Feature Article Symmetry Principles Have Been Applied at the Micro Scale Also

RITU PASRIJA AND SUNITA SRIVASTAVA

Nature loves symmetry. Whenever deviation from symmetry occurs there happens Article to be a certain imbalance. The appearance of this imbalance is what allows us to study nature, making symmetry an indispensable part of every Feature field of study. Beauty of Symmetry

Everything has beauty, but not everyone sees it. – Confucius

symmetry like a symmetric or a beautiful Symmetry is also a guiding principle YMMETRY is omnipresent in nature. flower, a symmetrical tree or any beautiful in scientific research. Scientists in different When humans began to observe shape. And so, humans have tried to apply fields understand symmetry differently but Snature they began to see various the principle of symmetry to design and ultimate the salient goal in many kinds of symmetries that were being followed for a build balanced and harmonious structures, research is to arrange things with a long time without deviation. Day and night and in science, to study unknown systems particular similarity in a group. In the follow each other without fail. It would from the known ones. language of mathematics, a group is a certainly be awkward if one of our ears The salience of symmetry in set of elements that satisfy certain functioned differently from the other or is design can be easily seen in its marvelous properties like associativity, identity and placed somewhere else on the face. use in a structure, a symbol or a building. Plants and animals have symmetrical Some religious symbols have certain bodies and shapes. When we remove the symmetry like the Judaic Star of David shell of a walnut, we see the wonderful (which has six-fold rotational symmetry), symmetry of walnut halves. Nature loves Christianity’s cross (which has bilateral symmetry and our notions of beauty are symmetry) and the Hindu Swastika (which closely tied to symmetry. has four-fold point symmetry). Taj Mahal is The term symmetry derives from the a beautiful example of a symmetrical Greek word ‘SUM’ (meaning ‘with’ or building. Monumental buildings like the ‘together’) and ‘METRON’ (meaning Egyptian pyramids and Cambodia’s ‘measure’). When we look at nature, Angkor Wat also use the principle of consciously or unconsciously our attention symmetry to enhance the beauty of their is attracted to objects that exhibit countenance.

19 SCIENCE REPORTER, AUGUST 2012 FFFeaeaeaturetureture Article Taj Mahal is a beautiful example of a symmetrical building. Monumental buildings like the Egyptian pyramids and Cambodia’s Angkor Wat also use the principle of symmetry to enhance the beauty of their countenance.

A wide range of biological systems possess approximate symmetries. We assume the organism to be symmetric in its ideal form.

invertibility – properties that are closely relations that are satisfied by various colonial algae called volvox exhibits this related to symmetry. Specific elements that form a class. symmetry. In radial symmetry the body is mathematical operations such as Further, group theory is related to cut into two similar halves by any vertical reflections, rotations and translations are representation theory. Representation plane passing through the centre of the used to describe the symmetric makes an abstract object concrete by body. Some special forms of radially interchange made within a system. One describing its elements by matrices and symmetric animals are sea anemone, jelly finds a relationship between symmetry and the algebraic operations in terms of matrix fish and sea stars. equivalence, between the elements of the addition and matrix multiplication making group – symmetry leads to certain it easier for us to understand the problem Symmetry in Mathematics leading to its solution. It is a prevalent In mathematics, symmetry principles or strategy in science to simplify a problem invariance under certain operations form by writing dynamical equations keeping in the basis of matrix representation theories mind specific symmetries in the system. A and also help us to find a solution to some representation of group helps reduce partial differential equations. Applied many theoretical problems in the physical mathematicians address complex natural sciences to problems in linear algebra, phenomena with mathematical tools; so which are reasonably well understood. there is often a blending of symmetrical Leaf with bilateral symmetry We consider examples of symmetry principles from mathematical operations in different scientific disciplines. and the natural phenomena it is trying to address. Symmetry in Biology Symmetric and numerical solutions for In biology, bilateral symmetry, spherical systems of nonlinear reaction-diffusion symmetry and radial symmetry is often equations find applications in biology and talked about. In bilateral symmetry or engineering in the study of transport of plane symmetry, the organism is symmetric oxygen from blood plasma to living tissue with respect to the plane passing through of the skeletal muscle or brain across the centre of the body, called the sagittal capillary walls. In these cases we need to plane, which divides it into right and left solve certain diffusion reaction equations. parts. The right part seems to be the mirror image of the left part (reflection symmetry). Most animals including humans are bilaterally symmetric, and have an identical shape on either side, as if cut by a mirror. In spherical symmetry, the organism or body is symmetric with respect to any Volvox aureus plane drawn through the body dividing it Sea anemone into exactly two equal halves. In nature,

SCIENCE REPORTER, AUGUST 2012 20 Feature Article Symmetry principles have been applied at the micro scale also. The symmetry properties of carbon nanotubes (CNTs ) are essential for interpretation of their Raman and IR spectra. Character tables for the symmetry groups satisfied by the CNTs are available in literature.

