The Intricate And Intriguing Dimensions And Manifestations Of Symmetry

My contributions to THE HINDU, one of India's most prestigious English-language newspapers, were spread over half a century in successive spells and columns which had several different characters and colors.  The initial spell in the 1960s featured articles cast in the mould of classical English essays ;  this was followed by reviews and sketches on the Carnatic and Western music scenarios in India's capital city New Delhi in the late '80s and early '90s. 

I also wrote a column titled 'Articulations' in 1990-93, recording the reflections of an articulate layman on the intricate manifestations of art and culture, as well as some psychological or philosophic aspects of life.  In the following recent blog I had explained how this column happened to be launched :   When & How Chairman Haksar Wrote His Monumental Essay OnArt & Culture.   

Earlier, in several successive blogs I had also recalled half a dozen essays on friendship, starting with the following one :  Frontiers Of Friendship : How, Beyond Natural Boundaries, Close Friendship Becomes Troublesome Bondage! 
            
Let me now  share with you a set of three seminal essays on the concept of symmetry, which were followed by a lively exchange of clashing ideas between an eminent Indian scientist and myself :-            

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THE HINDU

Sunday Magazine

17 November 1991

Reflections on symmetry

     In my last column I had expressed the view that other things being equal, the universality of music in terms of space and time depends on the degree of its symmetry.  This raises a pertinent issue : what exactly constitutes symmetry in music?  For finding a convincing answer one must naturally start with a proper definition of symmetry.  However, the effort to expain tends to become an exploration, and one is astounded by the far-reaching significance and apparently unlimited manifestations of symmetry.  So then, let music wait!

     The true meaning of symmetry is not limited to parallelism -- namely that two things (or two sides of the same thing) look alike in an absolute or inverted sense -- although that is the most popular connotation of the word.   In a much larger frame of reference, symmetry implies good proportion, balance, harmony and equilibrium ;  concordance, congruity, consonance and consistency ;  and even orderliness, grace and beauty.  And if one took a careful look at things,  one would realize that symmetry so defined exists on an intriguingly wide scale in nature, as well as in science, art and other concerns of civilized life.

Symmetry in universe

     Nowhere in nature is symmetry present as a more awesome phenomenon than in the meticulous order of the universe, where countless celestial bodies of colossal size exist in the form of noble spheres  -- either in a solid form or as a mass of energy -- and are endlessly tracing elliptical paths in the cosmos without  collision, indicating a perfectly balanced equilibrium.  The gravitational forces which are responsible for this amazing consistency in the perpetual motion and recurring relative positions of stars and planets do not operate in an arbitrary manner, but obey certain eternal laws of nature.  That is why astronomical research and predictions are able to reach out to infinity in terms of time and space.

     There is great symmetry in the way light behaves . . . in the infinite directions in which it tends to get dispersed unless obstructed . . .  in the constant and inexceedable speed at which light beams travel, and in the absolutely straight line of their course which can stretch over millions of light-years till they encounter something which stops, retracts or reflects them.  There is symmetry in the refraction of light . . .  in the way a prism breaks it up into the basic colors of the spectrum, or a wet sky conjures up a magnificent rainbow.  There is symmetry in the constant speed of sound, and in the echo which results when it is reflected.  There is symmetry, too, in the velocity of electricity and radio waves, and in the radiation of heat and other forms of vital energy. 
 
Symmetry on earth

     There is symmetry in the shape of the earth, and in the composition of its atmosphere.  One can see perfect order in the latitudes and longitudes which are man-drawn lines on paper or on a globe.  There is symmetry in the directions -- north, south, east and west -- and in the compass which indicates them.  There is consistency in the altitudes at which different kinds of clouds float or cruise in the sky ;  there is symmetry in the waves of the seas and their alternating tides.  There is symmetry in one's view of the horizon, which resolves itself into a perfect arc or circle as one moves up in the sky regardless of the nature of the landscape, in the way lighthouse beams constantly sweep and scan the horizon, and in the way they are observed from any given spot as flashes which recur with unerring regularity. 

     There is great symmetry to be found in crystals and snow-flakes.  Trees may not always be symmetrical in shape unless they are coniferous or palm ;  but leaves, flowers and fruits are almost always symmetrical in form.  Even where trees and plants grow in irregular shapes, the configuration of forests has an inherent pattern.  There is symmetry in the atoms and molecules of the elements, whether they are in a solid, liquid or gaseous state, and in the temperatures at which those conditions are altered.  There is precision in the annual cycle of the seasons :  spring, summer, rain, autumn and winter.  There is symmetry in the way water invariably finds its mean level no matter how intricately it is connected -- particularly in the way the surfaces of all the oceans have a common, immutable level which forms the basis for all geographical calculations of altitude and depth.  There is symmetry in the flame of a candle or oil-lamp, and even in the volatile flames of a blazing log-fire. 

