General aspects of symmetry
Some pages of this chapter have been taken from Dr. Horst Bögel

Certainly, anybody has a notion of what "symmetry" means. Investigations in nature and perceptions of the work of artists and architects can be summarized in a way that symmetry is based on a periodic repetition and variation of certain patterns and structures and that symmetry is often related with feelings of harmony and beauty.

Symmetry of molecules

The theory of symmetry provides a valuable set of terms to describe structures and phenonema widely distributed in nature, e.g. crystals, molecules, β-spin, materia and antimateria.

The symmetry of a molecule is of fundamental importance in quantum chemistry. This symmetry is tightly linked with properties of wavefunctions that describe the molecule, and in group theory, theoretical chemists and physicists analyse such correspondences.

A great deal of group theory is neither more nor less than some summary of our obvious experiences about the symmetry of objects. The group theory's approach is very schematic and systematic and sometimes, it leads to unexpected results. The methods of group theory often relieve from complicated calculations. In short, we need group theory as it allows to predict the behaviour or the properties of a molecule.

The roots of group theory

The term "symmetry" has been created in ancient Greece. In these days, for sculptors and other artists as well as architects, symmetry was linked to beauty and harmony. There are several examples that, in PYTHAGORAS' days (580-500 B.C.), the term was widely used in art, philosophy and science. It was not limited to the context of geometry but characterized as well the quality of being well proportionate and displaying harmony, of some concordance between parts and rules that unite them to something complete. For example, the Pythagoreans were interested in the correspondences within an object or underlined contrasts in the classification of things, i.e. image and mirror image, left and right.

In this context, we shall recall the Greek atomists LEUKIPP (presumably 500-440 B.C.) and DEMOKRIT (approx. 430-370 B.C.) and their thesis that unalterable atoms were the smallest units of matter. They stated that each atom is characterized by shape, size and weight and that they were subject to permanent motion.

ARISTOTELES his disciple PLATON (427-347 B.C.) founded another theory to explain the world. They assumed that anything that exists in the universe is composed of four "elements": Fire, earth, air and water. In contrast to the atomists that imagined atoms of all kinds of shape and symmetry, atoms, according to Platon, were object of simple geometry. The earth was said to be composed of cubes, water of icosahedrons, fire of tetrahedrons and air of octahedrons.

We can summarize that, beginning with the first contributions by the mentioned scholars, a theory of symmetry developed more or less independently in three directions: in philosophy, science and mathematics.

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