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Rivista di Biologia / Biology Forum90 (1997) , pp. 227-266.
Giuseppe Damiani and Paola Della Franca
Morph and Evolution
1. Introduction2. Evolutions Evolution3. Order, Fractals and Chaos4. Binary Genesis and Metabolic Hypercycles5. Cellular Automata and Morphogenesis6. Evolution as Optimization of Binary Processes According to the Principle of Parsimonia7. Conclusion
Key words: Hypercycle, fractal, thermodynamics, morphogenesis, metabolism.
Abstract. The Universe might be originated from two vacuum fluctuations with different size, emerged
from a fractal-chaotic distribution of fluctuating vacuum sets. A phase transition towards an isotropic
state between the macroscopic and the microscopic fluctuations might produce two opposite trends: an
expansive catabolic process of binary dissociation, and a complementary contractive anabolic process of
binary association. The relation between these two processes, described by the prey-predator model of
Lotka-Volterra, might be the basic mechanism of self-organisation phenomena, morphogenesis and
biological evolution. The equilibrium between the opposite binary processes leads to the formation of
fractal patterns which increase their complexity during the phylogenetic and ontogenetic development.
These fractal structures are typical of the most adaptable homeodynamical systems working at the
border between order and disorder.
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1. Introduction
Not content with an answer to the question How is it? we wish to know How did it come to be so?.
Man, wherever he awakens to ponder the riddles of existence, is prone to expect evolution to enlighten
him about the essence of things (Weyl, 1949).
With these words the German scientist Hermann Weyl began the appendix entitled The Main
Features of the Physical World; Morph and Evolution in his book Philosophy of Mathematics and
Natural Science (Weyl, 1949). Fifty years after Weyl, many of the deep problems analysed in his book are
still open questions, despite the impressive development of the scientific knowledge. As stated by the
publicity of an important scientific journal more information has been produced in the last 30 years than
in the previous 5000 years, but the search for a unitary scientific view of the world, for a theory of
everything, is still unsuccessful. History of science shows that different philosophical and methodological
premises result in different "ways of seeing" the world around us, which are often contrasting but at the
same time complementary. This is especially true in modern biology.
The formation of structures and shapes in living organisms and the origin of life on the Earth give
rise to many controversial explanations (Rieppel, 1988). Some of these dichotomies regard the
methodological aspects, others the substance of different theories. What is the intimate nature of physical
entities and processes? Are the natural structures and patterns parts of a single continuous entity or many
well separated individual entities? (continuity versus discontinuity). Is the biological evolution due to
gradual transformation or to rapid changes? (gradual evolution versus punctuated equilibria). Can a living
organism be considered as a functionally integrated continuos whole or it can be reduced to its discrete
constituent elements? (holism versus reductionism). What has generated the structures and patterns of our
Universe? Simple basic laws or complex transformations? (being versus becoming). Invariant symmetries
or random fluctuations? (order versus disorder). Is life the simple result of the well-known chemio-
physical laws or a manifestation of an unknown vital force? (materialism versus vitalism). Are there
real ortogenetic trends or not? Is evolution driven by a precise plane or by a blind chance? (finalism
versus casualness). Is the phylogenetic and ontogenetic complexification of living organisms the result of
an entropic or of a neg-entropic phenomenon? (entropy versus neg-entropy). Are internal ontogenetic
constraints more important than external selection for evolution of living organism? (structuralism versus
functionalism). Are evolution mainly due to the struggle for the life or to symbiotic association?
(competition versus collaboration).
A brief evaluation of historic development of these antithetical explanations could be very useful to
highlight some aspects of the evolutionary processes.
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2. Evolutions evolution
Evolution is not the foundation but the keystone in the edifice of scientific knowledge (Weyl, 1949).
The idea of evolution plays a predominant role in many cosmogonies and mythologies of primitive
cultures. In almost all ancient creation accounts, the biblical one included, the Universe was thought as
the progressive unfolding of an original primitive and mysterious entity. This fundamental entity was
without a defined form and was mysteriously related to the concepts of vacuum, nothing and chaos. A
creative process generated all the ordered and complex things of our Universe out of the very simple pre-
existing chaotic substratum. Since earliest antiquity, Egyptian, Hindu, Taoist, Hermetic and Greek
(Empedocle, Diodoro Siculo) philosophers have described the process of evolution of living organism by
means of natural selection. In many of the ancient cosmogonies, there is not qualitative difference
between matter and spirit, between organic and inorganic entities, between man and other creatures. In
the Timaeus, Plato wrote that ...this world is indeed a living being endowed with soul and
intelligence...containing inside itself all the other living beings.... For Plato the Universe is basically
simple and the forms of natural objects are the imperfect materialisation of geometric invariant
archetypes. Aristotle developed the first extensive classification system of living beings, according to the
Platos idea of a scale of nature, an interrupted series of organic forms, extending from the simplest to the
most perfect. In hisDe Animalibus Historia he wrote that Nature progresses so gradually from inanimate
things to animate beings, that because of continuity one cannot decide where the boundary between the
two division should be drawn....
The Christian Church accepted the Plato and Aristotle idea of the scale of nature but developed a
dogmatic and restrictive literally interpretation of the Genesis account of creation. The physical world,
each of the animal species, and even human beings were produced at the moment of the creation by the
direct intervention of God. This thesis, called creationism, led to a strong opposition to the rise of the
modern scientific view of the world. Unfortunately, creationism is a very common idea even in the
present time.
The mechanicistic explanation of the physical world developed by Galileo, Descartes, Newton and
many others, disturbed the traditionalists. The peaceful coexistence of Catholic religion and science was
permitted by a dualistic view of the world: on one hand the matter governed by deterministic physical
laws, on the other the spirit with different unknown laws.
Down to the beginning of the nineteenth century, the concept of the great chain of being, was
accepted by the great taxonomists and morphologists Linnaeus, Buffon, Bonnet, Cuvier, Owen, and
Geoffry St. Hilarie. They believed that there are laws of biological form and organisation; and they
provided much evidence from their study of taxonomy and comparative morphology to support this
belief. Among the conceptual tools, which they forged from their empirical studies there, are the notion of
typical form and the concept of homology. The scala naturae reflects the needs for a unique Creator:
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The unity of design reveals the unity of the intelligence which has conceived it. The harmony in the
Universe, or the affinities among the diverse contents of this vast building, prove that its cause is one...
The same general design embraces all parts of the Creation. A globule of light, a molecule of earth, a
grain of salt, a mould fungus, a polyp, a mollusc, a bird, a mammal, man are nothing but different
expressions of this design which represents all possible modification of matter on our globe... (Bonnet,
1764). The theological background and the social contest of these scientists have limited the development
of their evolutionary idea.
The emergence of the evolutionary theory broke the unity between religion and science. Pierre Louis
Moreau de Maupertuis was perhaps one of the first scientists of the modern time to propose an
evolutionary theory. Jean Baptiste Lamarck developed an evolutionary theory based on relationships
between environment and organisms and on the inheritance of acquired characters. In 1858 Darwin and
Wallace presented simultaneously the same theory of evolution, discovered independently. The neo-
Darwinian theory of evolution states that the natural selection of random hereditary variations is both
necessary and sufficient to explain evolution. An impressive amount of evidences of evolutionary
processes is furnished by palaeontology, comparative anatomy, embryology, genetics, biochemistry and
molecular biology. As Dobzhansky said, nothing in biology makes sense except in the light of
evolution.
