###### Abstract

Reduced fractals, with different levels of self-similarity and magnification,
are defined and used to introduce discrete spectra of reduced fractals. To
justify the concept of reduced fractals and their spectra, some specific
applications to biological systems, such as green algae, are performed and it
is demonstrated how these spectra can be used to classify biological systems.
The ranges of these spectra for the considered systems are determined and
their extension to other natural science systems is suggested.

Spectra of Reduced Fractals and Their Applications in Biology

D. T. Pham

Z. E. Musielak

##  1 Introduction

Mandelbrot1 originally defined fractals as objects that are self-similar on
all scales and whose dimensions are different than their topological
dimensions. From a mathematical perspective, fractals are defined by listing
their four basic characteristics,1-3 which are: self-similarity at all scales,
fine structure at all magnifications, too irregular to be described by
Euclidean geometry, and have non-topological (Hausdorff) dimension;
mathematical objects that have these characteristics are called classical
fractals.4 To make fractals applicable to Nature, Mandelbrot1 changed the
defintion to a more casual: “A fractal is a shape made of parts similar to the
whole in some way”. Thus, the main difference between classical fractals and
fractals that obey Mandelbrot’s casual defintion is that their self-similarity
is exact for the former and non-exact for the latter.3,4

Fractals have been used in numerous research topics ranging from biology and
biomedicine to physics, astronomy, geology, computer science, and in
epidemology, emerging diseases as well as in comparative studies.1-6 In the
previous applications of fractals to biology,7-11 the main emphasis was given
to population biology,12 plant structures,13 gene expression,14 heart rates,15
cardiovascular system,16 kidney structure,17 cellular differentiation,18
neuron branching,19 image rendering, image processing, mammography, images of
human brains,20,21 and classification of strokes in brains.22 Since fractals
in Nature are directly related to some growth process, therefore, methods such
as the Multiple Reduction Copy Machine (MRCM) and the L-systems have also been
used to generate plants, trees and bushes.4

Attempts have been made to introduce and use practical fractals,2 which
basically refer to fractals with limited range of self-similarity. On the
other hand, multi-fractals are used to descirbe phenomena, whose different
components may have different scaling exponent, which may require a spectrum
of exponents.2-4 Moreover, it was also shown that a different family of
fractals can also be defined by removing one fractal property from the list of
fractal basic characteristics; for example, in intelligent processing systems,
a fractal of limited scale range and partial symmetry is called semi-
fractal.23 Similar ideas were used in studies of images, structures and even
sounds,24,25 but to the best of our knowledge, they have never been applied to
biology and its systems.

Despite a broad range of applications of fractals to biology and biomedicine,
we propose here to refine the idea of fractals by introducing the concept of
reduced fractals with specific limited scale range and only partial self-
similarity (see Sec. 2). As a result, these reduced fractals form a spectrum
of reduced fractals, whose scales and self-similarities range significantly
for biological systems, such as green algae,26,27, considered in this paper.
Using green algae, we determine the form of their spectra and demonstrate how
to apply the obtained results to other more diversified biological systems. It
is suggested that the presented results are likely to be also applicable to
other natural sciences; however, such applications are out of the scope of
this paper.

Our paper is organized as follows: in Sec. 2, reduced fractals are defined and
applied to selected biological systems; fractal dimensions for the considered
biological systems and the resulting spectra of reduced fractals for these
systems are presented in Sec. 3 and 4, respectively; our conclusions are given
in Sec. 5.

