The World of Symmetry

The sky above is adorned with sun and stars the earth below by full of lives and that I found for me a place on it, flabbergasted is aroused, awakened my heart. 

An attempt has been made so far in the earlier chapters to introduce this symmetry present in crystals, which are three dimensional patterns, and the processes and techniques that are followed to know and find this symmetry. Now perhaps it is high time to make an attempt to visualize the symmetry present in nature. Some most common examples are cited only with the intention to get the readers induced if not fully but at least partially so that they would be tempted to peep through the window and have a glimpse of this world of symmetry that lies around us. Now, questions may arise in our mind that what is the domain of this world symmetry What are its upper and lower limits If the upper limit is taken as the galaxy our solar system belongs, the lower limit may be taken as the structure of DNA or even lower (Figs. 9.land 9.2). 

Let us start with the helical spiral symmetry that our Galaxy exhibits in the Domain of Symmetry. It is interesting to note that both these structures, that is, the Galaxy and DNA show the helical structures. Now within these limits let us examine the symmetry present and we first start with the symmetry present in living bodies. 

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Therefore, there is always some asymmetry in the world of symmetry. Some of the things are not perfectly symmetrical and even if they are symmetrical there is always some gradation of the order of syimnetry, that is, from no symmetry to perfect symmetry through the zone of partial symmetry. Sometimes the lack of perfect symmetry at least in some cases is an advantage as because materials having partial symmetry may exhibit some property features, which are special for this state of existence and which the perfectly symmetrical material or phenomena fail to deliver. 

Carbon 60 (C60, Buckyball) is this third form of carbon, discovered in 1985 by Richard Smalley, Harold Kroto, and Robert Curl for which they won the 1996 Nobel Prize in chemistry. It is named as Buckministerfuller to honor the architect of the geodesic dome, Buckminster Puller, because the dome s shell resembles the fullerenes hollow-core construction. Fullerene structure of carbon is face-centered cubic having carbon molecules at the corners and at the center of the faces and belonging to the fullerene family. In the world of symmetry it is definitely a new form of pattern created by the existing symmetry operations. 

Symmetry breaking is a universal phenomenon, from eosmology to the microscopic world, a perfectly familiar and daily experience whien should not generate the reluctance that it induces in some domains of Physics, and especially in Quantum Chemistry. In elassieal physics, the symmetry breaking of an a-priori symmetrical problem is sometimes refered to as the lack of symmetry of the initial conditions. But it may be a deeper phenomenon, the symmetry-broken solutions being more stable than the symmetrical one. 

Symmetry is a common phenomenon in tlie world around us. IT Nature abhors a vacuum, it certainly seems to love symmetry It is difficult to overestimate the importance of symmetry in many aspects of science, not only chemistry. Just as the principle known as Occam s razor suggests that the simplest explanation for an observation is scientifically the best, so it is true that other tilings being equal, frequently the most symmetrical molecular structure is the preferable one. More important, die methods of analysis of symmetry allow simplified treatment of complex problems related to molecular structure. 

The problem of symmetry breaking (SB) is well known and multiply discussed in literature. Briefly, we can formulate it as follows. The Hamiltonian of any system of particles forming the Universe is totally symmetric with respect to rotations and reflections in the isotropic space-time, as well as transmutations of identical and equivalent particles, whereas the real objects of the material world composed by these particles do not possess such symmetry. This is seen already from the examples that we live in a world of particles, not antiparticles, and in condensed matter, we have mostly low-symmetry structures. This circumstance can be expressed by the statement that the world is in a state of broken symmetry. An obvious explanation of the contradiction between the totally symmetric Hamiltonian and the broken symmetry of the real world is that the latter is not a solution of its Schrodinger equation. 

The world around us abounds in symmetries and they have been studied for centuries. More recently, research has probed into the role of symmetry in human interactions along with representatives of the animal kingdom. Special attention has been given to mate selection. One of the first appearances of this facet of symmetry in the popular press was an article in The New York Times, with an intriguing title, Why Birds and Bees, Too, Like Good Looks [3],... 

Many primitive organisms have the shape of the pentagonal dodecahedron. As will be seen later, pentagonal symmetry used to be considered forbidden in the world of crystal structures. Belov [82] suggested that the pentagonal symmetry of primitive organisms represents their... 

Symmetry considerations are fundamental in any description of molecular vibrations, as will be seen later in detail (Chapter 5). First, however, the molecular symmetries will be discussed, ignoring entirely the motion of the molecules. Various molecular symmetries will be illustrated by examples. A simple model will also be discussed to gain some insight into the origins of the various shapes and symmetries in the world of molecules. Our considerations will be restricted, however, to relatively simple, thus rather symmetrical systems. The importance and consequences of intramolecular motion involving relatively large amplitudes, will be commented upon in the final section of this chapter. 

It should be reemphasized that the above high-symmetry examples refer to isolated molecules and not to crystal structures. Crystallography has, of course, been one of the main domains where the importance of polyhedra has been long recognized, but they are not less important in the world of molecules. 

The concept of symmetry is of ancient vintage and in many ways almost identical with the equally elusive concepts of beauty and harmony, i.e. beauty of form arising from balanced proportions. Although symmetry can be described in mathematically precise terms, symmetry in the physical world, like beauty1, never absolutely obeys the mathematical requirements of group theory even the most perfect crystal has a surface that spoils the symmetry. 

To keep this proposition in focus, equivalence relationships, in the sense of symmetries in the physical world, may be defined in terms of a metric for the state space of a system. The metric [14] is a real non-negative function d(, ) with the following properties for all states u, v, w ... 

It should be noted that this book, both in its structure and in its content, is in many respects logically related to our earlier booklet I. S. Dmitriev, Symmetry in the World of Molecules, Mir Publishers, Moscow, 1979. This relation is not accidental since both topology and symmetry of molecules furnish a valuable qualitative and semi-quantitative complementary information about theif structure and properties. 

If a molecule possesses certain elements of symmetry, MOs are easier to calculate and classify using the group theory (see our hook Symmetry in the World of Molecules). 

See p. 28 in the book Symmetry in the World of Molecule mentioned above,... 

Symmetry is a phenomenon of the natural world, as well as the world of human invention (Figure 4-1), In nature, many types of flowers and plants, snowflakes, insects, certain fruits and vegetables, and a wide variety of microscopic plants and animals exhibit characteristic symmetry. Many engineering achievements have a degree of symmetry that contributes to their esthetic appeal. Examples include cloverleaf intersections, the pyramids of ancient Egypt, and the Eiffel Tower. 

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