HYLE--International Journal for Philosophy of Chemistry, Vol. 3 (1997), pp. 29-49:

*Dedicated to Achim Müller on the occasion of his 60th
birthday*

*Klaus Mainzer*

**Abstract**: Molecules have more or less symmetric and complex structures
which can be defined in the mathematical framework of topology, group theory,
dynamical systems theory, and quantum mechanics. But symmetry and complexity
are by no means only theoretical concepts of research. Modern computer
aided visualizations show real forms of matter which nevertheless depend
on the technical standards of observation, computation, and representation.
Furthermore, symmetry and complexity are fundamental interdisciplinary
concepts of research inspiring the natural sciences since the antiquity.

**Keywords: ***molecular structure, symmetry, symmetry breaking,
complexity, nonlinearity*

The *molecular structure hypothesis* states that a molecule is
a collection of atoms linked by a network of bonds. Since the 19th century
the molecular structure hypothesis has been a successful concept of ordering
and classifying the observations of chemistry. But this hypothesis cannot
directly be derived from the physical laws governing the motions of the
nuclei and electrons that make up the atoms and the bonds. It must be justified
that all atoms exist in molecules as separate definable pieces of the 3-dimensional
("real") space with properties which can be predicted and computed
by the laws of quantum mechanics.

The well-known models of molecules with different information for a
chemist are derived from the molecular structure hypothesis: a) The three-dimensional
ball-and-stick model with balls for the atomic nuclei, sticks for the atomic
bonds and their angles, b) its 2-dimensional representation as structural
formula, and c) its 1-dimensional representation as linguistic name which
can be derived from the structural formula. Graphic models are applications
of mathematical *graph theory* which is part of combinatorical topology.
This mathematical theory became fundamental for chemistry, when in the
midst of the last century the molecular structure of chemical substances
were discovered (e.g., Biggs et al. 1976, p.55). L. Pasteur recognized
that the relationship between symmetry of reflection and optical activity
is not a function of the crystal structure of a substance. With certain
water-soluble crystals, for example, the symmetry of reflection can be
demonstrated both in the solid state and in the liquid state. Pasteur investigated
tartaric acid and found a counterclockwise and a clockwise form, which
are called L-tartaric acid and D-tartaric acid (D = dextro = right) respectively.
He also isolated a third form of tartaric acid (meso-tartaric acid), which
cannot be separated into one of the other forms. To explain the optical
activity, it was therefore necessary to investigate more fundamental structures
than crystals, or even molecules and the orientation of atoms. R.J. Haüy
had already suspected that the form of crystals and their constituent components
were images of one another. Pasteur therefore inferred the symmetric form
of the crystal's components from the crystal reflections.

Another important step was A. Kekulé's investigation of quadrivalent carbon atoms, for whose multiple bonds he also introduced a structural formula notation which is still used in today's organic chemistry. An essential advance occurred in 1864, when the Edinburgh chemist A. Crum Brown introduced his version of the graphic notation. Each atom was shown separately, represented by a letter enclosed in a circle, and all single and multiple bonds were marked by lines joining the circles. Crum Brown's system is more or less the one in use today, except that the circles are now usually omitted. His notation was soon accepted everywhere, after some resistance from Kekulé and others. Its acceptance was partly due to its success in explaining the strange fact that there are pairs of substances which have the same chemical composition, although their physical properties are different. The graphic notation made it clear that this is because the atoms are arranged in different ways in the different substances. This well-known chemical phenomenon is called isomerism, and in many cases there are more than two isomers with the same constitutional formula. In 1874, the great British mathematician A. Cayley wrote a paper 'On the mathematical theory of isomers' inspired by the fusion of chemical and mathematical ideas.

But the experiments of J. H. van't Hoff and J. A. Le Bel were decisive
for the assumption of a *three-dimensional molecular structure*. In
1874, independently of one another, they established a relationship between
optical activity and the three-dimensional orientation of atoms. The initial
example was the carbon atom, whose four valences were arranged in the form
of a tetrahedron. A tetrahedral configuration with the carbon atom in the
center makes possible the existence of two different arrangements being
mirror images of each other. Tartaric acid has two carbon atoms which are
each connected to the atoms or groups of atoms H, C, OH and COOH. For this
combination, there are two possible arrangements (L- and D-tartaric acid)
which are mirror images of each other and one arrangement (meso-tartaric
acid) which is reflective symmetric in itself.

Van't Hoff's *stereochemistry* regarding the three-dimensional
structure of the atom must initially have appeared to be a highly speculative
idea, which betrayed a certain proximity to Platonic forms. Kekulé
may have been particularly adept at three-dimensional visualization as
a result of his prior study of architecture. Simultaneously with stereochemistry,
geometry and algebra were also undergoing a fruitful development (Mainzer
1980, p.135). Van't Hoff's success in experimental explanation and prediction
made his geometry and algebra of the molecule soon a method accepted by
chemists. But it lacked any definitive physical justification. At this
stage of development, stereochemistry remains a successful heuristic approach
which meets chemists' need for a means by which they can visualize their
structural analyses.

