**A CONTRIBUTION TO DEFINITIONS OF SOME FRACTAL CONCEPTS**

*
Dusan Ristanovic*^{1},
Gabriele A. Losa^{2},
^{1}Department of Biophysics, Faculty of Medicine, Belgrade, Serbia

^{2}Institute of Scientific Interdisciplinary Studies [ISSI], Locarno, Switzerland

**Submission date: **4 April 2013

**Acceptance date: **21 June 2013

**Pubblication date: **21 June 2013

ABSTRACT

**Background.
**The aim of the present study was to
address the dimensional imbalances in fractal geometry, define the modified
capacity dimension, calculate its value, relate its value with the scaling
exponent and show that such a definition satisfies basic demands of physics,
before all the dimensional balance in mathematical equations used in applied
sciences.

** Methods.
**The study was performed using the basic
rules from fractal geometry, l'Hôpital's theorem and other relevant facts from geometry and calculus.

**Results.**
It was offered the quantitative determination of the self-similarity using the
von Koch fractal set as an example.** **The main result was also the formula
for the modified capacity dimension and its relationship to the scaling
exponent and fractal dimension.

**Conclusion.
**The text includes some important
modifications and advantages in fractal theory and allows better communication
between fractal geometry and wide readership. It is important to notice that
these modifications and quantifications did not
affect already known facts in fractal geometry.

BACKGROUND

Fractal geometry has proven to be a useful tool in
analysis of various phenomena in numerous natural sciences [1]. Although it has
been widely applied and used for quantitative morphometric studies, mainly in
calculating the fractal dimensions of objects [2], there are still some
unresolved issues that need to be addressed. Since some concepts of fractal
geometry are determined descriptively and/or qualitatively, this paper offers
their more exact mathematical definitions or explanations.

Although standard quantitative methods in science
are based on classical Euclidean geometry [3], fractal geometry is developed as
a new geometry of nature [2, 4-6]. It was conceived in 1975 by Benoît B.
Mandelbrot [7], with the aim to describe the complexity and irregularity of
shapes and processes in nature [1]. Early work on fractal geometry showed that
most commonly biological patterns were characterized by fractal geometry [1,
8-12]. Up to now fractal geometry is being used in diverse research areas
[13-16] and is proving to be an increasingly useful tool.

When the physical or biological problem is stated in
mathematical terms, dimensional balance should be a routine part of the
solution of any problem. Exponents and logarithms are always dimensionless
[17]. When dimensioned quantities appear in exponents or in logarithms, they
must combine with other dimensioned quantities so that their quotient or
product is dimensionless. It seems that some of the authors who work in biomedical
sciences do not take care of these facts. The problem is particularly distinct
in defining the fractal dimensions [1, 11, 18, 19]. Fortunately, there are some
exceptions [20, 21]. One of the main aims of the present study is to define and
discuss physically correct modified capacity dimension, and relate it with the
scaling exponent and fractal dimension.

METHODS AND RESULTS

*The
fractal set*

Fractal
geometry was conceived in 1975 by Mandelbrot [7], after his extensive work
describing the complexity of forms and processes in nature [1]. Nowadays
fractal geometry is proving to be an increasingly useful tool for
characterizing biological and other patterns [2, 6, 15, 22, 23].

Fractals are classified into *geometrical* and *statistical*
[4, 5, 22, 23]. Each *geometrical fractal* should be considered as an
infinite ordered set of fractal pieces defined on a metric space. To determine
a fractal set* *we need to specify four things [1, 11, 22]: (1) the *shape*
of a starting object - the *initiator*, (2) the *iterated algorithm*
- enabling its iterative application on the initiator and then, repeatedly, on
all obtained objects (the *generators*), (3) the *conditions* - which
these generators should satisfy, before all, the properties of *self-similarity*,
*scaling* and *scale-invariance*, and (4) the *fractal dimension*
*D* - as the main quantifier to measure *complexity* of the objects.
In that case, such objects (generators) are called *prefractals* [1, 3,
5]. The final result of such infinite procedure is the *limit fractal* [4,
7]. The initiator, prefractals and limit fractal represent the *geometrical
fractal set* [11, 24]**. **

Basic definitions and laws of *fractal planimetry*
can be demonstrated on some classical fractal models [3, 11]. For that purpose
we chose the *triadic (snowflake) von Koch curve set*.