The evolutionary theory explains forms of living organisms mainly as adaptation produced by natural
selection. Biotic and physical factors present in the environment are responsible for selective reproduction
of living organisms. Selection is cumulative: each generation is built on the successful adaptations of the
former generation. Therefore a structural and functional increasing optimisation of biological structures is
achieved by evolutionary processes. The relationship between form and function supports the
adaptationist explanation for the biological shapes. For D'Arcy W. Thompson (1917) the forms of living
organism are determined mainly by physical laws: forms are diagrams of forces. However, forms are not
only moulded by functional selection and external physical forces, but are subjected to genetic, and
ontogenetic internal constraints, too (Russel, 1916). If functionalism presupposes the primacy of function
over form, on the other hand, structuralism emphasises the primacy of form over function. Unknown
internal factors might be more important than adaptation to the external selection (Gould et al., 1979).
Only few morphological transformations are permitted by the available genetic information and
developmental pathway. Only few types of genomic reorganisation are viable and economic solutions.
Moreover, relationships between different structures of an organism tend to vary during its growth. These
genetic and structural constraints limit the number of the possible morphogenetic changes which are the
substrate for adaptive selection.
A compromise solution between the functionalistic and the structuralistic viewpoint was proposed:
environment may select form, but only within the limits of the types created by ontogeny (Alberch, 1980).
Are internal factors more important than external selection? The scientific community rejects a discussion
about this point for the main reason that the proponents of the internal evolutive forces are often vitalist,
as Driesch, or finalist, as Teilhard De Chardin, Fantappi and Salvatore Arcidiacono. The major efforts of
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many scientists is to destroy any theological finalistic interpretation of the evolutive process. For Stephen
Jay Gould and Niles Eldredge non-adaptive and random processes, acting by rapid differential origin and
extinction of species, direct evolutionary trends within clades (macroevolution), while natural selection,
acting by differential birth and death of individual, directs evolutionary change only within populations
(microevolution): "The geometry of punctuated equilibrium means that we cannot extrapolate natural
selection within populations to produce evolutionary trends because trends are a product of the
differential fate of species considered as stable entities. Trends may occur simply because some kinds of
species speciate more often than others, not because the morphologies so produced have any advantages
under natural selection (indeed, such a trend will occur even if extinction is completely random)." (Gould,
1982). Raup and Gould (1974) showed with stochastic simulations of evolutionary processes that many
morphological patterns of apparent order arise at unexpectedly high frequency in random models.
Recently, Gould has launched his attack to one of the most important law of the biological world, the law
of the increasing complexity of living organisms during the course of evolution: Our strong and biased
predilection for focusing on extremes (and misconstruing their trends as surrogates for a totality), rather
than documenting full ranges of variation, generates all manner of deep and stubborn errors. Most notable
of these misconceptions is the false and self-serving notion that evolution displays a central and general
thrust towards increasing complexity, when life, in fact, has been dominated by its persistent bacterial
mode for all 1.5 billion years of its history on Earth. (Gould, 1997). Fortunately our brain is more
complex than a bacterial one, and, therefore, we can read the nice book of John Tyler Bonner entitled
The evolution of complexity by means of natural selection (Bonner, 1988).
Is the great chain of beings a psychological artefact? Are macroevolutionary trends the result of
random fluctuations? Is life the result of a very improbable random event? For Boltzman the emergence
of life is a very improbable event, a local fluctuation at the edge of the Universe, which will disappear
without significance in the thermal death of the whole Universe. Also for Monod (1970) the emergence of
life on the heart is the fortuitous result of a blind chance: The Universe was not preparing the birth of
life, and the biosphere was not preparing the birth of man. Our number is come out at the roulette: why,
therefore, we do not understand the exceptionality of our conditions in the same way of a man which has
just won a billion? (Monod, 1970). For P.W. Atkins and several scientists the future Theory of
Everything will not leave room for the existence of God: The only way of explaining creation is to show
that the creator had absolutely no job at all to do, and so might as well not have existed. We can track
down the infinitely lazy creator, the creator totally free of any labour of creation, by resolving apparent
complexities into simplicities, and we hope to find a way of expressing, at the end of the journey, how a
non-existent creator can be allowed to evaporate into nothing and to disappear from the scene. (Atkins,
1981). The fight between science and religion is at the end: science has defeated religion. At least man is
free from theological influence; at least man knows to be alone in the indifferent immensity of the
Universe, from which he emerged at random. (Monod, 1970).
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3. Order, fractals and chaos
Knowledge of the laws and of the inner constitution of things must be far advanced before one may
hope to understand or hypothetically to reconstruct their genesis (Weyl, 1949).
As suggested by Weyl the analysis of the primary structures encountered by science in its search for
order and law is an important tool to understand the genesis of life and of biological forms. The aim of
scientists is to discover the regularities in patterns of natural phenomena. These regularities can be
described by natural laws that are expressed in mathematical terms:The book of Nature is written with
mathematical symbols. (Galileo Galilei). Euclidean geometry and analytical calculus, typically in the
form of linear differential equations, are the standard way of modelling the laws of nature. These
mathematical tools allowed the formulation of laws describing simple phenomena with repeatable
behaviour. For Dirac and many other scientists the beauty and elegance of a mathematical model are
important guides for the formulation of correct theories of Nature : It appears as one of the fundamental
principles of Nature that the equations expressing basic law should be invariant under the widest possible
group of transformations. (Dirac, 1973). Weyl (1919, 1949), Fantappi (1973), and Dirac (1938, 1973)
have studied the group of transformations and the corresponding geometric symmetries underlying
physical laws. They discovered that the ultimate natural laws are invariant under rotations and gauge
transformation. Today's best working descriptions of the known forces of nature (the so-called standard
model) are all based on gauge symmetries. To an imaginary observer inside the internal space of these
theories, the interaction between a particle and an external gauge field looks like a simple rotation.
The discoveries of the modern physicists confirm the Platonic assumption that invariant geometric
blueprints are the fundamental cause of natural forms. But, also for Plato, there is a difference between
the irregularities of the real objects and the regular unchanging blueprints that governed them. Our world
is full of complex dissipative systems with an irregular and unpredictable behaviour. The traditional
development of science has carefully avoided the chaotic and non-reproducible aspects of our reality.
With the increasing amount of human and financial resources devoted to the scientific research, science
has been fragmented into many different specialised fields. The reductionistic approach of traditional
physics and chemistry has explained the basic molecular mechanism of life but it is unable to explain the
non-linear behaviour of complex systems, such as biological ones. Biology is thus very different from
physics. The basic laws of physics can usually be expressed in exact mathematical terms, and they are
probably the same throughout the universe. The 'laws' of biology, by contrast, are often only broad
generalisations, since they describe rather elaborate chemical mechanisms that natural selection has
evolved over billions of years (Crick, 1988). Why has there been a progressive increase in complexity
during the evolution of life on the heart? Organic forms and structures do not evolve randomly but
converge towards a relatively small number of patterns (Hildebrant et al., 1985; Thomas et al., 1993).