From an *experimental point of view* the shape of molecules can
be illustrated by an outer envelope of their electronic charge distributions.
These representations are similar to the pictures of atoms which we can
today obtain experimentally by the scattering of electrons in super microscopes
or from the scanning tunnelling electron microscope. It is the distribution
of charge that scatters the X-rays or electrons in these experiments. Thus,
it is the distribution of charge that determines the form of molecular
matter in 3-dimensional space.

Mathematical methods of *differential topology* enable us to identify
atoms in terms of the morphology of the charge distribution. The charge
density r(**r**) is a scalar field over 3-dimensional
space with a definite value at each point. Positions of extrema in the
charge density with maxima, minima, or saddles where the first derivatives
of r(**r**) vanish can be studied in the
associated gradient vector field Ñr(**r**).
Whether an extremum is a maximum or a minimum is determined by the sign
of the second derivative or curvature at this point. The gradient vector
field makes visible the molecular graph with a set of lines linking certain
pairs of nuclei in the charge distribution.

Local maxima of electronic charge distribution are found only at the
positions of nuclei. This is an observation based on experimental results
obtained from X-ray diffractions and on theoretical calculations on a large
number of molecular systems. Thus, a nucleus seems to have the special
role of an *attractor* in the gradient vector field of the charge
density. In short: The topology of the measurable charge density defines
the corresponding molecular structure.

In the mathematical framework of *dynamical systems theory* the
global arrangements of molecular forces can be represented by phase portraits
with *attractors* as nuclei and *trajectories* representing the
vector field. For example, Fig. 1 shows maps with nuclei and the symmetric
structure of the ethylene molecule. In Fig. 1a only those trajectories
are shown which terminate at the position of the nuclei. They define the
basins of the nuclear attractors. Fig. 1b includes the trajectories which
terminate and originate at certain critical points (denoted by full circles)
in the charge distribution. The pair of trajectories terminating at these
critical points mark the intersection of an interatomic surface with the
plane of the figure. The gradient paths which originate at these critical
points and define the bond paths are shown by the heavy lines. Fig. 1c
shows a superposition of the trajectories associated with these critical
points on a contour map of the charge density. These trajectories define
the boundaries of the atoms in the nuclear graph.

a b c

**Figure 1: **Molecular structures with nuclei as attractors in maps
of the gradient vector field of their charge densities (for the plane)
(Bader 1990, p. 30).

In general: the molecular graph is the network of bond paths linking pairs
of neighbouring nuclear attractors. An atom, free or bound, is defined
as the union of an attractor and its basin. Atoms, bonds, and structure
are topological consequences of a measurable molecular charge distribution.
In a next step, it is necessary to demonstrate that the *topological
atom* and its properties have a basis in *quantum mechanics*. Topological
atoms and bonds have a meaning in real three-dimensional space. But this
structure is not reflected in the properties of the abstract infinite-dimensional
Hilbert-space of the molecular state function. The state function y
contains all of the information that can be known about a nuclear quantum
system. From an operational point of view, there is too much and redundant
information in the state function because of the indistinguishability of
the electrons or because of the symmetry of their interactions. Some of
it is unnecessary as a result of the two-body nature of the Coulombic interaction.
Thus, there is a reduction of information in passing from the state function
in the infinite-dimensional Hilbert space to the charge distribution function
in the real three-dimensional space. But, on the other hand, we get a description
of the molecular structure in the observable and measurable space.

*Quantum chemistry* uses several mathematical procedures of approximation
to achieve this kind of reduction (e.g., Primas 1983). A well-known approximation
is the Born-Oppenheimer procedure which allows a separate consideration
of the electrons and nuclei of a molecule. We get the nuclear structure
of a molecule which is represented by its structural formula. In order
to distinguish the electrons as quasi-classical objects in orbitals, the
Hartree-Fock method is sometimes an appropriate approximation for the electronic
state function. The electronic charge density r(**r**,**X**)
with the space vector **r** of an electron and the collection of nuclear
coordinates **X** can be derived as the quantum mechanical probability
density of finding any of the electrons in a particular elemental volume.
In the case of molecules in stationary states, the probability density
is defined by the stationary-state function y(**x**,**X**)
depending on the collection **x** of electronic space and spin coordinates
and the collection **X** of nuclear coordinates (Bader 1990, p.6). This
state function is a solution of Schrödinger's stationary state equation
for a fixed arrangement of nuclei. The *coincidence of the topological
definition and the quantum definition* *of an atom in a molecular
structure* means that the topological atom is an open quantum subsystem
of the molecular quantum system, free to exchange charge and momentum with
its environment across boundaries which are defined in three-dimensional
real space. In this sense, symmetries of molecules referring to their topological
structure are real forms of matter which can be calculated by quantum chemistry.

Quantum chemistry and mathematical group** **theory are the modern
bases of symmetry considerations in stereochemistry. In quantum chemistry
the symmetries of molecular systems are represented by the symmetries of
the corresponding molecular Hamiltonian operators. In stereochemistry the
structure of molecules is classified by the *symmetry transformations
of point groups *(Mainzer 1996, p. 492):

The symmetries of a free molecule can be completely defined by a few
types of symmetry transformations. In general, the selection of the three
coordinates axes *x*, *y*, *z* is arbitrary. The trivial
symmetry transformation is *identity* I which leaves each molecule
unchanged. An additional symmetry element is the *axis of rotation*
C_{n} around which a molecule can be rotated
by the angle 2p/n without changing its position.
Linear molecules, in which all atomic nuclei lie on a straight line (e.g.
nitrogen NºN or carbon monoxide CºO),
can be rotated around the connecting axis by arbitrarily small angles and
have a continuous axis of rotation with infinite fold symmetry n ®µ.