Figure 1: Algae of class Zygnematophyceae and their dimensions (a) Euastrum
bidentatum $F_{d}=1.8408$ (b) Euastrum oblongum $F_{d}=1.8598$ (c) Euastrum
verrucosum $F_{d}=1.8739$ (d) Euastrum ansatum $F_{d}=1.8801$ (e) Euastrum
humerosum $F_{d}=1.8907$ (f) Euastrum crassum $F_{d}=1.8948$ (g) Micrasterias
americana $F_{d}=1.8117$ (h) Micrasterias truncata $F_{d}=1.8703$ (i)
Micrasterias rotata $F_{d}=1.8749$ Figure 2: Algae of Class Chlorophyceae and
their dimensions (a) Volvox globator $F_{d}=1.2288$ (b) Volvox aureus
$F_{d}=1.3701$ (c) Eudorina elegans $F_{d}=1.6975$ (d) Scenedesmus granlulatas
$F_{d}=1.7097$ (e) Pediastrum clothratum $F_{d}=1.7182$ (f) Pediastrum
angulosum $F_{d}=1.7806$ (g) Desmodesmus magnus $F_{d}=1.7447$ (h) Tetraedron
minimum $F_{d}=1.7087$ Table 1: Self-similarity and magnification in some
Chlorophyceae colonial algae.
Genus species | Colony Size | Magnification
---|---|---
Eudorina elegans | 16,32 | 2
Desmodesmus magnus | 4,8,16 | 3
Scenedesmus granlulatas | 2,4,8,16,32 | 5
Pediastrum angulosum | 4,8,32,64,128 | 5
Pediastrum clothratum | 8,16,32,64 | 4
Tetraedron minimum | 2,4,8,16 | 4
Table 2: Fractal dimension of unicellular algae of the class
Zygnematophyceae and its family Desmidiaceae.
Genus species | Cell shape | Fractal dimension
---|---|---
Euastrum bidentatum | Ellipsoid | $1.8408$
Euastrum oblongum | Ellipsoid | $1.8598$
Euastrum verrucosum | Ellipsoid | $1.8739$
Euastrum ansatum | Ellipsoid | $1.8801$
Euastrum humerosum | Ellipsoid | $1.8907$
Euastrum crissum | Ellipsoid | $1.8948$
Micrasterias americana | Spherical | $1.8117$
Micrasterias truncata | Spherical | $1.8703$
Micrasterias rotata | Spherical | $1.8749$
Table 3: Fractal dimension of colonial and multicellular algae of
the class Chlorophyceae and its three different families.
Family | Genus species | Form | Cell shape | Fractal dimension
---|---|---|---|---
Volvocaceae | Volvox globator | Multicellular | Spherical | 1.2288
Volvocaceae | Volvox aureus | Multicellular | Spherical | 1.3701
Volvocaceae | Eudorina elegans | Colonial | Spherical | 1.6975
Scenedesmaceae | Scenedesmus granulatus | Colonial | Ellipsoid | 1.7097
|  |  | Crescent |
Scenedesmaceae | Desmodesmus magnus | Colonial | Ellipsoid | 1.7447
Hydrodictyaceae | Tetraedron minimum | Colonial | Ellipsoid | 1.7087
|  |  | Spherical |
Hydrodictyaceae | Pediastrum clothratum | Colonial | Oval | 1.7182
Hydrodictyaceae | Pediastrum angulosum | Colonial | Cylindrical | 1.7806

##  2 Reduced fractals in biology

As first pointed out by Mandelbrot,1 most natural structures do not show self-
similarity at infinitely many stages, as classical fractals do, but instead
their self-similarity occurs only at a finite number of stages. Moreover there
can be imprefections in self-similarity resulting from the fact that a smaller
cluster is unlikley to be exactly the same as a larger cluster, in other
words, self-similarity is only approximate.3-6 If there are variations in
minature copies, then self-similarity is statistical;2,3 however, if minature
copies are distorted (skewed), then self-similarity becomes self-afinity.2
This shows that in natural structures the range of magnification is also
finite.4

The presence of these limitations leads us to believe that it is necessary to
provide an integrated framework towards a definition of fractals applicable to
natural structures, and such a framework is established in this paper by
introducing the concepts of reduced fractals and their corresponding, spectra
which for practical reasons must be discrete. Reduced fractals considered in
this paper have all four basic characteristics described in Section 1 but two
of them, namely, self-similarity and range of magnification are finite; this
makes our definition consistent with practical fractals introduced earlier2
but different than the concept of semi-fractal.23 Our definition allows also
for wild fractal or fractals whose self-similarity is limited to 1 or 2
stages, nevertheless, some scaling properties of these fractals can still be
identified.4 In the following, we identify reduced fractals in selected
biological systems.

##  3 Applications to biology

###  3.1 Selected biological systems and their self-similarity

As examples of biological systems considered in his paper, we select green
algae (Division: Charophyta and Chlorophyta), and consider two classes of
Charophyta, namely, Charophyceae and Zygnematophyceae, and three classes of
Chlorophyta, namely, Chlorophyceae, Ulvophyceae and Trebouxiophyceae.26,27 In
Fig. 1, we present 9 selected algae of the class Zygnematophyceae, and Fig. 2
shows 8 selected algae of Chlorophyceae; algae of other classes are used in
spectra of reduced fractals described in Sec. 3.3. As shown by the images
presented in Figs. 1 and 2, we consider only green algae that habitat mainly
in freshwater, but five different classes, with each class containing algae of
different sizes, shapes and nature. The considered sample of algae is
diversified and rich in its structure, and thus it is sufficient to illustrate
all main concepts and objectives of this paper.