Coronene molecule of D6h Point group. A unit face-centered cubic crystal

Such linear and nonlinear equations need 1936 by Rosenthal and Murphy. Normal In classical mechanics, the first study mathematical models that are being modes of vibration are decided on the applying this notion of symmetry was developed using Lie-symmetries of partial basis of the symmetry it satisfies. E. Bright made by Hamilton and Jacobi who used differential equations and classical Wilson has used them to predict the the method of applying transformations symmetries of systems of nonlinear diffusion symmetry of vibrational normal modes. that leave Hamiltonian or Lagrangian equations. Symmetry restrictions help chemists in unchanged leading to the solution of understanding chemical bonding and equations of motion. Using such Symmetry in Chemistry molecular dynamics like analyzing transformations, it was possible to reduce In chemistry, molecular symmetry vibrations of molecules, studying molecular difficult problems into simpler ones that describes symmetry in molecules. Group orbitals and their wave functions. could be solved using simple theory is an important and useful For a perfectly ordered three mathematics. Poincare brackets and framework for the study of symmetry in dimensional crystallographic structure, the canonical transformations were a result of molecular orbitals. The s, p and d orbitals the above procedure. unit cell contains atoms arranged in exist in certain symmetric shapes. Elements In relativistic dynamics, the characteristic configuration such that its like helium, neon and xenon and beryllium, assumption of symmetry between rest continuous repetition throughout the magnesium and calcium are very frame and moving frame of motion led to crystal space results in generation of the beautifully decorated in the periodic table, the prediction of principle of relativity which entire crystal structure. Common unit cells which is a result of grouping them with was given by Galileo (Galilean are simple cubic, face-centered cubic respect to common properties. The transformations). It made possible to derive and body-centered cubic. molecules are studied by placing them in laws of nature and test their validity using Molecules of considerable interest various point groups on the basis of their symmetry properties. Another turning point structure. Molecules of water and pyridine are fullerenes, which are a class of large was Einstein’s special theory of relativity 2- molecular carbon chain substances. are kept in C2v point group, CO3 anion is that was based on two fundamental Fullerene is considered to be the third placed in D3h point group whereas postulates: the light postulate and principle allotrope of carbon apart from diamond benzene molecule refers to D6h point of relativity. Einstein for the first time used group. The characters of point group and graphite. The fullerene molecule (C60) transformations that depended on space operations are used in the study of ligand has a truncated icosahedral ‘soccer-ball’ field theory. structure which is spherically symmetric with Many theories using various kinds of a total of 12 pentagonal rings and 20 symmetries have helped to explain hexagonal rings. different chemical processes. Substitution Irreducible wedge of reactions shift the molecule from a higher Symmetry in Physics graphite lattice to a lower symmetry; this is what is called Symmetry in physics has been generalized symmetry-breaking. In case of a complex to mean invariance – lack of any visible molecule, where it becomes tedious to change under any kind of transformation. know about the number of isomers it The concept of symmetry then can be possesses, the use of symmetry comes to applied to abstract things or objects or our rescue. Therefore, the chemistry of dynamical equations of physical symmetry lies in the molecular shape, importance. Then one could predict that which decides its behavior in chemical a particular system is symmetric with reactions. respect to laws rather than to objects or We now have a complete set of 32 phenomena, or one could say that the crystallographic point groups published in system has internal symmetry.