An additional symmetry element is the *reflection* s
on a plane in which the molecule does not change its position. For example,
if the *xy*-plane is the plane of reflection, then replacing all the
atomic *z*-coordinates by *-z* does not change the position of
the molecule. Depending on the selection of the plane of reflection, a
distinction is made between a vertical plane of reflection s_{n}
and a horizontal plane of reflection s_{h}.

The next symmetry element is *inversion* in which a molecule remains
unchanged during a reflection of all atomic coordinates (*x*,* y*,
*z*) at the point of inversion to (*-x*, *-y*, *-z*).
An additional symmetry element is *rotary reflection* S_{n}
= s_{h}C_{n}
in which a molecule is first rotated by an angle 2p/n
around the rotary reflection axis C_{n} and
is then reflected on the plane s_{h}
perpendicular to C_{n} through the centre
of the molecule, without changing its position.

The remaining symmetry element is *rotary inversion* in which a
molecule does not change its position in spite of rotation followed by
inversion. It should also be noted that the compound symmetry transformations
of rotary reflection and rotary inversion do not presuppose the partial
transformations of rotation, reflection or inversion as symmetry elements
of the same molecule. The symmetry transformations of a molecule, when
executed one by one, produce symmetry transformations in turn and define
as a whole the symmetry structure of the molecule by the mathematical group
of these symmetry transformations.

In general, mathematical symmetries are defined by so-called *automorphisms*
that means self-mappings of figures or structures whereby the structure
remains invariant (example: rotation or reflection of polygons in the plane).
The composition of automorphisms satisfies the axioms of a mathematical
group. So the symmetry of a molecular structure is defined by its group
of automorphisms. There are continuous groups of symmetries (for instance,
circles and spirals) and discrete groups (for instance regular polygons,
ornaments, Platonic bodies).

On account of the finite number of combinations of symmetry elements,
it is clear that there can only be a finite number of *point groups.*
Thereby many different molecules can belong to the same point group, i.e.
they can have the same symmetry structure. The classification of point
groups also makes it possible to explain the relationship of optical activity
and molecular structure in terms of group theory. According to Pasteur
a compound had optical activity, if the molecule in question could not
be made to coincide with its reflection. In that case, Pasteur spoke of
*dissymmetry*. Other terms are "enantiomery", which in the
Greek translation means opposite shape, or "chirality", which
alludes to the left and right-handedness of the reflective orientation.
In terms of group theory, it is a matter of determining the elements of
symmetry which lead to *optical activity*. In general, 1) a molecule
with any axis of reflection S_{n} cannot be
optically active, and 2) a molecule without an axis of reflection is optically
active.

Point groups describe the symmetries of *stationary* molecules
in the equilibrium state. Reduced symmetries may be present in the *non-stationary*
case of translation, rotational motions, oscillations etc. Scalar characteristics
such as mass, volume or temperature, which have only an amount but no direction,
are apparently independent of the symmetry operations. But characteristics
which have not only an amount but also a direction can affect the symmetry.

So far we have discussed the symmetries of the structures of molecular
nuclei. What *symmetries* determine the *electron orbitals* of
the molecules? Molecular orbitals y are frequently
introduced by approximation as linear combinations of the atomic orbitals
c_{i} of the individual
atoms of the molecule (Linear Combination of Atomic Orbitals = LCAO method)
with y *= *S_{i}**
**c

a b

**Figure 2:** Orbital symmetry of Kekulé's ring structure of
benzene (Mainzer 1996, p. 498).

The fourth valence electron corresponds to the p_{z}
orbital, which is above and below the plane with its two dumb-bells, each
perpendicular in the nodes of the carbon atom. The p_{z}
orbitals overlap with their respective neighbors and form a p-bond.
Figure 2b shows a p-orbital of benzene. In contrast
to the s-bond, the p-bond
is weak, so that the p-electrons can be easily
influenced by extremal forces, and thus determine many of the spectroscopic
characteristics of benzene. s and p
orbitals of benzene can be distinguished by their symmetry behavior in
a reflection on the xy plane. While s orbitals
f_{s} do not change
their sign during the reflection z ®-z and
are therefore symmetric, antisymmetry occurs with the p
orbitals f_{p}:

f_{s}(*x,y,
-z*) = f_{s}(*x,y,z*)

f_{p}(*x,y,
-z*) = -f_{p}(*x,y,z*)

The system of p electrons offers a simplified
way to calculate the energy levels of the benzene molecule. In the *Hückel
model*, we first consider p-electrons, since
it is assumed that p molecule orbitals are significantly
higher in energy than s orbitals and can therefore
be considered separately. Calculating y orbitals
according to the LCAO method is therefore restricted in the Hückel
model to the atomic orbitals c_{i}
which form p-molecular orbitals. That is another
major simplification, of course, but one which has proven valuable in actual
practice, e.g. in the calculation of benzene orbitals.