What is the ordering principle, which Kauffman (1991, 1993) calls antichaos, leading to these patterns?
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Can science achieve a unified theory of both the simple physical world and the complex biological
systems?
Recent developments in fractal geometry and in many other interdisciplinary scientific areas
(cybernetics, information theory, analysis of complex and dissipative systems, emergence of auto-
organisation, chaos theory, and cellular automata) are creating new ways of thinking about Nature. The
Euclidean geometry and the differential equations are powerful analytical tools, but for understanding the
intimate nature of the reality, they turn out to be wrong kind of abstractions. Clouds are not spheres.
Mountains are not cones, coastlines are not circle, and bark is not smooth, nor does lightning travel in a
straight line. (Mandelbrot, 1982). Mandelbrot coined the word fractals, from the Latin fractus
meaning irregular, to describe these shapes that repeat themselves at many different sizes (scale
invariance). Fractals are everywhere in nature, in topics as diverse as noise transmission, price variation
in economics, galactic clustering and shapes of natural objects. The twigs of a tree have the same
structure as the trunk and branches of the tree itself. A piece of rock resembles the mountain from which
it has been broken. There are many objects with the same level of irregularity over a smaller and smaller
scale as you come closer to them. Computer simulations have opened the exploration of fractal-irregular
shapes that defy conventional geometry. They have shown that iteration of simple rules may produce
fractal structures. Moreover an interesting relation between fractals and chaos was discovered. Computer
simulations of iterative processes have shown how chaos may arise in a deterministic system. Chaotic
behaviour is produced by sensitivity to initial conditions: trajectories starting very close together, diverge
rapidly. Even if real systems are controlled by a set of completely deterministic rules, their behaviours
may be unpredictable because real measurements are not precise enough to distinguish nearby trajectories
that may later diverge.
Surprisingly, as discovered by Feigenbaum, the chaotic behaviour has a fractal form: many different
systems show the same universal properties in the scenario of trajectories bifurcations. Why the transition
to chaos has a fractal shape? The study of complex systems, has shown that simple interactions
sometimes lead to very complicated structures and behaviours (May, 1976). Conversely, complex
interactions can lead to simple results. This is particularly true for systems formed of many interacting
subunits, such as liquids, solids or gases, which are poised at a 'critical point' where two or more
macroscopic phases become indistinguishable. At a critical point, many of the precise details of the
interactions between constituent subunits play virtually no role whatsoever in determining the bulk
properties of the system. To understand these surprising phenomena (known as 'critical' phenomena), 25
years ago scientists developed the twin concepts of 'scaling' and 'universality'. Systems near critical points
exhibit self-similar properties; thus, to some degree, they are invariant under transformations of scale.
This is the property of scaling. The word universality connotes the fact that quite disparate systems
behave in a remarkably similar fashion near their respective critical points.... (Stanley, 1996). This
citation explains in a clear way one of the most universal and mysterious law underlying the form of each
transformation happening in our Universe: each phase transition, at its critical point, has a fractal-chaotic
pattern. The characteristic behaviour of real physical systems at the critical points is a mix of regularity
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and irregularity, of order and disorder. This is true at all levels, from the micro-world of quantum
mechanics to the macro-world of cosmology. Fractal-chaos, at the border of the phase transition between
order and chaos, is everywhere. The scale invariance at the critical points is described by simple power-
law correlations between the different elements of the systems, because there is no characteristic scale
associated with a power law. The range of systems that apparently display power-law correlation has
increased dramatically in recent years, ranging from base-pair correlations in DNA, lung inflation and
interbeat intervals of the human heart, to complex systems involving large numbers of interacting
subunits that display 'free will', such as city growths and even economics. (Stanley, 1996).
A wide range of natural structures has been quantitatively characterised using the idea of a fractal
dimension. In particular many biological structures have fractal, branching and vascular morphologies:
trees, venation of leaves, lungs, liver, kidney and in general most circulatory and transport systems in
living organisms, suture lines of ammonoids, aggregated structures produced by myxobacteria and
myxomyceta, different types of cells (neurones, lymphocytes, cromatophores, ecc...) and many others
(Weibel, 1994). The description of biological objects by means of fractal geometry is an useful tool in
many biological and medical applications, as for example in tumor diagnosis (Losa et al., 1996). The
Mandelbrot geometry of Nature provided an indispensable language and a catalogue of pictures of
nature. As Mandelbrot himself acknowledged, his program described better than it explained. He could
list elements of nature along with their fractal dimensions - seacoasts, river networks, tree bark, galaxies -
and scientists could use those to make predictions. But physicists wanted to know more. They wanted to
know why. There were forms in nature - not visible forms, but shapes embedded in the fabric of motion -
waiting to be revealed. (Gleick, 1987).
4. Binary genesis and metabolic hypercycles
For the moment we can say no more that the construction of the world seems to be based on two
pure number, 1 / 137 and z 11041
, whose mystery we have not yet penetrated(Weyl, 1949).
Cosmologists suggested that our Universe was born out of a critical and unstable state of the vacuum
by a phase-transition process analogous to the formation of bubbles in boiling water. For an intuitive
description of the evolution of this process we use a scenario constituted by a Euclidean pluridimensional
"superspace". A limited set of the superspace is a quantised vacuum set. The vacuum is the most simplephysical entity. Experimental evidences, as for example the Casimir effect, show that the vacuum is not
empty but is filled with randomly fluctuating fields having the zero-point isotropic spectrum. The
intensity of this spectrum at any frequency is described by a cubic curve, which is shape-invariant, while
the magnitude of the intensity depends on Planck's constant (Boyer, 1985). One may suggests that the
value of the Plank constant could be a "frozen accident," a scale factor, characteristic of the microscopic
vacuum fluctuations. Several researchers have suggested that the entire Universe may be a vacuum
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fluctuation according to quantum field theory (Tryon, 1973). If the Universe is a vacuum fluctuation, then
its size might be another "frozen accident," a scale factor characteristic of the macroscopic vacuum set.
As noted by Weyl (1919, 1949), Eddington (1946), Dirac (1938, 1973), Recami (1992, 1994),
Arcidiacono (1979), Caldirola (1978a, 1978b), De Sabbata (1979, 1980) and many others, our Universe is
characterised by two space-time metrics, one at the microscopic atomic level and the other at the
macroscopic cosmological level. If these two scale factors have not a special state, then the superspace is
filled with a self-similar fractal-chaotic distribution of shape-invariant and scale-invariant fluctuating
vacuum sets, without any privileged frame or scale of reference.