21 SCIENCE REPORTER, AUGUST 2012 Feature Article Nobel Prize in Physics for 2008 being The term symmetry derives Interestingly, this implies awarded to Yoichiro Nambu of Enrico Fermi Institute at University of Chicago for from the Greek word ‘SUM’ that the word ‘symmetric’ discovery of the mechanism of (meaning ‘with’ or should not be taken as an spontaneous broken symmetry in ‘together’) and ‘METRON’ subatomic physics. Symmetry breaking absolute truth – there is does not imply that all of the symmetry of (meaning ‘measure’). When hardly anything that the system is lost. Rather that the system we look at nature, shifts to a lower symmetry subgroup related possesses absolute to original symmetry group through some consciously or unconsciously symmetry. transformation relation. The symmetry- our attention is attracted to breaking concept has helped us to objects that exhibit symmetry theoretically explain the existence of mass- and time referred to as local symmetries. less bosons. According to Higgs, when like a symmetric or a Noether in 1918 derived a connection internal symmetry is promoted to a local between global symmetries and beautiful flower, a symmetry, the mass-less bosons disappear conservation laws. Since then, many and the gauge bosons acquire mass. This symmetrical tree or any problems in physics derive solutions to local phenomenon was said to be responsible symmetries by taking global symmetry as beautiful shape. for giving masses to gauge vector bosons a subgroup of some local symmetry. like W and Z. In quantum theory, symmetry It is also believed that an unexplained are applied to it or its environment is principles were found to be most effective. broken symmetry of some kind lies behind changed due to some reason, a loss of Wigner and Weyl were the first to apply the very origin of the cosmos in the Big symmetry occurs. This loss of symmetry symmetry of groups to quantum physics. Bang some 14 billion years ago. If equal changes the system’s properties, which Quantum states are related to each other amounts of matter and antimatter were enables the system to be studied. through different transformations that created, they ought to have annihilated Interestingly, this implies that the word implicitly conserve the symmetry of the each other. But this did not happen; there ‘symmetric’ should not be taken as an system. The transformations are operations was a tiny deviation of one extra particle absolute truth – there is hardly anything that relating the states corresponding to of matter for every 10 billion antimatter possesses absolute symmetry. A wide operators, which in turn, are related to particles. It is this broken symmetry that range of biological systems possess physical observables or quantities. seems to have caused our cosmos to approximate symmetries. We assume the Eigenvalue spectra can represent an survive. However, what exactly happened organism to be symmetric in its ideal form irreducible representation of the symmetry still remains unanswered. May be some and then exploit the system to study the group and the symmetry of Hamiltonian new experiments would explore an answer mechanics and dynamics by making slight then would correspond to particular to this in the future. changes in the ideal form. In this way, we conserved quantity. Symmetry principles have been are able to draw conclusions of what Permutation symmetry is a discrete applied at the micro scale also. The might happen if the symmetry is close to symmetry that says that a state is symmetric symmetry properties of carbon nanotubes the ideal case and not exact. We have if we swap indistinguishable quantum (CNTs ) are essential for interpretation of already seen from the CTP theorem and particles, which led to quantum statistics their Raman and IR spectra. Character symmetry breaking examples how new of bosons and fermions. It is related to the tables for the symmetry groups satisfied insights have been derived even if the initial indistinguishable nature of particles that by the CNTs are available in literature. assumption of symmetry is not entirely could be applied to Bose-Einstein statistics Helical and rotational symmetries are used correct. and Fermi-Dirac statistics. The famous CPT to define different types of graphite tubules Symmetry truly reveals the beauty of theorem postulated by Luders in 1952 constructed by rolling up a single graphite the natural world or scientific states that C (charge conjugation), P sheet. Using such symmetries graphite representation. The famous Victorian (parity) and T (time reversal) in combined tubules have been predicted to be designer Owen Jones said, “That which is manner are a general symmetry of moderate bandgap semiconductors. beautiful is true; that which is true is physical laws. In other words, the laws beautiful.” We can add “and that which is governing gravitational forces, Symmetry: The Essence of true and beautiful is symmetric!” electromagnetic forces and strong forces are invariant with respect to charge Nature or a Useful Lie for conjugation, parity and time reversal Science? Dr Ritu Pasrija is Assistant Professor in Physics at independently. Weak interactions violate Nature appears to love symmetry! DAV College, Abohar (Punjab). She is presently parity (symmetry of reflection) as seen in Whenever deviation from symmetry doing research in Nano Science at the Department the case of beta decay. However, beta occurs, there also occurs a certain of Physics, Panjab University, Chandigarh. decay respects a combination of C and P imbalance. As long as the system is Dr Sunita Srivastava is Assistant Professor in symmetry. symmetric or invariant, no change is Department of Physics, Panjab University, A new concept of ‘symmetry observed in its behaviour or properties. Chandigarh. Her area of research is Condensed breaking’ is popular in particle physics When the system tries to interact with the Matter Physics. Email: [email protected], nowadays. Passion for symmetry led to the surrounding environment or some forces [email protected] SCIENCE REPORTER, AUGUST 2012 22