The p orbitals of benzene are eigenfunctions
of a Hamilton operator of the p-electrons, which
is invariant with respect to symmetry operations of the point group D_{6h}
of the regular hexagon with horizontal reflection s_{h}.
Physically, therefore, the potential energy of the p
electrons is not changed when the benzene molecule is rotated, e.g. by
60° around the center. The Hückel model and the orbital symmetries
thereby assumed are also used to predict chemical reactions, as expressed
in the *Woodward-Hoffmann rules*. One requirement is that the orbital
symmetry is conserved during reactions, i.e. the symmetry of all occupied
orbitals remains unchanged during the reaction with respect to each symmetry
element shared by the reacting and resulting molecules.

In contrast to low-molecular chemistry, high-molecular or macromolecular
chemistry is concerned with compounds which are composed of a great many
atoms, and therefore have high molecular masses. From the standpoint of
symmetry, *polymerizations* are nothing more than polyadditions of
monomers, the structural formulas of which form certain chains like those
which are known from the Fries groups. These structural formulas recall
the artful friezes in mosques, "structures of altogether unusual simplicity,
unity and beauty" (W. Heisenberg). But with regard to the symmetries
of the friezes of chemical structural formulas, we have to consider that
these are only two-dimensional projections of three-dimensional structures.
For crystal polymers in particular, X-ray diffraction spectra reveal stable
conformations with well-defined symmetries.

The significance of *macromolecules* in nature becomes clear when
we investigate the structure and metabolism of living organisms. For example,
their high molecular masses makes it possible to construct solid and simultaneously
flexible structures. On the other hand, their complex atomic structure
makes it possible to regulate metabolic processes and to store information.
From the standpoint of symmetry, *proteins* are of fundamental interest.
These are macromolecules of many amino acids of 20 different types in nature.
Protein analysis shows that amino acids have an antisymmetrical carbon
atom and occur only in the left-handed configuration in nature. If we investigate
the three-dimensional conformation of various amino acid units in the protein,
we encounter a characteristic *antisymmetry* of the protein, in which
the antisymmetry of its components is continued.

L. Pauling, who detected a spiral structure in certain crystal protein
fibers, called it a*-helix*. The a-helix
consists of 18 monomer units on 5 revolutions each, which, among other
things, are stabilized by intramolecular hydrogen bonds. One of the characteristic
symmetry breakings of biopolymers is therefore the fact that proteins in
nature only form left-handed spirals. Of course, reflections of the protein
components also occur, but they cannot be fitted into the molecular chains
of proteins. Certain proteins differ from the regular helix structure.
One well-known example is *hemoglobin*, whose stereochemical structure
was reconstructed by the Nobel Prize winner M. Perutz, among others. To
be precise, hemoglobin consists of a spherical protein (globin) and a complicated
compound of iron (heme) which is not a protein. Hemoglobin is characterized
by the double axis of rotation of its molecular chains.

X-ray crystallography now makes it possible to systematically analyze
the symmetry structures of crystallized proteins and to explain them in
terms of group theory. The mathematical structure of crystals is altogether
independent of their physical or chemical interpretation. In biochemistry,
atoms are not selected as structural units, but molecules. Since amino
acids naturally occur in proteins only as left-handed configurations, certain
symmetry elements of crystals which require an equal number of left-handed
and right-handed configurations, such as the plane of reflection, glide
reflection and center of inversion, are *a priori* excluded. Mathematically
speaking, from the 230 possible crystal groups only the first 65 ones with
intrinsic movements remain. These 65 discrete intrinsic motion space groups
in which biological macromolecules such as proteins can crystallize are
therefore also called *"biological" space groups* (Mainzer
1996, p.507). They are used in the investigation of enzymes, for example.

*Nucleic acids*, which are primarily responsible for transmitting
characteristics through generations of living organisms show also characteristic
symmetry breakings. Nucleic acids are macromolecules which are formed by
linear polymerization of certain units (nucleotides). According to the
double helix model of J.D. Watson and F.C. Crick, the DNA molecule consists
of two strands of DNA which are intertwined in a regular double helix around
a common axis. The two strands are parallel, but in opposite directions.
The sequence of the bases in the one strand determines the sequence in
the other strand, so that an A is always opposite to a T and a G is always
opposite to a C. *Antisymmetry* is of fundamental importance for the
transmission of genetic characteristics, which can be explained on the
molecular level of DNA helix.

The genetic information of an organism is encoded in its set of chromosomes
(genome) in the form of DNA. The mechanism of *reproduction*, which
makes possible a clear duplication (replication or reduplication), can
be illustrated very clearly: An enzyme enables to break the hydrogen bonds
separating the double helix into two strands. Each strand reproduces its
exact opposite, to which it is reconnected by hydrogen bonds, thereby forming
a new double helix. On account of the complementary base pair formation
of A-T and C-G, which is expressed in three dimensions in the twisting
of the double helix, the accuracy of the copy is guaranteed even after
many reproductions. If the two strands were orientated parallel and symmetric
to one another like a ladder, then the *accuracy of the reproductive
process* would not be guaranteed, with disastrous consequences for the
respective organisms. This case demonstrates that antisymmetry or parity
violation can have a decisive biological function.