According to this hypothesis, a Binary theory, based on a simple model of vacuum decay, may be
developed. This model allows a mathematical, geometrical and graphical description of the process of
vacuum fluctuation and can be simulated by simple computer programs. In addition to the properties of
the Euclidean superspace, three simple principles are needed to explain the process of vacuum decay: the
principle of conservation, the principle of action and reaction, and the principle of parsimony. The first
two principles are a generalisation of the analogous principles of the Newtonian dynamics, and the last is
similar to the Maupertuis principle of minimal action. The principle of conservation dictates that the
superspace and its limited subsets are quantitatively invariant: only complementary and opposite entities
and processes can come into existence, according to the principle of action and reaction. The principle of
parsimony states that among several possible solutions, the most economical one is selected. According
to this principle, the probabilistic process of vacuum fluctuation is directed by the minimisation of its
trajectory in the superspace. Therefore, the simplest vacuum fluctuations are monodimensional sets. An
emergent expanding fluctuation is connected in its central region to a complementary contracting
fluctuation in the perpendicular dimension of the superspace. The reaction of the superspace leads to the
annihilation of an emergent pair of fluctuations (Fig. 1).
The existence of a pair of macroscopic fluctuations may be stabilised by a resonant tunnelling with a
complementary pair of fluctuations at a microscopic scale (see Fig.1 and 2). Each monodimensional
fluctuation has two linear gradients of "stresses" with the maximum of intensity at the two extremities.
These "stresses" may be released by a rotation, which produces a polarisation of the vacuum set in two
symmetric entities. The behaviour of the polarizable vacuum is analogous to that of an electroreologic
fluid between two opposite electric charges: two "repulsive" regions diverge and two "attractive" regions
converge towards a central point. This situation may induce the emergence of an internal pair of
microscopic fluctuations. The two hierarchies of vacuum fluctuations are different for a scale factor z.
The polarisation of the microscopic fluctuation leads to its rotation opposite to that of the macroscopic
fluctuations. When the two fluctuations are perpendicular, the two opposite poles of the microscopic
fluctuation move towards the external part of the repulsive regions, while the two poles of the
macroscopic fluctuations move towards the innermost part of the attractive region. The result is the
replication of the microscopic rotating entities.
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Fig. 1 -A model of the morphogenesis of our Universe from a fractal-chaotic vacuum. The unstable equilibrium
state of the vacuum is represented as a scale-invariant bundle of bubbles (a). Each bubble is a different vacuum
set with different size. A schematic view of the process of vacuum decay for a single bubble is shown. The
perpendicular bars represent two perpendicular interconnected vacuum fluctuations, the circle indicates the
horizon of events of the bubble universe and the white and black arrows show respectively the directions of the
expansion or contraction of the fluctuations. A pair of monodimensional vacuum fluctuations starts the decay
process (b). The black fluctuation is lifted out, thus leaving a hole in the perpendicular white fluctuation. The
reaction from the disturbed superspace tends to restore the starting equilibrium state. The stresses produced by
the superspace reaction may induce a polarization of the fluctuations (c). The positively and the negatively chargedextremities of the black fluctuation are indicated respectively by the circles with the plus and minus signs. The
white diagonal lines delimit the attractive and repulsive regions produced by the polarization. The polarization of
the fluctuations may induce the formation of an opposite pair of polarized fluctuation at a smaller scale (d). A
metabolic hypercyclic relation between the macroscopic and the microscopic fluctuations leads to the formation of
two complementary coupled sets of expanding and contracting vascular network of vortices. The dynamics of the
rotating fluctuations is the same of a photon model developed by M.H. Mac Gregor (1995).
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Iteration of this process, called binary microscopic dissociation, at each breaking point of the new
generation produces a Peanian network of vortices with a geometry similar to those of the Tartaglia-
Pascal triangle. After n generations, the numberNof the microscopic entities increases exponentially with
a power law (N= 2n). The expansion process is analogous to that of a fluid that breaks into a many-
branched fractal and vascular structure when forced under pressure into another immiscible fluid of
higher viscosity. The branching process is coupled to an opposite process, called binary macroscopic
association. The macroscopic vacuum set can be considered an incompressible fluid, which breaks up
during the binary process like a Cantor dust. The "size" of each macroscopic vortex, connected with all
the others N replicating microscopic vortices, is inversely proportional to N (1/N= 2-n). The feedback
relation between the two opposite binary processes is a particular type of hypercycle (Eigen, 1979), called
metabolic hypercycle, which can be described by a simplified version of the Lotka-Volterra equation
(Fig.2).
Fig. 2 - The metabolic hypercycle. A simple feedback hypercyclic relation between the X prey and the Y
predator is shown. The autocatalic anabolic growth of X (binary dissociation) is symbolized by an expanding
branching structure while the autocatalitic catabolic decay of Y (binary association) is represented by a
contracting branching structure. Below the hypercycle is reported a simplified version of the Lotka-Volterra
equations describing the complementary variations of X and Y. This hypercycle give rise to ordered spatio-
temporal patterns with a dynamic behaviour characterized by an oscillating rhythmicity.
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When the expansion front of the microscopic entities reaches the macroscopic horizon of events,the
generated structures bounce back, and the inverse processes of binary microscopic association and binary
macroscopic dissociation take place. At the end of these processes the starting conditions are restored and
the vacuum fluctuations can disappear.
The process of binary dissociation leads to an isotropic spread of the microscopic entities in the
maximum available region of the superspace around the breaking point. On the contrary, the
complementary process of binary association limits the surface and the volume of the expanding
structures. The four-dimensional hypersphere is the isotropic minimal boundary region containing the
maximum amount of Euclidean superspace. A cosmological model of a constant curved four-dimensional
space-time was developed by De Sitter and is related to a group discovered by L. Fantappi (1973). This
group describes the possible rotational self-movements of a four-dimensional space-time and is the base
for the development of the final relativity, or projective relativity (Arcidiacono, 1979; Caldirola et al.
1978a). The four dimensional vacuum decay produces four different interconnected expansion and
contraction fields which are related to the gravitational (FG)and electric (FQ) forces at the macroscopic
level, and to the strong (FC) and weak (FH) forces at the microscopic level (Damiani, 1997). The
quantistic microscopic vortices corresponding to the gravitational, electric, strong and weak charges are
the fundamental components of the space-time and of the microscopic particles (Harari, 1979; Shupe,
1979). All the fundamental mathematical relations and numerical coincidences of classic and quantum
physics can be deduced from the geometric structure of the proposed mechanical model, the length units
of the microscopic and the macroscopic fluctuations, and the present status of the contraction and
expansion processes. As proposed by Weyl (1919; 1949) the Compton radius of an electron
re 3.86 10
-13m
is the fundamental unit of a projective spatial measurement of the microscopic entities. The light velocity
c links the unit of space to the rotational unit of the temporal quanta te = 2 re /3 c 2.73 10-22
[sec]. Many mysterious physical constants are only conversion factors due to arbitrariness of the chosen
units of measure. Starting from these quantistic values of space and time and the geometric relationships
of Fig. 2, we can calculate the values of the quantistic unit of mass me = te2
/ 2 re 9.1110-31
[kg] ( me
is the electron mass ), angular momentumJe = te re = h /2
1.05 10
-34
[kg m sec
-1
] (h is the Plankconstant), energy Ee = me c
2 3 / 2 = re 3.86 10-13[kg m2 sec-2], and thermodynamical entropy Se
=Ee/ T k loge2 [kg m2 sec-2 K-1] ( Tis the temperature, kis the Boltzmann constant ). As time goes
by, the strength of the forces and of the charges increases or decreases according to the actual number n of
cyclic iterations of the binary processes and to the number z which is the scale factor between the size of
the microscopic and the macroscopic fluctuation. The strengths of the forces are correlated, in a
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complementary way, to the radii of the horizons of events, which are the borders of expanding or
contracting binary processes:
FG rG = FQ rQ = FH rH= FC rC = re
In these equations, rG = re n is the radius of the visible part of the Universe corresponding to about 137
times its present Schwarzschild radius, rE= re z / n is the radius of the hydrogen atom, rC= re n /z is
about the Compton radius of the nucleon, and rH= re /n is approximately 137 times the Schwarzschild
radius of the nucleon. Comparing these equations with experimental data, it is possible to calculate the
approximate values ofn and z:
n 11039
z 11041
where = n /z 1 / 137 is the fine structure constant and the ratio between the present age of our
Universe and its total length of life. According to Dirac (1937) all the very large dimensionless numbers
occuring in Nature are simple powers of the epoch, with coefficients of the order unity.
