as the calculated fractal dimensions, can be considered to be
approximately true of the original, non-fractal form. The circumstances in which such
an approximation is appropriate depend on the purpose of the analysis. For instance,
it may turn out to be useful when the object’s smallest details are smaller than is
deemed relevant or measurable. In such cases, the application of fractal geometry can
turn out to be convenient, since it enables a description of general traits of the forms
without having to measure each and every detail of them.
However, one must bear in mind that fractal geometry can only be approximately
applied to natural forms. It can be applied for the sake of convenience in the
29This presentation of the concept of a natural form is not without problems, but it suffices presently.
See Hans Reichenbach (1957) The Philosophy of Space and Time, trans. M. Reichenbach and J. Freud
(New York: Dover).
978 Studies in History and Philosophy of Science
appropriate circumstances, but, strictly speaking, these forms are not fractals. This
last point cannot be overemphasized, since neglecting to remember it has actually led
to gross misunderstandings with regard to the status of fractal geometry in the
scientific analysis of natural forms. It has led to the mistaken opinion that the ability
to approximately apply fractal geometry in some cases indicates a deep natural
structure, hitherto ignored, and profoundly different from the prevailing theories
regarding natural forms. This, in turn, led to the conclusion that fractal geometry is
to be the basis of a new physics relating to natural forms. The failure to discern the
approximation from the approximated has led to far-reaching conclusions that
groundlessly challenge the atomistic paradigm, as we shall shortly see.
My answer to the above objection is that one can call the approximation of a fractal
‘a fractal’, as long as one abstains from relating infinite complexity to finitely complex
forms. It is necessary to insist on discerning between fractals and their approximations
exactly because a neglect of this discernment has led to mistaken conclusions with
regard to the geometry of nature.
5
. The Lack o
f
a Phy
s
ical Interpretation o
f
Mandelhrot
’
s
Hypothe
s
i
s
Generally speaking, scientific theories connect mathematical functions to nature
by interpreting them. An interpretation of a function relates to the function as
embodying a law of nature, through its structure and parameters. The interpretation
has nothing to do with the shape of the function’s plot, as plots heavily depend on
the choice of coordinate systems. In the scientific literature, graphical representations
of mathematical functions are used only for the sake of clarification or for didactic
purposes. A scientist presented with a plot without the function that generates it, can
make nothing of the plot unless the function is specified; it is scientifically
meaningless. A case close to this is when the scientist is presented with a plot and
its generating function, but that function does not belong to any theory. This time,
the meaninglessness is due to lack of context and lack of interpretation. The last case
is the case of so-called fractal images.
Kadanoff, in the above quotation, is well aware of this fact. He recommends a
search for interpretation. In which direction can it be sought? The resemblance
between these plots and pictures of natural objects cannot serve as an interpretation
of their generating functions, because it does not relate to these functions’ structure
and parameters. Can an interpretation be found within existing science? The
increasing popularity of Mandelbrot’s hypothesis regarding natural forms has not
been accompanied by serious attempts to make it coherent with scientific theories that
are otherwise widely corroborated. Probably not without reason: such a coherence
cannot be obtained. Let us briefly discuss this point.
If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one
sentence passed on to the next generation of creatures, what statement would contain the
Fractal Geometry is not the Geometry of Nature 979
most information in the fewest words? I believe it is the atomic hypothesis that all things
are made of atoms3’
These words of Richard Feynman embody one of the most basic ideas of modem
science. In its original context, at the introduction to The Feynman Lectures on
Physics, the word ‘atoms’ is not meant literally, but refers to the objects decomposable
into protons, neutrons and electrons. However, if we interpret the word to mean that
there are things such that all other things are composed of them, then Feynman’s
words express the paradigm on which the whole scientific conception of the material
world is based. I shall emphasize that the paradigmatic idea consists of the compact
statement that elementary particles exist; their number and nature is not part of the
paradigm.
Under the atomistic paradigm, the geometrical characteristics of natural forms are
determined as follows. First, each elementary particle is mapped to an abstract space,
in which it is ascribed a geometrical form, whether as a point or as having some
volume, depending on its specific properties.31 A macroscopic object is an aggregate
of elementary particles, and the mapping mirrors their relative positions. The natural
form of a large object is, then, an aggregate of the forms ascribed to its elementary
particles, with their appropriate arrangements and relative positions.