In general, the symmetry of reflection (*inversion*) means that
right-handed and left-handed structures of chiral molecules can be distinguished
in space. But energetically they seem to be completely equivalent. It was
van't Hoff who found the geometrical explanation of chiral and optically
active molecules. Mathematically we can use coordinate systems with right
and left orientation to distinguish both forms of chirality. In quantum
chemistry the symmetry of chirality is represented by a quantum number
('*parity*') with two possible values +1 for positive parity and -1
for negative parity.

But the symmetry of chirality is violated by observations and measurements
in the laboratories of biochemists. Macromolecules like, for instance,
L-amino acids or D-sugars which are building blocks of living systems possess
a characteristic homochirality or dissymmetry. Sometimes enantiomers (i.e.
the reflections of isomers) can be distinguished by simple tests of taste:
S-asparagin has a bitter taste, while R-asparagin has a sweet taste. We
can perceive this kind of symmetry breaking, because our body is a handed
(*chiral*) biochemical system. In the 19th century Pasteur already
presumed that living systems are characterized by typical dissymmetries
of their molecular building blocks which have emerged during biological
evolution (e.g., Pasteur 1861). Then the handed receptor molecules of our
taste organs fit the chiral forms of the tasted molecules such as the right
or left hand fits the right or left glove. But it cannot be explained why
the actual molecular form of symmetry breaking was realized during the
evolution and why the other form was unable to survive.

As usual in *classical physics,* the two stable enantiomers can
be illustrated by two minima of a symmetric potential curve *V*(*q*)
where *q* is the reaction coordinate for the chemical transformation
of the molecular substituents. Mathematically the potential curve of the
reaction equation is assumed to be completely symmetric with respect to
inversion. There are three solutions as equilibrium points with the two
stable minima of the left- and right-handed forms and an unstable solution
of a symmetric achiral form. The symmetry is broken by the actually realized
stable form with respect to peculiar supplementary conditions.

In *quantum chemistry* the framework of classical physics must
be replaced by the principles of quantum mechanics. Molecular states are
described by wave functions which can be superposed as pure entangled states
according to the superposition principle. Thus for every temperature and
energy there is not only the possibility of chiral molecules with either
a left- or a right-handed form, but also a third possible form which is
both left- and right-handed. Spontaneous symmetry breaking in quantum chemistry
can be introduced by superselection rules forbidding the symmetric achiral
superposition states which can be realized by a special physico-chemical
environment (e.g. certain radiation fields).

The classical and quantum mechanical concept of *spontaneous symmetry
breaking *can only explain *that* a chiral molecule must emerge
under some supplementary conditions. But it cannot explain *why* the
*actual* form was realized instead of the other possibility. An explanation
has been suggested with respect to the *parity violating weak interaction
*which can be evaluated at least numerically in chiral molecules. In
case of parity (*P*)-symmetry the right- and left-handed forms would
be energetically exactly equivalent, transformed into each other by inversion.
But parity was violated by the symmetry breaking of weak interaction during
the cosmic evolution. Thus, if the parity violation can be measured by
a small energy difference D*E _{PV}*,
we get the non-equivalence of the two isomers or enantiomers which are
no longer simple mirror images of each other. The corresponding potential
curve is no longer symmetric, but the two minima differ with the energy
difference D

Of course, these energy differences in molecules are extremely small.
Even if these differences increase proportionally during polymerization,
they still remain very small under laboratory conditions. But in *evolution*,
nature itself was the laboratory. For amino acids, for example, we can
accurately calculate the prebiotic evolutionary conditions in which homochirality
can be selected, e.g. in a lake with a certain volume of water and over
a certain period of time. These calculations are based on an *ab initio*
method (Hartree method) of numerical quantum chemistry, which currently
has the best claim to accuracy. Therefore homochiral biochemistry can be
interpreted as a direct result of the parity violation of weak interaction.

Pasteur's suspicion of a universal dissymmetrical force in nature is
therefore reasonable, at least in terms of quantum chemistry. We could
go even further and classify the *chirality of biomolecules in a sequence
of symmetry breakings* which took place in the cosmological growth of
the universe. Elementary particle physics intends to unify all the known
physical interactions by deriving them from one interaction scheme based
on a single symmetry group. Physicists expect to arrive at the actually
observed and measured symmetries of fundamental forces and their elementary
particles of interaction by spontaneous symmetry breaking processes. Electromagnetic
and weak forces could already be unified by very high energies in the laboratories
of high energy physics (for instance the accelerator ring of CERN). That
means that at a state of very high energy the particles of weak interaction
(electrons, neutrinos, etc.) and electromagnetic interaction cannot be
distinguished any longer. They can be described by the same symmetry group
*U*(1) x *SU*(2). At a particular critical value of lower energy
the symmetry breaks down in two partial symmetries *U*(1) and *SU*(2)
corresponding to the electromagnetic and weak force.

The most successful method to achieve this kind of spontaneous symmetry
breaking is provided by the Higgs' mechanism. After the successful unification
of electromagnetic and weak interaction physicists try to achieve the *big*
unification of electromagnetic, weak and strong forces (*GUT*), and
in a last step the *superunification* of all four forces. There are
several research programs of superunifications such as supergravity and
superstring theory. Technically the unification steps should be achieved
along growing values of very high energy. Mathematically they are described
by extension to richer structures of symmetry (*gauge groups*). On
the other hand, the variety of elementary particles is actualized by spontaneous
symmetry breaking. The emergence of weak interaction with its particular
violation of parity was a result of cosmic symmetry breaking during the
expansion of the universe. "C'est la dissymétrie qui crée
le phénomène”, said Pierre Curie (Curie 1894).