All considered algae of the class Zygnematophyceae are unicellular and they
belong to the family Desmidiaceae; each one of them splits into 2 parts that
have perfect symmetry; however, if we scale down, then they show no self-
similarity. Nevertheless, these algae are examples of reduced fractals defined
in Sec. 2. On the other hand, two selected algae of the class Chlorophyceae
that are multicellular show more prominent self-similarity than the
unicellular algae but still their self-similarity is limited, and their range
of magnification is finite, which means that these algae can also be
represented by reduced fractals.

The limits on self-similarity and on the range of magnification in the class
Chlorophyceae of colonial algea are presented in Table 1, which contains algae
of different families, namely, Eudorina elegans belongs to the family
Volvocaceae, the next two algae belong to the family Scenedesmaceae, and the
remaining three algae are from the family Hydrodictyaceae. For the algae in
Table 1, it is seen that the replication of colony (shown by magnification) is
an indicator of limited self-similarity. Since both self-similarity and the
range of magnification are finite, these algae can also be described by
reduced fractals.

The above description shows that the algae selected for this paper have self-
similarites ranging from very limited, as is in the case of the unicellular
algae of the class Zygnematophyceae, to more moderate, as observed in the
multicellular algae of the same class and in the colonial algae of the class
Chlorophyceae; similar limitations and variations are observed in the range of
magnification. The observed self-similarity is limited and it is not perfect
as there are variations in minature copies, so self-similarity observed in the
selected algae is limited and statistical. For these reasons, the algae
considered in this paper are well represented by reduced fractals.

###  3.2 Fractal dimension by box-counting method

The sample of selected algae allowed us to identify reduced fractals as the
best way to represent them and describe their physical properties; one such
property is irregularity, or complexity, of their surface and structure.
According to Mandelbrot,1 the complexity can be measured by the so-called
fractal dimension, which can be identified with either self-similarity
dimension, or compass dimension, or box-counting dimension, with all of them
being special forms of Mandelbrot’s fractal dimension.2-4 It is well-known
that different dimensions may give different results, and that the box-
counting dimension is the most commonly used in scientific applications
because it can be applied to objects that do not show self-similarity but only
some scaling.4

Since in this paper, we consider some algae that show very limited self-
similarity, all results presented below are obtained by perfoming calculations
of fractal dimension by using the box-counting method. The computed fractal
dimension by this method is a metric that characterizes algea complexity or
space-filling characteristic. As already pointed out,4,28 most previous
studies failed to evaluate the assumption of statistical self-similarity that
underlies the validity of the method. Another source of error is arbitrary
grid placement, which is strictly positive and varies as a function of scale,
which may make the procedure’s slope estimation step non unique.4 In our
calculations performed in this paper both errors were eliminated.

The results presented in Table 2 show that the unicellular algae of the the
class Zygnematophyceae and the family Desmidiaceae have high fractal
dimensions ranging approximately from $1.8$ to $1.9$; this narrow range of
fractal dimensions implies that the level of complexity (or irregularity) of
surfaces and structures is very similar for all these selected objects. The
results are also consistent with the fact that the unicellular algae have very
limited range of self-similarity and magnification that are observed in these
objects as already is pointed out in Section 3.1. It must be also noted that
the unicellular algae in Table 2 are separated into two groups called here
Euastrum and Micrasterias, and that within each group algae are ordered based
on their increasing fractal dimension. Since all algea shown in Table 2 belong
to the same family Desmidiaceae, similar ordering can be made for other
familes of green algae.

Fractal dimension in Table 3 is computed for multicellular and colonial algea
of three families (Volvocaceae, Scenedesmaceae and Hydrodictyaceae) of the
class Chlorophyceae, and that the results are presented for for the selected
members of each class, which are ordered based on their increasing fractal
dimension. The presented results show that fractal dimension of the two
multicellular algae (Volvox globator and Volvox aureus) are the lowest, which
means that complexity of these two algae is the highest among the selected
objects. Interestingly, the fractal dimension of Eudorina elegans is more
similar to one member of the family Scenedesmaceae and one member of the
family Hydrodictyaceae, which is caused by the fact that these algae are
colonial despite being members of different familes.

Similarly, fractal dimension of Tetraedron minimum, which belongs to the
family Hydrodictyaceae is closer to that of the members of the family
Scenedesmaceae. The reason is likely caused by the fact that these algae are
colonial but also by similarities in their cell shapes. However, the two
remaining members of the family Hydrodictyaceae have significantly higher
fractal dimensions that may be caused by both different cell and colony shapes
between Tetraedron minimum and these two members.