The hypotyzed variations furnish a rationale for the numerical and behavioural coincidence between
the quantistic world of elementary particles and cosmology discovered by Weyl (1919, 1949), Eddington
(1946), Dirac (1938, 1973), and many others, for the geometric meaning of the Maxwell equations
(describing both retarded diffusing waves and anticipated contracting waves), for the physical meaning of
the Schrodinger equations, for the dualistic corpuscular (in the contraction phase) and wavelike (in the
expansion phase) nature of elementary particles considered as microuniverses, for the Caldirola quantumtheory of the radiating electron (Caldirola, 1978b), for the analogy between the gravitational and strong
force (Caldirola et al., 1978a; Recami et al., 1992,1994), and for the relation between the mechanical and
the magnetical spin of the particles and that of the astronomical objects (de Sabbata et al., 1979, 1980).
The proposed model agrees with the fundamental mathematical relations and numerical coincidences of
classic and quantum physics, with the suggested geometrodynamical origin of physical entities (de
Sabbata et al., 1995), with the thermodynamical description of dissipative systems (Prigogine, 1979), with
the Kibble-Zurek description of phase transition in superfluid Helium (Zurek, 1996), and with the fractal
structure of mass cluster (Sylos Labini et al., 1996). More technical and precise descriptions of different
specific aspects of the described processes are available in the quoted articles and in the referencestherein.
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5. Cellular automata and morphogenesis
The morphological laws are today understood in terms of atomic dynamics : if equal atoms exert
forces upon each other that make possible a definite stable state of equilibrium for the atomic ensemble,
then the atoms will of necessity arrange themselves in a regular system of points (Weyl, 1949).
Some fundamental aspects of the hypothesized binary processes can be investigated through simple
computer simulation based on cellular automata (Fig. 3).
Fig. 3 -An Ising model of a toy universe. From left to right: a renormalized cellular automaton model of the toyuniverse (only the macro and micro levels are shown), a more detailed version of it (the hypermicro level is
shown, too), and a graphic representation of the evolution of the stress distribution in this structure (these
patterns are analogous to the basic morphology of ammonoid septa). A big macro-cell is composed by small
micro-cells, which are composed by smaller hypermicro-cells. The cells can be either receptive (white cells) or
active (black cells). The temporal evolution of the toy universe is from top to bottom: the initial, the intermediate
and the final steps of a diffusive process of binary dissociation are shown. The information required to specify the
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distribution of the active cells is related to the number of receptive and active cells. The information in the initial
condition about the state of the macro-cell is converted into information about the state of the micro-cells. The
evolution of the expanding toy Universe involves three separate steps: an initial step characterised by macroscopic
order and microscopic disorder, an intermediate step of fractal-chaos, and a final step of thermodynamic-chaos
characterised by macroscopic disorder and microscopic order. For the description of a contraction process is
sufficient to revert the temporal order.
A cellular automaton is a set of elements (the cells) that can interact and change their states
generation after generation (von Neumann, 1966). The evolution of a cellular automaton depends on the
initial configuration of the cells and on the rules for the calculation of the next state of each cell in each
generation (Wolfran, 1984; 1986). Iteration of simple rules may produce very complex structures. Many
cellular automata are able to produce different kinds of many-branched, fractal structures. A general
characteristic of these programmes is that when uniformly distributed entities are diffused or concentrated
by a repulsive or attractive force, they respectively produce divergent or contracting fractal and vascular
structures. For example the dielectric breakdown model (DBM) simulates a diffusion process with tipsplitting (Niemeyer et al., 1984) and the diffusion limited aggregation (DLA) describe the fractal growth
by means of random aggregation (Witten et al., 1981).
A simple Ising model based on cellular automata of the microscopic binary dissociation process in a
bidimensional toy Universe is shown in Fig. 3. A large cell, representing the macroscopic fluctuation, is
divided intoz small microscopic cells, representing the microscopic fluctuations. Only two states of the
cells are possible: the active black state, and the receptive white state. An active cell induces the adjacent
receptive cell in the active state. The result is a diffusion process that leads to a complexification of the
system. According to the theory of information, the degree of complexity of our toy Universe from the
point of view of the active cells, can be measured by the entropy S = n, or by the configurationalinformation, negative entropy, I=z - n (Layzer, 1975). The entropy is a measure of the systems disorder,
while the information is a measure of its order. They are coupled by a simple conservation law S+I =z .
At each activation of a single cell, a bit of entropy is created and a bit of information is destroyed. The
system starts with a minimum of entropy and a maximum of information, then it goes through an
intermediate stage of fractal-chaos, and it ends with a maximum of entropy and a minimum of
information. Dividing each microscopic cell into smaller cells, one can show that the initial information
about the macroscopic state of the system may be converted into information about the microscopic state
if a process of hypermicroscopic binary association is at work inside the microscopic cell. Therefore, in a
closed toy Universe, the variations of entropy and information at the macroscopic level arecomplementary and these variations may be balanced by an inverse variation at the microscopic level. In
the real Universe the entropy production is related to the diffusion of electromagnetic energy which is
complementary to the gravitational collapse of matter. The analogy between the toy and the real Universe
suggests that their evolution is a phase transition between different critical points. When a couple of
macroscopic and microscopic fluctuations appears, it has an external isotropic distribution in the
superspace and a dysequilibrium between the two different scale levels. The hypercyclical interaction
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between the macroscopic and microscopic levels leads to the production of an isotropic internal fractal-
chaotic distribution of the two phases in the intermediate steps. The dynamics of the described
mechanical model is accessible to experimental study and computer simulations in many different
systems with analogous rules and behaviours. In particular, diagrams of phase transitions in Ising models
of magnetic systems may provide a useful graphic representation of the described binary processes (Fig.
4).