*Von
Koch fractal set*

The sequential construction of the von Koch curve
set begins with the *initiator,* which is an
equilateral triangle of the side length *r*_{0} (Fig. 1A). The *iterated
algorithm* to generate the set of the von Koch curves (known as *prefractals*)
is to recursively reduce the straight line segment (or the *scale*) by 1/3
exchanging repeatedly the middle third of each side of the initiator, or a
preceding generator, with two sides of a smaller, equilateral triangle whose
side is one-third the length of a previous side. The result after the first
iteration (the *stage of construction* *z* = 1) is shown in Fig. 1B,
and that after the second iteration (the stage of construction *z *= 2) in
Fig. 1C. Continuation of this process results in the *limit von Koch curve*.

For the von Koch generators, the length of a segment
at the *z*^{th} stage of construction (*r*_{z}) and
the number of segments at the same stage (*N*_{z}) are,
respectively,

, , (1)

where
*r*_{0} is the side length of the initiator (Fig. 1A). The length
of the fractal curve, actually the perimeter since the curve is closed, (*L*_{z}),
is defined as a product of the number of segments and the length of a segment,
at the *z*^{th} stage of construction,

. (2)

* *

*The
inverse power law*

From Eq. (2) it is evident that the perimeter of the
von Koch prefractals diverges as *z* approaches infinity. If along the
horizontal coordinate axis we put the values *r*_{z} = 1/3^{z}
for *z* = 0,1,2,3, ... and along the vertical coordinate axis the values
of *L*_{z} = 3·(4/3)^{z} also for *z *= 0,1,2,3, ...,
and fit the power function to these data using Microsoft Excel software, the
graph shown in Fig. 2 is obtained. The corresponding fitting parameters are
also carried out and shown in

(3)

with
the coefficients of determination *R*^{2} = 1 and where *r*_{0}
= 1 cm for visual clarity.* *Generally, if we wish to express *L* as
a function of *r* for similar fractal set, two constants of
proportionality (*α* and *β*) should be used, thus the
length can be written as

. (4)

The
value *β* is the *prefactor* and *α* is the *scaling
exponent*. From Eq. (3) it is seen that for the von Koch fractal set *β*
= 3 and *α* = 0.262. We say that the length of a curve (a prefractal)
*scales* with the length of the corresponding fractal segment. Or,
mathematically, *L*_{z} is a unique function of *r*_{z}.

The simplest scaling relationship has the power law
form [1]. The mathematical form of such scaling is an *inverse* *power
law scaling* (Eq. (4)). It describes how a property *L* of the system
(say a perimeter of the von Koch set) depends on the scale *r* at which it
is measured [1]. Thus, this scaling relationship shows how the perimeter of a
prefractal depends on the length of its segment: the smaller the length of the
segment, the larger the perimeter. Since *L*_{ }= *N*·*r*
(the expression known as the Richardson-Mandelbrot equation [20]), from Eq. (4)
it follows that

(5)

* *

*Geometrical
self-similarity*

The object’s property known as *self-similarity*
was first coined by Mandelbrot [7] and can be *geometrical* or *statistical*
[1, 11, 24].

A fractal pattern is said to be *geometrically
self-similar* if each small piece of it is an exact replica (i.e.
‘‘duplicate’’) of the whole object [1, 4]. Thus, the self-similarity
qualitatively means that every small piece of an* *object resembles the
whole object [1, 11]. This definition of the concept ‘geometrical
self-similarity’ should be quantified since small pieces that constitute
geometrical or natural objects are rarely identical copies of the whole object
[22].

We have offered a more exact interpretation of this
descriptive definition introducing a *generating element* of a generator
as a “small piece” [5]. A generator is usually made up of straight-line
segments (for instance, see Fig. 1). A particular and logical concatenation of
some segments of a generator could be thought of as the *generating element*
[11] of a generator if the whole object can be completely built with such
elements by their translations and/or rotations.

In our example shown in Fig. 1 the generators of the
von Koch prefractals, at the first and second stages of construction, have the
generating elements made up of four equal segments each, as shown as details
below the drawings in Fig. 1B and C. For example, the drawing in Fig. 1B can be
subdivided into three generating elements, that in Fig. 1C into 12 elements,
and so on.

Two generating elements of any two generators of a
fractal set can be geometrically similar or not. According to the definition of
similarity in Euclidean planimetry, two generating elements of the generators
at stages *z* and *z* + 1 (say, those in Fig. 1B and C) are similar
to each other if (a) the ratio of the measure of a segment of the generating
element at stage *z* + 1 and the measure of the corresponding segment of
the element at stage *z* is constant for all pairs of corresponding
segments (e.g., for the four pairs of segments of the two mentioned details in
Fig. 1) and for all *z*, and (b) the angles between the pairs of
corresponding segments of the two generating elements are congruent. If the
generating elements of the generators at stages *z* and *z* + 1 are
similar for every *z*, we say that the whole class of the generators is
geometrically self-similar, or, that this set has the property of *geometrical*
*self-similarity*.