     There is a strong element of symmetry in the physical features of most living creatures . . .   in the limbs and bone structures of mammals ;  in the twin eyes and ears which make three-dimensional vision and stereophonic hearing possible ;  and in the almost invariable matching of the left and right sides of animate bodies.  There is remarkable balance in the shape of fish, which are streamlined by nature to overcome the resistance of water, and in the wings which sustain the airborne condition of birds.  There is symmetry in the traits which characterize any given species on this earth ;  and even between different species there is a striking similarity in the sexual division and functions.     

Symmetry in shapes

     So far as shapes are concerned, symmetry invariably  concerns two dimensions, and sometimes three.  It will hardly be disputed that the circle and the sphere are the most perfectly symmetrical shapes which exist.  Symmetry can be axial, radial or bilateral, depending on whether it has a reference to a bisecting line, central point or dividing plane.  When we consider only two dimensions, we observe the symmetry of the circle, ellipse (or oval), square, rectangle, and equilateral triangle, pentagon, hexagon, and so on.  

     In three dimensions, we find symmetrical structures in the sphere, cone, pyramid (which you might call a 'square cone'), cylinder and cube.  The whole exercise of solving the intricate problem set up by the Rubik's Cube is a quest for restoring a symmetrical scheme of things from an extremely disturbed state.

Symmetry in science

The fundamental connection between nature and science is obvious, and has two distinct aspects.  Science is concerned with the discovery and interpretation of natural phenomena or laws ;  it also seeks to counter or harness the forces of nature for the purposes of mankind, whether these are constructive or destructive.  And since symmetry figures strongly in nature, so it does in science also.

Thus, the discipline of physics is concerned with deciphering the symmetry which governs the universe as well as the microscopic atoms of the elements.  Physics and astronomy together deal with complex issues concerning gravity, relativity, displacement in space and time, homogeneity of space, properties of heavenly bodies -- all of which raise vital questions of symmetry.  In chemistry, we come across equations and formulas which indicate the reactions which ensue when elements interact with one another, altering the symmetry of things.  Geology studies the symmetry of the earth's structure.  In biology scientists are deeply rooted in nature, studying the symmetry of plant and animal life ;  research in microbiology is essentially  concerned with the symmetry of living cells.

     No less significant is the status of symmetry in mathematics.  It has a conspicuous presence in geometry, which deals with magnitudes in space and the visual shapes of things ;  it is also present, though less perceptibly, in other branches of maths like algenra and calculus.  Of course, it is difficult to mark the borderline between mathematics and certain other sciences like physics and astronomy which are concered with the symmetry inherent in the laws of nature.  In statistics, there is always symmetry in the compilation and tabulation of data and their systematic analysis.

Symmetry in technology

     Symmetry can be found in the elementary tools devised by primitive men as well as in the sophisticated machines produced by high technology today.  There was symmetry in the blunt instruments of the stone age and in the original wheel whenever it was invented, as there is in the complicated gears or mysterious electronic systems of modern machinery.  There is symmetry in the way a sensitive gyroscope works, or in the way a simple bicycle in motion balances itself.

     The physical shapes which vehicles have assumed from early times to the present day provide sriking examples of symmetrical forms.  The streamlined structure of boats and ships, aeroplanes and rockets is the result of the necessity to overcome the resistance set up by natural forces.  The basic objective of hydrodynamics and aerodynamics is to discover the most effective symmetry of objects moving in water and air.  Of course, it took researchers condiderable time to realize that automobiles and rail-bound trains moving on the ground also need streamlining. 

Symmetry in standards

     Symmetry is inevitable in standardization, whether it concerns instruments, weights and measures, currency or the diverse products of modern industry.  A forceful illustration of this is provided by the identical specifications of gramophone records, photographic and cinematic film and magnetic recording tapes adopted all over the world.  We tend to take things for granted when we casually play a long-playing record or video cassette made in India or Bulgaria on a reproducing machine made in America or Japan.  The remarkable symmetry of the whole system which makes this possible may not be perceived unless one consciously looks for it, but it is there on a global scale all the time.   By contrast, the spread of the computer culture in the world today is perhaps retarded to some extent because of the inadequate symmetry of the existing software, although the computer itself represents a marvelous symmetry in science and technology. 

                                                               (to be continued)  
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PostScript, 2018
Symmetry in articulations

Looking at this tightly-packed text written more than 25 years ago, I can't help feeling that it couldn't be so relevant and readable today -- as I am sure it's likely to be even after a hundred years from now -- if there hadn't been a substantial element of symmetry in the whole composition.  This impression is strongly reinforced by the two essays which followed, as you will see.  But  then, of course, that must be true of almost all my reflections as an articulate layman!