Fractals are defined above as being infinitely complex, as having an infinite number
of details within a finite volume of the embedding space. However, if an object is
made of elementary particles, then the number of details its geometrical form has is
finite. This number is determined by the number of the elementary particles and their
relative positions. It may be very large, but it is always finite. The inevitable
conclusion is that according to the modem scientific conception of natural forms,
natural objects cannot be fractals. The very idea of an object having an infinite number
of details contradicts the atomistic paradigm and therefore is inconsistent with modem
science. Mandelbrot’s hypothesis cannot then receive an interpretation within known
science. It needs to be interpreted within a radically new, non-atomistic science. As
the atomistic paradigm is otherwise invariably corroborated, the sole motivation for
rejecting it in search of a ‘physics of fractals’ (in Kadanoff’s words) is theprima facie
resemblance of pictures. This resemblance itself we have shown to be superficial and
misleading. It seems that the fact that a satisfactory non-atomistic interpretation was
not offered, not even in the most general terms, is not without a reason.
The proponents of Mandelbrot’s hypothesis did not mean, perhaps, to reject or
challenge the atomistic paradigm, or else they would have provided deeper
arguments. Nevertheless, the hypothesis taken by itself implies such a challenge.
Lacking interpretation, the status of Mandelbrot’s hypothesis is that of a mere
30Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963) The Feynman Lectures on Physics
(Reading, Massachusetts: Addison-Wesley) Vol. 1, pp. 1-2.
“Although I cannot defend the point in detail here, I think it is clear that the points in this paragraph
and the next are not affected by the differences between the ontologies offered by the various interpretations
of quantum theory.
980 Studies in History and Philosophy of Science
collection of abstract functions, no different from any other such collection. It has
no scientific status, not even the status of a candidate for becoming a scientific theory.
6.
C
onclu
s
ion
s
The impressive so-called fractal images offered a great temptation to science. On
the one hand, while science has it that natural forms are in principle reducible to the
structure and processes at the level of elementary particles, the computational
difficulties make such a reduction impossible. Thus science lacks a practical theory
of macroscopic forms, in all but a few simple cases. On the other hand, successful
imitation of natural forms seemed attainable by using simple algorithms. The
temptation to see in these images a thread that can lead to a theory of macroscopic
forms was so great, that basic principles of scientific methodology seem to have been
forgotten.
The first forgotten principle is the need to have at least an approximate empirical
agreement between the hypothesis’ predictions and reality. The experiment of
comparing imitated magnifications of fractal images and real magnifications of
natural forms conspicuously reveals their deep differences. This has not led to
abandoning the fractal structure hypothesis. It has not led to restricting the application
or scope of fractal geometry to just approximations.
The second forgotten principle is the need for consistency between various
scientific theories. The fact that the concept of a fractal object is profoundly
inconsistent with one of the most basic postulates of modem science, the atomistic
paradigm, did not lead to emphasizing the restricted role and approximative nature
of fractal geometry in the analysis of natural forms. Instead, scientists and
non-scientists alike advocate the view that this geometry offers a novel method
of analyzing nature that reveals a deep structure hitherto ignored. This radical view
was not supported by any argument other than the superficial resemblance of
pictures.
The third forgotten principle concerns the difference between a system and its
approximation or idealization. Whenever the two are so qualitatively different that
they seem governed by different scientific paradigms, the paradigm relating to the
real system is of course to be preferred to that relating to the approximation.
Kadanoff, in the quotation in Section 1, is aware of the problem that a superficial
resemblance is far from being enough, and that unless the fractal functions receive
a proper interpretation they cannot be said to have a scientific meaning. He
recommends the creation of a physical theory within which such functions will be
meaningful. This actually implies the search for a radical theory not based on the
atomistic paradigm. However, Kadanoff’s very recommendation rests on the
assumption that forms of natural objects can be successfully imitated by fractals, an
assumption that is empirically wrong. A physics of fractals cannot then be born,
because fractal geometry is not the geometry of nature.
Fractal Geometry is not the Geometry of Nature 981
Acknowledgements-The author wishes to thank Itamar Pitowsky, Yemima Ben Menachem,
Mark Steiner and the referees who read this article for their helpful comments and criticisms.
