Numbers that encode deepest secrets of nature
It is well known that the discipline of mathematics makes our lives orderly and prevents chaos. It also helps us develop the faculties of reasoning,...
It is well known that the discipline of mathematics makes our lives orderly and prevents chaos. It also helps us develop the faculties of reasoning, creativity, abstract thinking and even improves our ability to communicate.
The dominant role mathematics has to play in subjects like engineering and architecture is only too well known.
Mathematics has also deep and wide applications in the fine arts such as painting where parallelism, symmetry and accuracy play an important role – arts such as music and painting. Music, also, is highly mathematical. Harmony and chord structures essentially denote pleasing sounds produced when the ratio of their frequencies can be written as a fraction with a small denominator – basically a mathematical result. Both mathematics and music use patterns. The legendary composer Bach was able to write in every key so successfully because mathematicians had found better ways of determining the square 12th root of two, a very important number in the theory of Western music as it represents the frequency ratio (music interval) of semi-tone in 12 – tone equal temperament. Mathematics is also widely used in art. In three-dimensional art for instance, it is used in drawing flat objects which give the illusion of being three-dimensional objects when viewed from certain points of view. The legendary da Vinci used mathematical principles of linear perspective-parallel lines, the horizon line and the vanishing point – to create the illusion of depth on a flat surface. His famous painting "The Annunciation" is a classic example of this method.
Embedded deeply in the recesses of the mysterious architecture of the organisation of natural phenomena are many relationships which way denote by numbers. Perhaps the most common of these is the relationship between the circumference of a circle and its diameter whose value is approximately ratio 22/7 and is represented by the Greek letter 'pi'. The relationship holds respect of the size of the circle. 'pi' is, therefore, a universal constant.
Then there is the number 'e' called Euler's number which represents the base of natural logarithms. To understand what this means consider
10 x10 x10 =1000.
In other words 10 to the power of three equals 1000.
In that case it is said that the logarithm of 1000 to the base 10 is 3.
The natural logarithm of a number is its logarithm with the natural number e as the base.
Without going into too many boring technical details suffice it to say that logarithms are powerful mathematical tool which greatly reduce the drudgery of long and tedious mathematical calculations.
Another number that is not so commonly known but is of equal significance in natural phenomena is what is known as the Feigenbaum number. It occurs in the context of the study of properties of dynamical systems such as flowing rivers, the behaviour of clouds, in the context of making meteorological forecasts and yes(!) studying and predicting the variations of prices in the stock market.
Let us consider a familiar example.
Water is in a stable state when in the liquid form, and also in a stable state when it turns into steam. But it has to pass through the chaotic process of boiling to transit from one state to the other. Similarly, many natural systems travel through a chaotic phase while moving from one stable state to another. And, example is the case of a river flowing towards a drop where it becomes a waterfall. It is during this transition, from one stable phase to another, that, while approaching the phase of chaos, a process called 'periodic doubling' takes place. And the value at which the process of doubling finally ends up is, remarkably enough for the variety of phenomena in which it happens, the Feigenbaum number!
The next number we encounter on our journey is the universal constant 'G', known as the gravitational constant and discovered by Sir Isaac Newton. The gravitational force between two bodies is given by the product of their masses divided by the square of the distance between them and multiplied by the gravitational constant. Another remarkable example of how mathematical numbers convey the deepest secrets of the universe.
A related constant is 'g' which represents the acceleration due to gravity. An interesting fact associated with this constant is that it remains the same no matter what the mass of a body is, as was (at least apocryphally) demonstrated by Galileo Galilee by dropping metal and wooden objects, including gold and silver coins from the top of the leaning tower of Pisa.
Another number that attracts one's attention in the study of natural phenomena is Planck's constant. A number fundamental to the discipline of quantum mechanics, that constant figures in the relationship between the energy of light and its frequency, the former having been shown to be proportional to the latter by Max Planck, the discoverer of the constant. The discovery (namely that light behaves both as particles and waves), incidentally, signalled the end of the classical view of the universe which had, until then, held that light comprised particles which travelled in a straight line.
Another fascinating number is Fibonacci number. The Fibonacci series is obtained by constructing a sequence in which each number is the sum of the two numbers preceding it. The series look like this.
The Fibonacci number is the ratio of a term in this sequence and its predecessor and is a constant (1.618). This is also known as the Golden Ratio the Golden Mean or, sometimes, even the Divine Proportion.
The human body is also based on patterns of five. There are five appendages to that torso, the arms, the legs and the head. There are five appendages on each of these, the fingers and the toes. And openings on the face and five sense organs sight, sound, touch, taste and smell.
Strangely enough, the concept of the Golden Mean figures in such diverse disciplines as dentistry and cardiology.
"An equation has no meaning unless it expresses the thought of God" – Ramanujan. John D Barrow has written "constants of nature– the numbers that encode the deepest secrets of nature- fundamental concepts of cosmic importance which our universe, make existence possible and will ultimately decide the ultimate fate of the universe.
'C' the speed of light which Michaelson and Morley showed that was independent of direction and Einstein postulated that it represents the upper limit of velocity in the universe. R' The ideal gas constant that represents the proportion between the products of pressure and volume and temperature of a gas.
'R', the ideal gas constant that represents the proportion between the products of pressure and volume and temperature of a gas. Avogadro's number or the number of molecules in mole of a substance.
At the risk of being accused as being biased in favour of mathematics, having been a student of that discipline until the post-graduation stage, this columnist must point to the obvious connections of a fundamental nature between mathematics and physics, physics and chemistry, chemistry and biology and, then, biology and life itself.
All life, in other words, can be explained by mathematics! Little wonder, then, that Bertrand Russell said of mathematics that it possesses "......not only truth but supreme beauty, a beauty cold and austere…."
(The writer is former Chief Secretary, Government of Andhra Pradesh)