Fig. 4 - Graphic representation of the fundamental morphogenetic processes. Many mathematical (a, d, g),
physical (b, e, h), and biological (c, f, i) systems exhibit analogous patterns produced by analogous systems of
forces. The process of binary dissociation (a, b, c), some branching structures produced by its iteration (d, e, f),
and some spiral-like structures produced by rotation of branching structures (g, h, i), are shown. The pictures a, d,
and g are computer graphic representations of the phases of a magnetic system. They are described by the equation
xn+1=((xn2+c-1)/(2xn+c-2))
2, where c is a constant. This equation is derived from a renormalized Ising model of the
phases transitions in a ferromagnet (Peitgen et al., 1986). The patterns in a, d, and g are characteristic of different
mathematical structures and in particular of the Julia sets. In dis evident the fractal nature of the border between
an internal expansive process of binary dissociation and an external contractive process of binary association. In b
there is the magnetic field produced by a magnetic dipole; in e the patterns resulting from the collapse of a viscous
substance, in h the galaxy NGC 5194, in c a dividing sea urchin egg, in fthe alga Chondrus crispus, and in i the
shell of a Nautilus. The analogies between the different structures are the results of the topological properties of
the Euclidean superspace: only few types of stable structures can be produced by iterative processes.
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Very simple cellular automata based on few cellular states and simple rules of binary dissociation and
association can simulate almost all the complex morphogenetic processes of living organisms. The
question of how cells estimate their location within the body has preoccupied embryologists for the past
century. The concept of gradients of unknown diffusible substances, first propounded by Morgan in the
1905, has been invoked as determinant of polarity in a wide variety of organisms. DArcy Thompson
(1917) suggested that mechanical stresses, as tension and pressure, can determine the shape of biological
structures. Alan Turing (1952) hypothesized that the morphogenetic development of several chemical and
biological systems is produced by diffusion and reaction processes. At any given position cells are able to
feel the local concentration of a morphogenetic gradient and they may move and/or differentiate in reply
to it. In some cases cells are sensitive to physical stresses and therefore the ontogenetic development is
determined by interaction between genetic and environmental elements. Oster et al. (1980) suggest that
mechanical deformation influencing both the viscosity and the elasticity of the cellular cytogel are the
major factor determining morphogenetic transformation. An interesting example of a biological structure
constructed in response to mechanical stresses is the case of the ammonoid septum (Damiani, 1984). The
tension distribution in the ammonoid septum is analogous to that of a bidimensional vacuum fluctuation
(Fig. 3). Therefore the ontogenetic and phylogenetic evolution of complex ammonoid septa is a direct
representation of the morphogenetic dynamics of the branching structures produced by binary processes.
According to Turing model, a general model to explain the ontogenetic and phylogenetic upward
trend of complexity of biological branching structures was proposed (Damiani, 1994). The function of
these structures in living organisms is to distribute or to gather biological material and physical entities
(West et al., 1987). Optimisation processes produced by natural selection led to the minimisation of the
networks length (the Steiner problem) (Bern et al., 1989) and of the amount of information needed for the
construction of these structures. What are the recursive rules chosen by nature to generate minimal
networks? Experimental data and computer simulations suggest two complementary ways to construct
Steiner networks: discrete entities are spread or concentrate in a surface or in a volume responding to
gradients of physical forces (as bending stresses) or of chemical morphogenetic elements (Damiani,
1994). In cellular automata models of diffusive processes, the activated cells go away according to a
repulsive force and when one cell is too isolate, it splits in two divergent cells (binary dissociation). In
contraction models, distributed cells converge according to an attractive force and when two cells are very
close, they stick together (binary association). These simple recursive local rules leads to the formation of
fractal, branching, vascular and global networks. Slight changes of morphogenetic gradients and of
responsiveness of sensitive entities influence the shapes of produced patterns. These basic models, with
few modifications, can simulate the Turing reaction-diffusion process, too. A simple automaton of the
Turing reaction is based on three types of cells: activated, receptive and quiescent. An active cell
stimulates the adjacent receptive cells into activity (binary dissociation) and became quiescent for one
generation. This simple automaton reproduces all the morphological features of the Belousov-
Zhabotinsky reaction (Madore et al., 1987). These simulations may explain the formation of a wide range
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of patterns and in particular of those characteristic of the morphogenesis of the slime mould
Dictyostelium discoideum. The transition from the wavelike pattern of Belousov-Zhabotinsky reaction to
a convergent vascular structure requires only few changes of the same fundamental mechanism.
The most interesting cellular automata are generated when the active cells can be both induced into
activity and reduced to the receptive state. These rules simulate the behaviour of metabolic reactions in
living organisms since both the anabolic and the catabolic reactions are present. For example, an active
cell induces the adjacent receptive cells into activity (binary dissociation) and becomes quiescent for one
generation (as in the simulation of the Turing reaction), but an active cell induces the adjacent active cells
to the receptive state (binary association). The resulting dynamic patterns are very complex and are
similar to fractal-chaos patterns. The fascinating behaviour of Life, the most famous cellular automaton
(Gardner, 1970), is due to the fact that both the anabolic and the catabolic reactions are produced in a
balanced proportions by its simple iterative rules. As in Life, in the metabolic cellular automata the
oscillating dissipative behaviour is very common and different types of replicating complex structures
can emerge with high probability.
The metabolic cellular automata with both expansion and contraction of activities are the same thing
of the NK automata of Kauffman (1991; 1993). These NK cellular automata are composed by N
elements with two binary states. Each element is linked to others by K inputs, according to simple
Boolean functions. These systems were called Boolean NK networks. Phase transitions between ordered
"solid" states and chaotic "gaseous" states can occur in self-regulating NK networks, depending on their
local characteristics. If Boolean functions are biased or if each element has only two inputs (K = 2), then
a network in which all the elements can initially vary will eventually become stable and hence a
crystallised "solid" systems. Such ordered systems consist of a large web of frozen elements and isolated
islands of variable elements. If the functions are unbiased or the interconnectedness of elements is high
(K> 3), the system becomes a "gas" and behaves chaotically. Only small islands of elements will be
frozen in this conditions. (Kauffman, 1991). The complexity that a network cans co-ordinate peaks at the
"liquid" transition between solid and gaseous states: this state is the fractal-chaos. Networks on the
boundary between order and chaos may have the flexibility to adapt rapidly and successfully through the
accumulation of useful variations. In such poised systems, most mutations have small consequences
because of the systems' homeostatic nature. A few mutations, however, cause larger cascades of change.
Poised systems will therefore typically adapt to a changing environment gradually, but if necessary, they
can occasionally rapidly. These properties are observed in organisms (Kauffman 1991). Kauffman
suggested that the NK networks are analogous to genetic networks (each NK elements represents a gene).
A genetic network could be defined as a set of genes which interact on the basis of cis and trans
regulatory elements. A necessary and sufficient condition to form a genetic network is that several genes
should be controlled by functionally identical cis elements and that at least some of them should code for
regulatory trans acting proteins. Both physically and functionally linked genes form such networks.
Kauffman concludes: Taken as models of genomic systems, systems poised between order and chaos
come close to fitting many features of cellular differentiation during ontogeny - features common to
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organisms that had been diverging evolutionarily for more than 600 million years... (Kauffman 1991). If
the analogy between the Kauffman networks and the genetic networks is true, biologists may have the
beginnings of a comprensive theory of genomic organisation, behaviour and capacity to evolve
(Kauffman 1991).
6. Evolution as optimization of binary processes according to the principle of parsimony
An evolutionary interpretation is suggested when one realizes that large parts of the system display a
tree-like iterated ramification in one direction (Weyl, 1949).
