## Introduction

Everyone has some intuitive idea of what symmetry is: we recognize bilateral symmetry in our bodies, we enjoy the rotational symmetry of many flowers, admire the fragile regularity of snowflakes, and are fascinated by the complex but at the same time simple polyhedral shape of crystals. As we encounter plenty of symmetrical objects in our everyday life, it should be no surprise that symmetry arguments have entered, either consciously or not, as a basic ingredient in the conceptual models developed to explain the physical world. Symmetry acquired its basic place in science through the deep influence of Greek philosophy in western culture. Prominent examples of this important role of symmetry along the history of science are, for instance, Plato's attempt to relate the properties of matter with those of the highly symmetrical regular solids,[1] the polyhedral model of the universe described by Kepler in his *Mysterium Cosmographicum*,[2] or the actual models of particle physics, all of them relying heavily on symmetry-based arguments.[3]

The influence of symmetry in science has been profound at all levels, not only in theories trying to explain the overall behavior of nature. The definitive boost to this tendency came at the end of the XIX century with the formalization of symmetry through the development of group theory.[4] In this respect, crystallography was probably the first scientific field where group theory was applied systematically to rationalize and explain the origin of the geometrical regularity in crystals which turned out be strongly related to the behavior of matter at the atomic level. These ideas permeated into related fields such as chemistry where the postulate of van't Hoff[5] and Le Bel,[6] that the four valencies of carbon where arranged in a tetrahedral disposition opened definitively the door for applying symmetry in chemistry. The prominent role of symmetry in modern chemistry came with the development of quantum mechanics and its application to explain chemical bonding. In modern quantum chemistry, symmetry is not only applied to molecular geometry but also to molecular vibrations, operators, wavefunctions, or to molecular orbitals. For instance, simple arguments based on the symmetry of molecular orbitals along a reaction path allow the classification of certain reactions as being either symmetry-allowed or forbidden. The proposal of the Woodward-Hoffmann rules and the concept of “Conservation of Orbital Symmetry,”[7] some 40 years ago established in this sense an indissoluble link between theoretical chemistry and symmetry that led to the introduction of group theory and its applications in any modern standard chemistry curriculum.

From the point of view of a modern chemist with access to the wealth of structural information contained in data bases, it should be remarkable that these theories derived for idealized symmetric molecular geometries can be applied without major revisions to obtain rational explanations for many experimental observations, even if we know that the vast majority of molecules occurring in nature or artificially created in a laboratory have no symmetry at all. It is precisely when trying to understand this apparent contradiction that one of the major shortcomings of the traditional approach to symmetry becomes evident: symmetry has been usually defined as a “black or white” property, that is, a molecule either has a certain symmetry or not. In this respect, small atomic displacements with an imperceptible influence in the physical properties are, however, sufficient to destroy its symmetry. A possible solution to this problem is based on the generalization of the concept of symmetry to describe it as a continuous property allowing a progressive series of “gray shades” between the “black” or “white” situations.