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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.