Besides spatial symmetries chemists are involved in the fundamental
problems of *time symmetry*. While the laws of classical physics and
quantum chemistry assume symmetry with respect of time inversion, the factual
chemical reactions in the laboratories proceed only in one direction to
the chemical equilibrium. Chemical processes are *irreversible*. Their
reversion seems to be unnatural. It is a question of current research,
if the violation of time inversion can be experimentally observed for isolated
quantum systems in analogy to the parity violation of isolated chiral molecules.

Since Boltzmann's statistical interpretation of the 2nd law of thermodynamics,
irreversible processes have been discussed for complex molecular systems
like gases, fluids, etc. The 2nd law states that closed systems irreversibly
approach the thermal equilibrium of maximal entropy. It is remarkable that
I. Prigogine explains the irreversibility of dissipative processes far
from thermal equilibrium by a universal symmetry breaking of time. *Time*
has now the status of a *mathematical operator* which only allows
physically asymmetric states. While the spontaneous symmetry breaking of
elementary particles in high energy physics assumes the symmetry of its
laws with respect to unitary (gauge) groups, Prigogine's time operator
delivers (non-unitary) semi-groups representing both directions of time
(Nicolis, Prigogine 1989). The 2nd law of thermodynamics is a kind of selection
principle for the realized symmetry breaking process. In short: The law
itself has become asymmetric.

The emergence of *dissipative structures far from thermal equilibrium*
is an irreversible process of symmetry breaking which can be geometrically
illustrated by a bifurcation scheme (Mainzer 1997). In other words: The
bifurcation tree of a dissipative system represents the growth of forms
in an irreversible time direction. Chemical reactions provide paradigmatic
cases of complex dissipative systems. In the Belousov-Zhabotinsky reaction,
for instance, the forms of concentric rings or moving spirals appear when
specific chemicals are poured together at a critical value. The competition
of separated ring waves shows the nonlinearity of these phenomena very
clearly, because in cases of a superposition or linearity the ring waves
would penetrate each other like optical waves. The BZ-reaction is not only
an example of a time symmetry breaking process, but a space symmetry breaking,
too. The kinetic laws of chemical reactions in a (in principle unlimited)
medium are invariant with respect to the group of all translations, rotations,
and reflections.

In the framework of *dynamical systems theory*, this kind of space
and time symmetry breaking refers to phase transitions of complex open
(dissipative) systems far from thermal equilibrium. Macroscopic patterns
('*attractors*') arise from the nonlinear interactions of microscopic
elements (i.e. atoms, molecules) when the energetic and material interaction
of the dissipative (open) system with its environment reaches some critical
value ('*dissipative self-organization*'). Phase transitions of closed
systems near to thermal equilibrium are called *conservative self-organization*
creating ordered structures with low energy at low temperature. A physical
example of conservative self-organization is provided by a ferromagnet
consisting of very many atomic elementary magnets. At a temperature *T*
greater than a critical value *T _{c} *,
these magnets point to random directions. When

In *supramolecular chemistry*, conservative self-organization plays
a tremendous role (Müller et al. 1996). In this case molecular self-organization
means the spontaneous association of molecules under equilibrium conditions
into stable and structurally well- defined aggregates with dimensions of
1-100 nanometers (1 nm = 10^{-9} m = 10 Å).
Nanostructures** **may be considered as small, familiar, or large, depending
on the view point of the disciplines concerned. To chemists, nanostructures
are molecular assemblies of atoms numbering from 10^{3}
to 10^{9} and of molecular weights of 10^{4}
to 10^{10} daltons. Thus, they are chemically
large supramolecules. To molecular biologists, nanostructures have the
size of familiar objects from proteins to viruses and cellular organelles.
But to material scientists and electrical engineers, nanostructures are
the current limit of microfabrication and thus are rather small.

*Nanostructures* are complex systems which evidently lie at the
interface between solid-state physics, supramolecular chemistry, and molecular
biology (Mainzer et al. 1997). It follows that the exploration of nanostructures
may provide hints about both the emergence of life and the fabrication
of new materials. But engineering of nanostructures cannot be mastered
in the traditional way of mechanical construction. There are no man-made
tools or machines for putting together their building blocks like the elements
of a clock, motor, or computer chip. Thus, we must understand the principles
of self-organization which are used by nanostructures in nature. Then,
we only need to arrange the appropriate constraints under which the atomic
elements of nanostructures associate themselves in a *spontaneous self-construction*:
The elements adjust their own positions to reach a thermodynamic minimum
without any manipulation by a human engineer.

Historically, the idea of supramolecular interactions dates back to the famous metaphor of Emil Fischer (1894), who described a selective interaction of molecules as the lock and key principle. Today, supramolecular chemistry has by far surpassed its original focus. Molecular self-assemblies combine several features of covalent and non-covalent synthesis to make large and structurally well-defined assemblies of atoms. Single van der Waals interactions and hydrogen bonds are weak relative to typical covalent bonds and comparable to thermal energies. Therefore, many of these weak non-covalent interactions are necessary in order to achieve molecular stability in self-assembled aggregates. In biology, there are many complex systems of nanoscale structures such as proteins and viruses which are formed by self-assembly. Living systems sum up many weak interactions between chemical entities to make large ones. How can one make structures of the size and complexity of biological structures, but without using biological catalysts or the informational devices coded in genes?