Weyl believed in the unity of nature and in the importance of the knowledge of the symmetries of the
basic natural laws for the understanding of the evolution of both the physical and the biological world.
The validity of this belief is confirmed by the discussed ideas. The concepts of binary processes and of
metabolic hypercycles would greatly facilitate the understanding of the dynamics of complex physical and
biological systems. The most important metabolic and homeostatic processes are often described by
hypercyclic feed-back relations. The ordered spatio-temporal developmental pathways of living
organisms may be the results of maps and clocks based on metabolic hypercycles. The control of the
anabolic and catabolic activities is very important to avoid waste of material and energy. The critical
points of equilibrium between the anabolic and the catabolic phase are characterised by fractal-chaotic
distributions and by long-range correlations of the different elements of the system. At these critical
points a biological system is very sensitive to small perturbations. A small pulse at a critical time or place
can direct a reproducible transition of the system towards a defined state. Therefore, the amplification of
small events may be a very useful tool for a sensitive, economical and efficient control of complex
systems (Skinner, 1994). The very non linear world of chaotic dynamics and fractal geometries seems
me to justify substitutions of the term homeodynamics for homeostasis. Homeodynamics carries with it
the potential for a deeper understanding of what it means for a complicated system to be stable and yet
show very rich behaviours, including the possibilities of development of individuals and evolution of
species. (Yates, 1992). A simple relation of negative feedback between two complementary entities may
result in steady state equilibrium, while a metabolic hypercycle result in dynamic prey-predator
equilibrium based on oscillating rhythms.
A "cosmic metabolism", produced by the coupling and competition between the anabolic processes
of binary association and the catabolic processes of binary dissociation, controls the evolution of all
physical systems, both organic and inorganic. At the macroscopic level, these two opposite irreversible
trends are related to the diffusion of electromagnetic energy and to the gravitational contraction of matter.
At the molecular level, the evolutionary dynamics of chemical systems are the result of the competition
between repulsive and attractive electromagnetic forces, between catabolic reactions degrading energy
and chemical compounds, and anabolic reactions increasing the complexity and the configurational order
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of material structures. The entire Universe is a complex dissipative system, in a non equilibrium state,
similar to a living organism, as hypothesised by Plato. The evolutionary dynamics of every dissipative
system is directed by the increase of entropy of the energetic component and by the increase of
configurational information of matter (Prigogine et al., 1979). The progressive increase in complexity and
information content of living entities at the molecular, cellular, individual, social, and ecological level is
complementary to the entropic decay of electromagnetic energy. Replicating structures appear
spontaneously when a relatively constant flux of energy passes, for a long time, through a large system of
reactive molecules, which are able to perform cyclic chemical reactions. The liquid elements, and in
particular their interfaces with solid and gaseous elements, are the environments more propitious to life
for their fractal nature, between order and disorder. Therefore, it is very probable that life, perhaps similar
to the terrestrial one, is very frequent and diffused in planets with liquid elements, as the Earth.
In this scenario, the origin and evolution of life on Earth is not a very improbable event, as
suggested by Boltzman and Monod, but is the unavoidable consequence of a more general natural law
as hypothesised by Darwin in 1881. This general law is the principle of parsimony. According to this
principle, evolution is a process of optimisation of those structures which have the ability to wield and
store information (negentropy), matter, and energy to maintain their identity and reproduce. The
ontogenetic and phylogenetic development of living organisms requires the following steps: (1)
reproduction of discrete units; (2) occurrence of differences among these units; (3) different rates of
differentiation and reproduction of the different units. The first two steps, replication and mutation, are
binary dissociation processes. The third step, natural selection, is a binary association process. The most
important molecules for the evolution of life on Earth are nucleic acids, which carry the genetic
information that is the algorithmic information content of living organisms. The increase of genetic
information may be quantitative (nucleic acid replication) or qualitative (nucleic acid mutation). The
decrease of genetic information is due to natural selection, operating by means of the death of organisms
(and the consequent nucleic acid degradation). A cladogram is a graphic representation of the evolution of
life, a divergent vascular structure (growth of genetic information) interacting with a convergent vascular
structure (decay of genetic information) which terminates the growing branches. Moreover the
phylogenetic tree is produced not only by splitting branches (binary dissociation) but also by splicing
branches (binary association). These cross-linking of branches, technically known as anastomosis, are
generally underestimated but recent researches on horizontal gene transfer and symbiosis have stressed
the importance of these phenomena (Margulis, 1993).
The theory of games suggests that not only separation and competition but also aggregation and
cooperation are important factors for the elaboration of successful strategies of survival (Axelrod et al.,
1981). We can look many evolutionary, ecological and behavioural dynamics in the spirit of a computer
game between opposing complementary processes (Sigmund, 1993). Evolution searches for a balanced
compromise between alternative and alternating needs. An example of this process is the optimization
between specialisation and plasticity. Successful adaptation produces specialised structures but the
genetic and ontogenetic hierarchical complexification reduces the possibility of new types of adaptation
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in a changed environment. A maximum increase of the phylogenetic complexity is counterbalanced by a
principle of minimum increase of ontogenetic complexity (Saunders, 1984). The most adapted organism
is able to produce the maximum amount of useful phenotypic diversity with a minimum amount of
genetic information. Since the utility of phenotypic variation is relative to the changing environments, the
most adapted genetic program is able to anticipate environmental changes and to interact with them.
These considerations suggest that a genetic network has the capacity to reorganise itself and to induce
programmed mutagenesis in response to environmental stress. Is the internal programmed self-
mutagenesis, and not random mutations, the most important mechanism of genetic evolution? Until now,
many surprising processes of programmed and intelligent self-mutagenesis, as the S.O.S. system of
prokaryotes or the rearrangements and somatic mutation of the immune system genes, have been
discovered. The small organisms have plastic genomes and are able to exchange genetic material by
horizontal transfer. A frequent genetic exchange may explain why is not possible to reconstruct the
universal root of Bacteria, Archea, and Eukarya. Preliminary results, based on the assumption that the
amount of mutations detected by random amplified polymorphic DNA (RAPD) analysis is an indication
of the genome plasticity, suggest that there is a correlation between the body size and the amount of
genetic variation in wild animal populations (see Fig. 5).
Fig. 5 - Relation between body size and amount of genetic similarity (Dice index) for several animal
populations. For small animals the value are referred to Italian populations sampled at the end of Summer, since
the amount of genetic variability in these species changes according to the geographic localization an climate
variation. The genetic similarity was estimated from standard RAPD analysis with the same primers and reaction
conditions as described in Damiani et al. (1996). The measure of genetic variability, based on RAPD data, may be
related to the degree of genomic plasticity. The adaptive response of small organisms to unexpected challenges
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may be a programmed generation of diversity in a large number of offspring (r strategy of reproduction). With
increase in size and development of physiological adaptation, organisms become more resistant to environmental
stresses and therefore the generation time increases, the number of offspring decline (k strategy of reproduction),
and the genomic plasticity is lost. The balance between the drawbacks and the advantages of a rapidly changing
genome probably depends on the trade-off between selection at individual and at population levels.