The results presented in Tables 2 and 3 demonstrate that algae can be ordered
within each family by using their fractal dimension. This may be useful in
classification of algae and their studies, since typically algae within a
given family are neither ordered nor organized. 29 The proposed order of
decreasing complexity (see Tables 2 and 3) can be replaced by increasing
complexity, which would require to reverse orders in Tables 2 and 3.

Figure 3: Spectra of reduced fractals for algae of differennt form (panel A),
class (panel B), shape (panel C), family (panel D).

###  3.3 Spectra of reduced fractals and their implications

The fractal dimensions calculated for different green algae can now be used to
obtain spectra of reduced fractals (SRFs). The spectra are generated by
plotting the fractal dimensions versus selected characteristics of algae.
Among different characteristics, we consider forms and shapes of algae
described in Sec. 3.1. Moreover, we also demonstrate how to generate SRFs for
algae of different classes and families. In the panels A, B, C and D of Fig.
3, we show the SRFs for algae of different forms, classes, shapes and
families. Let us now describe each panel of Fig. 3 and discuss biological
implications of the presented SRFs

The panel A of Fig. 3 shows the form of the SRF for one unicellular alga of
the class Zygnematophyceae and the family Desmidiaceae, and one colonial and
one multicellular algae of the class Chlorophyceae, whose families are
Scenedesmaceae and Volvocaceae, respectively. As expected, the presented SRF
demonstrates a rapid decrease of fractal dimension with increasing algae’s
complexity. The observed almost linear spectrum allows us to establish the
following criteria for classifying green algae: $F_{d}>1.8$ for unicellular,
$F_{d}>1.6$ for colonial, and $F_{d}<1.4$ for multicellular, where $F_{d}$ is
the fractal dimension.

The SRF presented in the panel B of Fig. 3 shows high fractal dimensions
$F_{d}>1.7$ for all considered algae, which are two algae of the class
Charophyceae, one alga of the class Ulvophyceae, and two algae of the class
Trebouxiophyceae. The resulting spectrum shows that differences between
freshwater and saltwater (or land) within the same class are practically
negligible, which is shown by flat parts of the SRF. Moreover, the entire
spectrum remains practically flat, which implies that differences in the
fractal dimension (or complexity) between the considered three classes are
small.

The fact that algae have different shapes is well-known.26,27 Using several
most commnoly known shapes of green algae (see also Figs. 1 and 2, and Table
3), we obtained the RFS shown in the panel C of Fig. 3. The spectrum shows its
maximum for cylindrical algae, and smaller values of the fractal dimension for
oval and ellipsoid algae; both oval and ellipsoid algae have comparable
fractal dimensions. An interesting result is that spherical algae have the
lowest fractal dimensions, which implies that their complexity is the heighest
among algae considered for this spectrum.

The panel D of Fig. 3 shows the SRF for different familes. The presented
results are consistent with the SRFs given in the previous panels,
specifically, in the panel A, as the SRF of the panel D mainly reflects
differences between algae being unicellular (Desmidiaceae), colonial
(Hydrodictyaceae and Scenedesmaceae) or multicellular (Volvocaceae). The
spectrum demonstrates (similar to the SRF of the panel A) a rapid decrease in
the fractal dimension, and related increase in algae complexity, for colonial
algae as compared to unicellular ones. Even sharper decrease in the fractal
dimension is observed for multicellular algae as compared to colonial. As
expected, there is only a small difference in the fractal dimension between
the families Hydrodictyaceae and Scenedesmaceae as members of these families
are mainly colonial (see Table 3).

The above results demonstrate that the SRFs can be used as a new tool to
investigate properties of green algea and also to classify them within their
families as currently algae of a given family are neither ordered nor
organized. 26,27 Moreover, SRFs can also be generated for other types of
algae, namely, Macroalgae (red and brown) or Microalgae26,27 as well as for
other biological systems and natural science systems. We do hope that
biologists and other natural scientists find the SRFs useful in their work and
apply them to different natural systems.

##  4 Conclusions

The concept of reduced fractals, with specific limited scale range and only
partial self-similarity, is introduced and used to generate spectra of reduced
fractals. To demonstrate applicability of these spectra to biology, the
spectra are generated for selected green algae, which include Charophyta and
Chlorophyta algae, and their classes Charophyceae, Zygnematophyceae,
Chlorophyceae, Ulvophyceae and Trebouxiophyceae. By showing how these spectra
can be used to investigate physical properties of algae and to classify them
within their familes, we hope that the spectra will become a new tool to study
algea, including also red and brown algae, as well as microalgae. It is also
suggested that the spectra can be used for other biological systems and that
they may provide biologists a tool to brigde over to physics, electro-sensory
artificial life and synthetic biology. The spectra may also become a useful
tool in other natural sciences.

References

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