Many non-biological systems also display self-organizing behavior and
furthermore provide examples of useful interactions. Molecular crystals
are self-organizing structures. Liquid crystals are self-organized phases
intermediates between crystals and liquids with regard to order. Micelles,
emulsions, and lipids display a broad variety of self-organizing behavior.
An example is the generation of cascade polymers yielding molecular bifurcational
superstructures of fractal order. Their synthesis is based on the architectural
design of trees. Thus, these supramolecules are called *dendrimers *(from
the Greek word *dendron* for tree and *polymer*). The construction
of dendrimers can follow two basically different procedures. Divergent
construction begins at the core and builds outward via an increasing number
of repeating bifurcations. Convergent construction begins at the periphery
and builds inward via a constant number of transformations. The former
displaces the chemical reaction centers from the center to the periphery,
generating a network of bifurcating branches around the center. The bifurcations
increase exponentially up to a critical state of maximal size. They yield
fractal structures such as molecular sponges which can absorb smaller molecules,
which can then be dispersed in a controlled way, e.g. for medical applications.

Examples of cave-like supramolecules are the *Buckminsterfullerenes*,
forming great balls of carbon atoms. The stability of these complex clusters
is supported by their high geometric symmetry. The Buckminsterfullerenes
are named after the geodesic networks of ball-like halls which were constructed
by the American architect Richard Buckminster Fuller (1895-1983). The cluster
C_{60} of 60 carbon atoms has a highly Platonic
symmetry of atomic pentagons forming a completely closed spheroid.

Cave-like supramolecules can be arranged using chemical templates and
matrices to produce complex molecular structures. Several giant clusters
comparable in size to small proteins have been obtained by self-assembly.**
**Figure 3 shows a ball-and-stick model of the largest discrete cluster
(700 heavy atoms) ever characterized by X-ray structure analysis. This
cluster containing 154 molybdenum, 532 oxygen, and 14 nitrogen atoms has
a relative molecular mass of about 24 000. The highly symmetric "*big
wheel*" was synthesized by Achim Müller and co-workers (Müller
et al. 1995). Giant clusters may have exceptional novel structural and
electronic properties: There are planes of different magnetization which
are typical for special solid-state structures and of great significance
for material sciences. A remarkable structural property is the nanometer-sized
cavity inside the giant cluster. The use of templates and the selection
of appropriate molecular arrangements may well remind us of Fischer's lock
and key principle.

**Figure 3: **Supramolecular cluster in a ball-and-stick representation
as an example of a complex near-equilibrium system (Müller 1995, p.2293)

Molecular cavities can be used as containers for other chemicals or even
for medicaments which need to be transported within the human organism.
An iron-storage protein that occurs in many higher organisms is ferritin.
It is an unusual host-guest system consisting of an organic host (an aprotein)
and a variable inorganic guest (an iron core). Depending on the external
demand, iron can either be removed from this system or incorporated into
it. Complex chemical aggregates like polyoxometalates are frequently discovered
to be based upon regular convex polyhedra, such as Platonic solids. But
their collective electronic and/or magnetic properties cannot be deduced
from the known properties of these building blocks. According to the catch-phrase
"*from* *molecules to materials*” supramolecular chemistry
applies the "*blue-prints*" of conservative self-organization
to build up complex materials on the nanometer scale with novel catalytic,
electronic, electrochemical, optical, magnetic and photochemical properties.
Multi-property materials are** **extremely interesting.

The growth of modern natural science is characterized by increasing
mathematization and computerization. After physics in the 17th and 18th
century, chemistry has been involved in that process at the latest since
the 19th century. In a famous quotation Kant (1786) stated that without
mathematical laws chemistry could only be a collection of experimental
rules ('*Experimentallehre*') and no science ('*Wissenschaft*').
Since Kant's time the degree of mathematization in chemistry has completely
changed. But there is not only an application of mathematical methods which
are well-known from physics. Specific topics of chemical research has been
developed with specific mathematical methods and models.

In the first chapter, I started with molecular models as typical topics
of chemical research and applications of mathematical *graph theory*
since the 19th century. A physical justification of molecular models is
suggested in *quantum chemistry*. Furthermore, quantum chemistry provides
computational procedures for chemical features which are correlated to
specific molecular structures. Symmetry, dissymmetry and asymmetry are
essential properties of molecular structures which are analyzed in mathematical
*group theory*. The formation of very complex molecular systems has
tremendous importance for biochemistry with the exploration of living processes
and for supramolecular chemistry with the technical development of new
materials. These phase transitions can be modelled in the *theory of
complex dynamical systems*, which is a rapidly growing discipline of
modern mathematics. The mathematical *topology* of molecular structures
allows algorithmic procedures to compute correlated chemical features.
But modern *computer science* does not only provide increasing capacities
of computation, but also new insights of chemical research. *Computer
aided molecular design*, for example, is a technique that enables chemists
to simulate the structure, behavior, and interactions of molecules on computers,
in order to design new drugs, chemicals, and materials such as plastics
and ceramics and to test them in *computer experiments*.