The ontogenetic constrains of large organism lead to a stabilisation of the genetic programs, with a
loss of the useful genetic plasticity. But large organisms can escape many environmental changes by
means of physiological plasticity (homeostasis).
The dynamics of binary processes explains the structural and functional optimisation of biological
structures achieved by evolutionary processes. The results of computer and mechanical simulations
strongly support the hypothesis that many biological forms, as the branching structures, are the most
functional design in relation to the material properties of the Universe. The biological forms converge
repeatedly on a limited number of architectural designs. Convergent evolution is evident at the
morphological level (as in the case of radiative evolution of mammals and marsupials) but is generallyignored at the molecular level since its detection in DNA sequences requires careful examination of the
sequences contest and of the ratio between the synonymous and non synonymous mutations. For example
the evolution of the DRB genes in mammals is a surprising case of convergent evolution (Gustafsson,
1994) which is misinterpreted by the transspecies theory as persistence of ancient allelic genealogies
(Klein, 1987). The optimal forms and structures discovered by natural selection "are topological attractors
that evolution cannot avoid" (Thomas and Reif, 1993). These morphological archetypes often are fractal,
branching and vascular structures. The most important property of these structures is the scale invariance.
The change of scale during growth is an important structural problems of pluricellular organisms
(McMahon, 1973). Many physiological processes are influenced by the size. For example the mass of anorganism increases as the cube of its size while the surface area only as the square. A fractal structure is
endowed with the same pattern, more or less ordered, at each level - that of the cell, the multicellular
organism, the animal society, the ecological community, and the entire biosphere. A power law describes
this type of relation. The exponent of the allometric equations of Tessier (1931) and Huxley (1932) is the
fractal dimension of Mandelbrot (1975) and describes the self-similarity at different scales of the
biological structures and processes. The dimensional analysis explains the meaning of the value of this
exponent. McMahon and Bonner (1983) wrote an extensive review of the studies on the power law
relationships between size and shape in biology. Also the geographical and taxonomic distributions of
living organisms follow power laws (Minelli et al., 1991) (Burlando, 1993). Scale-invariance and long-range power-law correlations are features not only of physical and biological structures (Fig. 6) but also
of many dynamic processes. Most of the chemical reactions and physiological processes of living
organisms work near their critical and instable equilibrium points, which define the borderline between
deterministic order and unpredictable chaos. All complex systems are alike in having fluctuations
over a vast range of size and similar universal behaviour near their critical points. These points are
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instable equilibrium states with mixed phases characterised by fractal-chaotic patterns and structures with
multiple scales of length. If the Binary theory will be confirmed we could explain the emergence of
"universal" properties and structures in different mathematical, physical, and biological systems as the
consequence of the fractal geometry of the vacuum fluctuations and binary processes.
Fig. 6 - Power law relation between sizes and masses of the principal structures of our Universe. The biological
systems do not show a special status. Many universal power law relationships describing both physical and
biological systems can be found in nature. These relations are characteristic of fractal-chaotic distributions
produced by the balance between opposite and complementary processes. This picture is redrawn from Barrow
(1991).
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7. Conclusion
Whether or not the view is tenable that the organizing power of life establishes correlations between
independent individual atomic processes, there is no doubt that wherever thought and the causative agent
of will emerge, especially in man, that power is increasingly controlled by a purely spiritual world of
images (knowledge, ideas) (Weyl, 1949).
The variety of living organisms is the result of evolutionary processes. But the reductionistic neo-
Darwinian point of view is insufficient to explain the complexity and the dialectic nature of these
processes. Apparently dichotomic explanations of biological evolution (as many of the controversial
points cited in the introduction of this article) may be alternative and complementary aspects of the same
reality, as the two different faces of the same coin. Many important factors, as the ontogenetic trends, the
internal self-organising ability of the genomes, the interaction between different organisms and
environments, the convergent evolution at the molecular level, the symbiosis and in general the co-
operative relationships are not taken in the right account by many evolutionary biologists. After the defeat
of the theological orthodoxy a new type of scientific orthodoxy has arisen. All the scientific hypothesis
which do not fit with the standard orthodoxy, as many of the ideas discussed in the present work, have a
difficult life (Sermonti, 1971). But successful evolution requires the right balance between innovation and
conservation (this is true also for evolution of scientific theories). Even if it is impossible to be free from
unconscious prejudices, the scientists should be in the search for objective reality. The rejection of any
theological and finalistic interpretation of evolutionary processes do not exclude the existence of
negentropic and self-organising phenomena.
The evolution of our Universe is driven by a cybernetic plane, a universal Bauplane, coded in the
geometric and mathematical structure of the fundamental physical laws. The emergence of dissipative
replicating structure increasing their complexity is the unavoidable result of both the deterministic
macrophysical laws and the unforeseeable microphysical laws. Evolution can not escape the general
ortogenetic trends, but the result of a single quantistic event is unforeseeable. A microscopic event, as for
example a point mutation, may have unexpected and dramatic macroscopic effects on the dynamics of
very complex and sensitive systems, as the biological ones. In the living systems the freedom of the
microphysical world is amplified at a macroscopic level. An important point is that we do not know if the
quantistic and probabilistic processes are driven by a blind chance or by metaphysical entities. God and
our soul, if they exist, can play dice. A single quantistic result can be chosen even if the sum of the results
must be in accordance with the probabilistic laws. The physical laws do not explain many aspects of the
individual brain behaviour, such as the Schrodinger equation do not explain the behaviour of an electron
in a single quantistic event. The evolution of our superior intellectual abilities, as the abstract thinking
and conscience, are clearly related with the evolution of our brain, but are very strange immaterial
things. Moreover also cultural evolution is a strange immaterial thing which influences, with a feedback
relation, the biological evolution.
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The scientific approach is a useful tool to develop models, but does not explain our conscious
perception of reality. Scientist would be wrong to ignore the fact that theoretical construction is not the
only approach to the phenomena of life; another way, that of understanding from within, is open to us
(Weyl, 1949). Reason and emotion are different things, but are not incompatible and both are important
for our life. As stated by A. Einstein all the religions, arts and sciences are branches of the same tree.
For example, the creative processes described by the Binary Theory are essentially the same hypothesized
by Atkins (1981) in his book entitled The Creation. Atkins believes to be a very reductionistic and
materialistic scientist, but his modern description of the Creation is very similar to the ancient stories of
Hermetic, Greek, Hindu and Taoist philosophers. In the Taoist cosmogonies the evolution of the world is
produced by a binary process of polarisation of a fundamental primordial entity (Tao) in two opposite,
complementary and hypercyclically interrelated entities (Yin and Yang). Iteration of these binary
processes leads to the increasing complexity of our Universe. Order and disorder, determinism and
indeterminism, reductionism and holism, analysis and synthesis, reason and intuition, materialism and
spiritualism, science and mysticism, all the opposites run after each other and seem to amalgamate into a
single reality by means of hypercyclic relations, as in the ancient Yin and Yang symbol of the Taoist
philosophers.
Acknowledgements
We are grateful to E. Recami, G. Sermonti, R. Morchio and F. Scudo for interesting and useful
comments.
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