*The similarity dimension*

Mandelbrot [7] thinks that 'the plethora' of
distinct definitions of the fractal dimension should be reduced to two: the
similarity dimension and Hausdorff dimension. The *similarity dimension* (*D*_{s})
is basic dimension related to all self-similar (fractal) sets. We shall define
this dimension using the von Koch set as an example. If all the scales of the
prefractals at the stage of construction z are reduced by a factor *F *=
3, the number *N*_{z} _{+1 }of the scales at the stage *z*
+ 1 becomes four times larger than at *z*. Indeed, since

* ** * (6)

it
follows that

for
every z. Generally, since *N* >*F* for every *z*, that fact
can be presented by the expression

(7)

where
*D*_{s} must be larger than 1. This relationship represents the
definition of *the similarity dimension D*_{s}. This also can be
presented as

* * (8)

For
the von Koch set this dimension is *D*_{s} = log 4/log 3 = 1.262.

*The
Hausdorff dimension*

One of the main differences between the methods of
fractal geometry and fractal analysis is in treating the number *N*_{z}.
In fractal geometry this value is the number of segments (scales) counted at *z*^{th
}stage of construction carried out using the scale length *r*_{z}*
*at the *z*^{th} stage of iteration. In fractal analysis this
value, shown as *N,* represents the minimum number of the 'balls' of a
given size (*r*) necessary to completely 'cover' the border (or a line) of
an image. A ball consists of all points within a distance *r* from a
center. In one dimension balls are line-segments, in two dimensions balls are
circles, and in three dimensions balls are spheres. If one covers the object
with balls of radius* r* it must be done* *so that every point of the
object is enclosed within at least one ball. This may require that some of the
balls overlap. One should find the minimum number of balls *N*(*r*)
of* *size* r *needed to cover the object*. *

Consider the number *N*(*r*) of balls of
radius at most *r* required to cover an object completely. When *r*
is small, *N*(*r*) is large (similar relation exists between number
of segments and scale length in fractal geometry). The *Hausdorff dimension*
*d* is a number found such that *N*(*r*) grows with 1/*r*^{d},
as *r* approaches zero. The precise definition requires that the dimension
*d* so defined is a critical boundary between the growth rates that are
insufficient to cover the object, and the growth rates that are overabundant
[18, 19]. The Hausdorff dimension *d* is theoretically rather complex but
it is a successor to some less sophisticated but in practice very similar
dimensions such as the capacity dimension.

*The
modified capacity dimension*

The
similarity dimension can only be used to analyze geometrical (self-similar)
fractal sets. Therefore, it would be necessary to generalize the similarity
dimension. The most important results of such generalization are the Hausdorff
and capacity dimension. These two dimensions are quite similar [1], but the
Hausdorff dimension is rather sophisticated, being a subject of mathematical
measure theory, and not suitable for practical use in fractal geometry. The
capacity dimension is given by

(9)

It
is suitable for determining irregularly shaped geometrical or natural objects.
Since the logarithm is defined only for dimensionless values not for physical
quantities like 1/*r*, we shall define *the modified capacity dimension*
as

(10)

where
*r*_{0} is a *reference scale* [20, 21]. We will show that
this quantity does not influence the common determination of the capacity
dimension ((Eq. (9)). Such quantity *r*_{0} ((Eq. (10)) is
included into the definition of the capacity dimension, because the ratio *r*/*r*_{0}
is dimensionless quantity.

If
we submit Eq. (5) into the last definition ((Eq. (10)) and put *r *, one can see that the
expression in Eq. (10) represents an undetermined form,
but nevertheless the limit of the quotient in Eq. (10) may exist. Application
of the corresponding l'Hôpital's rule often converts an undetermined form to a
determined one allowing easy evaluation of the limit. Let *Q*(*r*) = *F*(*r*)/*G*(*r*).
Suppose, generally, that lim *F*(*r*) and
lim G (r) when*
r* .
If these functions are differentiable on an open interval containing 0, il lim
[*F'*(*r*)*/G'*(*r*)] exists and *G*'(*r*) 0
for all *r* from the interval, then

* * (11)

If the derivatives of *F*(*r*) = *N*(*r*)
and *G*(*r*) = -log (*r/r*_{0})

are
inserted into Eq. (10), the limes of this expression (constant!) is

*D*
= (12)