Molecular models are typical topics of chemical research and applications
of combinatorical topology and graph theory in mathematics. They correspond
to, what I call, the *'structural view'* of modern mathematics. Topology
and group theory are typical mathematical disciplines which allow to classify
molecular structures and their correlated chemical properties (e.g., symmetries
of crystals, chirality of biomolecules, topological indices).

Chemistry does not only explore static structures, but dynamic processes
like chemical reactions as applications of kinetic equations. They correspond
to what I call the *'dynamical view'* of applied mathematics. In complex
molecular systems chemical interactions may be highly nonlinear and need
very sophisticated mathematical instruments such as the theory of complex
dynamical systems, bifurcation and chaos theory, or fractal geometry. Their
application indicates a clear tendency of research from the chemistry of
molecules to a chemistry of complex molecular systems. At the frontiers
of modern chemistry to biology, medicine, and material sciences the dynamics
of complex molecular systems plays a dominant role.

The prediction or determination of chemical events and properties needs
highly developed computational procedures of numerical mathematics, approximation
and algorithmic theory. Remember the ab initio computations in quantum
chemistry. Theses applications correspond to, what I call, the *'numerical
view'* of applied mathematics. Today, the numerical procedures become
more and more efficient by the increasing capacities of computer technology,
for example, the power of massively parallel computers.

However, chemistry is not only interested in numerical procedures, but
also in the construction of three-dimensional geometric models and the
derivations of linguistic terms such as chemical formulas. Today, these
activities can be assisted and even simulated by knowledge based expert
systems, three-dimensional computer aided molecular design (CAMD)-programs
and computer aided knowledge processing of AI-programs. Models, structures,
dynamical equations, numerical, graphical and symbolic processing are combined
and converted to computer programs. I call this aspect of modern computer
mathematics the *'program view'*. There is a clear tendency of research
in all natural sciences that the traditional experiment in the laboratory
is assisted by computer experiments. They are not only supplementary visualization.
Their programs provide strategies to refine scientific conceptions and
to focus the research in a productive manner. They help to prevent and
to select less productive, expensive, or even dangerous experiments in
the laboratory. But, of course, research cannot do without lab experiments,
inspiring and confirming research.

The epistemic perspectives of the structural, dynamical, numerical,
and program view demonstrate some typical topics of chemical research.
Furthermore, they indicate changing research tendencies. Frontiers between
the disciplines are removed or even broken down and new topics emerge.
Chemistry is involved in a growing *network of mathematical methodologies
and computer-assisted technologies* with increasing complexity. Thus,
chemistry is a *science* in the sense of Kant, but with dynamically
changing frontiers.

But molecular symmetry and complexity are by no means only theoretical concepts of mathematical methodology. Since the antiquity the structure of matter and the whole universe was represented by regular and symmetric models. In Platonic chemistry the variety of mathematical phenomena were reduced to the regular bodies of Euclidean symmetry. In Greek astronomy celestial movements were described by models of rational symmetry. Since the 19th century symmetries are not only defined as properties of geometric models, but as properties of physical and chemical structures, too. In this sense, symmetry means the invariance of a chemical structure with respect to spatial or time coordinates. The growths of new complex chemical structures can be understood as phase transitions close to or far from thermal equilibrium which often means symmetry breaking of spatial or time order.

In this sense, *symmetry* and *complexity* are fundamental
concepts of chemical research which refer to observable and measurable
features of matter. They are by no means only models in the head of a human
chemist or in the virtual reality of a computer. The symmetries or dissymmetries
of complex atomic structures obtained experimentally by the scattering
of electrons in super microscopes or from the scanning tunnelling electron
microscope can be observed by everyone. As once the Aristotelian opponents
of Galileo, one could object that there might be only some effects in the
measuring instruments causing our impressions. Of course, our knowledge
of molecular structures depends on our technical standards of measurement,
observation and theoretical representation. In the case of molecular structures,
these effects are well-known. Thus, molecular structures are as real as
Galileo's craters on the moon. In the antique-medieval philosophy of '*hyle*',
we can summarize these results of modern chemistry in the following way:
*Forms*, more or less *symmetric* and *complex*, are '*in
rebus*', although we can only construct representations '*in mente*'.

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Curie, P.: 1894: 'Sur la Symétrie dans les phénomènes
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Mainzer, K.: 1980, *Geschichte der Geometrie*, B. I. Wissenschaftsverlag,
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Mainzer, K.: 1997, *Thinking in Complexity*, 3rd ed., Springer,
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Mainzer, K.; Müller, A.; Saltzer, W.G. (eds.): 1997, *From Simplicity
to Complexity: Information, Interaction, and Emergence, *Part II, Vieweg,
Braunschweig.

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-} : ein wasserlösliches Riesenrad mit mehr als 700 Atomen
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New York.

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Paris.

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2nd ed., Springer, Berlin.

Quack, M.: 1993, 'Die Symmetrie von Zeit und Raum und ihre Verletzung
in molekularen Prozessen', *Akademie der Wissenschaften (Berlin) *Jahrbuch
1990 - 1992, de Gruyter, Berlin, p. 469.

Thom, R.: 1975, *Structural Stability and Morphogenesis*, Benjamin,
Reading MA.

*Copyright *Ó*1997
by HYLE and Klaus Mainzer*