For the von Koch set *D*
= 1.262 which is in accordance with the value obtained using the similarity
dimension method. The expression given by Eq. (12) sometimes is presented as a
definition of the fractal dimension: This statement is only a consequence of
the definition given by Eq. (10). For that reason Eq. (5) may be presented as

(13)

Equation (13) reveals as decreasing straight-line
plots when the results of counting the numbers of segments are plotted on
log–log axes against the values of the segment length. Thus if these data are
presented in doubly logarithmic paper and fitted by a decreasing straight line,
its coefficient of decreasing represents the fractal dimension *D* (Fig.
3). It should also be noticed that it will be wrong to enter the logarithms of
the data into Descartes co-ordinate system, because, as we have noticed,
logarithms of physical quantities have no physical sense.

*Fractal
dimension of biological objects*

Unlike geometric fractal objects, biological and
natural elements do express statistical self-similar patterns and fractal
properties within an interval of scales, termed the ‘‘*scaling* *window*"
[20]. It is defined experimentally by upper and lower limits in which a direct
relationship between the observation scale and the measured size/length of the
object or the frequency of a temporal event can be verified and in turn
quantified by a peculiar *D*. The straight line fitting ceases out of the
scaling limits to become a sigmoid curve with an asymptotic or even
bi-asymptotic course, as revealed by several authors [26-29]. In other words,
the measured dimensional parameters remain unaffected by further changes in
resolution exceeding both limits. However, real ‘‘fractality’’ exists only when
the experimental scaling range covers at least two orders of magnitude,
although fractality over many orders of magnitude has been observed in various
natural fields [30].

DISCUSSION

It is noticed before that *geometrical
self-similarity* means that every small piece of an object resembles the
whole object. Many authors tried to demonstrate graphically this definition [1,
4, 14]. We showed that such set of objects can be self-similar if their
generating elements are geometrically similar. Any two prefractals of the von
Koch set are not geometrically similar, contrary to their generating elements.
This is obvious from Fig. 1B and C: these two images could only be visually
alike, but not geometrically similar.

Losa and Nonnenchamer [20] used the expression *N*(ε)
= *l*_{0}^{D} ε^{-D}, where ε is the *unit
length* (corresponds to our *r*) and *l*_{o} is the *reference
scale*. The value *l*_{0}^{D }corresponds to the
prefactor* β* (Eq. (13)). It is important to notice that these
authors underlined that *β* depended on *D*,* *which is not
directly visible in the present study. Using this method they presented their
dimensionless *power law scaling*.

Bassingthwaighte et al. [1] noted that the capacity
dimension tell us how much balls needed to cover the object as the size of the
balls (*r*) decreases. In the analytical definition of capacity dimension
there exists log (1/r). West and Deering [11] defined the similarity dimension
as the ratio of two logarithms ln N/ ln (1/r). In defining the box-counting
dimension Falconer [18] showed lower and upper box-counting dimension by the
expression containing log δ where δ is the diameter of a
corresponding set. Edger [19] also use log (1/r) in the definition of the upper
box-counting dimension, where *r *is a diameter of the set.

CONCLUSION

Since
some concepts in fractal geometry are determined descriptively and/or
qualitatively, this paper provides their exact mathematical definitions or
explanations, and Richardson’s coastline method.

REFERENCES

1.
Bassingthwaighte JB, Liebovitch LS, West BJ. Fractal physiology. New York &
Oxford: Oxford University Press; 1994.

2.
Smith TG, Lange GD, Marks WB. Fractal methods and results in cellular
morphology – dimensions, lacunarity and multifractals. J Neurosci Meth.
1996;69:123–136.

3.
Panico J, Sterling P. Retinal neurons and vessels are not fractal but
space-filling. J Comp Neurol. 1995;361:479-490.

4.
Fernández E, Jelinek HF. Use of fractal theory in neuroscience: methods,
advantages, and potential problems. Methods 1998;24:9-18.

5.
Ristanović D, Nedeljkov V, Stefanović BD, Milošević NT,
Grgurević M, Štulić V. Fractal and nonfractal analysis of cell
images: comparison and application to neuronal dendritic arborization. Biol
Cybern. 2002;87:278-288.

6.
Losa GA, Di Ieva A, Grizzi F, De Vico G. On the fractal nature of nervous cell
system. Front.Neuroanat. 2011;5(45):1-2.

7.
Mandelbrot BB. The fractal geometry of nature. NW Freeman and Co, New York:
1982.

8.
Flook AG. Fractal dimensions: their evaluation and significance in
stereological measurements. Acta Stereol. 1982;1:79-87.

9.
Peng CK, Buldyrev SV, Goldberger AL, Havlin S et al. Long-range correlations in
nucleotide sequences. Nature 1992;356:168-170.

10.
Hoop B, Kazemi H, Liebovitch L. Rescaled range analysis of resting respiration.
Chaos 1993;3:27-29.

11.
West BJ, Deering W. Fractal physiology for physicists: Levi statistics. Phys
Rep. 1994;246:1–100.

12.
Caserta F, Eldred WD, Fernández E, Hausman RE, Stanford LR, Buldyrev SV, et al.
Determination of fractal dimension of physiologically characterized neurons in
two and three dimensions. J Neurosci Meth. 1995;56:133–144.

13.
Bizzarri M, Giuliani A, Cucina A, Anselmi FD, Soto MA, Sonnenschein C. Fractal
analysis in a system biology approach to cancer. Semin Cancer Biol. 2011;
21(3):175–182.

14.
Thamrin C, Stern G, Frey U. Fractals for physicians. Paediatric
Respiratory Rewiews 2010;11:123-131.

15. Perrotti
V, Aprile G, Degidi M, Piattelli A, Iezzi G. Fractal analysis: a novel method
to assess assess roughness organization of implant surface topography. Int J Periodontics Restorative Dent.
2011;31(6):633-639.

16.
Tălu S.
Fractal analysis of normal retinal vascular network. Oftalmologia.
2011;55(4):11-16.

17.
Simon W. Mathematical techniques for physiology and medicine. Academic Press,
New York and London 1972.

18.
Falconer K. Fractal geometry. Mathematical foundations and applications. 2nd
Edn. John Wiley and Sons, New York and London 2003.

19.
Edger G. Measure, topology and fractal geometry. 2nd Edn. Springer, New York,
2008.

20.
Losa GA, Nonnenmacher TF. Self-similarity and fractal irregularity in
pathologic tissues. Mod Pathology 1996; 9 (3): 174-182.

21.
Losa GA. Fractal morphology of cell complexity. Biology Forum, 2002; 95: 239-258.

22.
Grizzi G, Ceva-Grimaldi G, Dioguardi N. Fractal geometry: a useful tool for
quantifying irregular lesions in human liver biopsy specimens. J Anat Embriol.
2001;106 (Suppl.1):337-346.

23.
Nakayama H, Kiatipattanasakul W, Nakamura S, Miyawaki K, Kikuta F et al.
Fractal analysis of senile plaque observed in various animal species. Neurosci
Lett. 2001;297:195-198.

24. Feder J. Fractals. New York: Plenum Press,
1998.

25. Losa GA.
The fractal geometry of life. Theor Biol.
2009;102(1):29-59.

26.
Paumgartner D, Losa GA, Weibel ER. Resolution effect on the stereological
estimation of surface and volume and its interpretation in terms of fractal dimensions.
J. Microsc. 1981; 121: 51–63.

27.
Rigaut JP. An empirical formulation relating boundary length to resolution in
specimens showing 'non-ideally
fractal' dimensions. J. Microsc.1984; 13: 41–54.

28.
Nonnenmacher TF, Baumann G, Barth A, Losa GA. Digital image analysis of
self-similar cell profiles. J Biomed Comput. 1994; 37: 131–138.

29.
Dollinger JW, Metzler R, Nonnenmacher TF. Bi-asymptotic fractals: fractals
between lower and upper bounds. J Phys. A: Math Gen.1998; 31: 3839–3847.

30. Mandelbrot B. Is Nature fractal ? *Science* 1998; 279
(5352): 783–784.

LEGENDS TO FIGURES

**Figure 1**.
Sequential construction of the von Koch curve set. (A) The initiator is the
equilateral triangle of the side length *r*_{0}. (B) The first
stage of construction (*z* = 1). (C) The second stage of construction (*z*
= 2). Details below the drawings B and C represent the generating elements of
the von Koch prefractals shown in B and C.

** **

**Figure 2. **Hyperbolic
decrease of the perimeters for the von Koch curves set. The perimeter (*L*_{z}
) of the von Koch prefractals plotted against the segment length (*r*_{z}).

**Figure 3. N**umbers
of segments *N*_{z} shown on log-log axes against the length of a
segment *r*_{z}. The graph is obtained using equation inscribed on
the pictures. *R*^{2} is the coefficient of determination.

FIGURE 1

FIGURE 2

FIGURE 3