Sequence and order permeates all spheres of life. One remember the orderliness and maniacal love of geometrical angles of Hercule Poirot. We are discussing mechanisation of creativity, an apparent contradiction in terms. Creativity is the essence of that which is not mechanical. Yet every creative act is mechanical – it has its explanation no less than a case of the hiccups does. The mechanical tranche of creativity may be hidden from view, but it exists.
The fourteen lines in Shakespeare sonnet are written in a meter called iambic pentameter, a rhyme scheme in which each sonnet line consists of ten syllables. The syllables are divided into five pairs called iambs. A iamb is a metrical unit made up of one unstressed syllable followed by one stressed syllable. Unrhymed iambic pentameter is called blank verse. Michael Madhusudan brought this rhyme in India while writing his sonnets in blank verse in Bengali.
Lest start with simple concepts like Fibonacci series : 1, 1,2,3,5,8,13,21,34,55,89,144, 233,……,.After the initial pair of ones, each succeeding number is simply the sum of the previous two. Sunflowers have seeds spiralling around the centre. Some spirals contain seeds leaning clockwise while other spirals going the other way. On most of them, the number of clockwise and anticlockwise spirals are adjacent Fibonacci numbers. Ancestry of bees, structure of pine cones, topology of spiralling galaxies to modern day applications of wealth & insurance management reflect Fibonacci sequence. The ratio of each successive pair of numbers in Fibonacci sequence approximates phi ( 1.618..). The Great Pyramid, Partheon, Colosseum incorporate phi, the Golden Ratio.
Tenney conceived his piece ‘ For Ann’, which plays twelve computer generated tones that glissando upwards, as having each tone as a golden ratio below the previous tone ( in between an equal tempered minor and major sixth). The combination tones produced by consecutive tones are a lower or higher pitch than that played already, or soon to be produced. We see reflection of golden ratio in Debussy’s “ Reflections in Water”, where the sequence of keys is marked by the intervals of 34,21,13 and 8 ( a descending Fibonacci sequence) and the main climax sits at the ‘phi’. This has been beautifully explained by Hofstadter.
Try dividing N, a natural number, by 2, then by 3, then by 3,4,5,6 till N-1. If N survives without being divisible, we call N a prime number. Each step in this test of dividing N is similar to one another but not the same. This is known as a loop. A fixed set of instructions to be repeated , need not be actually fixed. It may vary in some predictable way. All managers of corporate know this as do all line managers on the shop floor. A simple C Major scale in one octave is a small loop. I can play this in two octaves, in three and so one, perhaps displacing pitch every time. The last movements of Prokofiev’s fifth piano concerto and Rachmaninoff’s second symphony has passages in which fast, medium and slow scale loops are played simultaneously by different ensemble of instruments. The effect is awesome ! We clearly see this in our Ragas. Avaroha in Darbari is sung or played as S’ d~ n P m P n g~ m R S. The table enters in Asavari that after the initial prelude of Aaroha. Darbari requires “andolan” or “a light shake/oscillations” for komal ga and komal dha (marked as g~ and d~). If one has a draw a parallel to the oscillation then we think of Sine Curve. Intensity of a sound wave is similar to oscillation of a spring. We colloquially mention of Sine Curve in a long career.
Strange loop ( a term coined by Hofstadter) is manifested in the canon’s of Bach’s ‘ Musical Offerings’. In canon, one theme is played against itself. ‘ Three Blind Mice’ and ‘ Row, row, row your boat’ are the straight forward rounded canons. The theme enters in the first voice and after a fixed time delay , a copy of it enters in the same key. After a fixed time delay, the third voice enters the same theme and so on. Each of the notes of a canon must be able to serve in a dual, triple or quadruple role : it must be part of a melody and it must be part of the harmonisation of the same melody.
Pythagoras observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer. He also was intrigued by the fact that if the length of the two metal strings are in relation to each other 2:3, the difference in pitch with the same hammer is called a fifth. If the length of the strings are in relation to each 3:4, then the difference is called a fourth. Music was created by him thereafter in geometrical shapes and designs. We see congruence in Fine Arts. ‘ The Last Supper’ was constructed in a tight ratio of 12:6:4:3. Dali incorporated geometrical shapes in ‘ Crucifixion’ and ‘ The Swallos’s Tail.’ We see strange loops in several of Escher’s etchings.
In corporate, we often mention ‘ lets go back to the drawing board’ when we want to crack a impasse. This phenomenon resembles a loop. We see leaders displaying the same style of management day after day on different pitches. Hierarchies tangle in corporates and government. Two disputants arguing their cases to the same manager and the manager adjucating the matter. The manager is on a different level from the disputants leading to a loop. CBI investigating its own wrong doings, a minister going to jail while in office, you committing the same type of mistakes on at different levels of career and so on, are examples which can be explained by the loop theme of mathematics.
Another theme we see reflection in life is recursion. Solve a larger problem breaking it up into smaller and smaller pieces until you solve it and then combine the results. A simple example is ‘ Every boat has a life boat.’ Collaboration is also a recursive process. Self Mirroring’ or understanding the mind is a recursive process. The self comes into ‘being’ at the moment it has the power to reflect itself. Many voiced fugue of the human mind. Recursion will comprise of some loops at different levels and all synchronised. Stories inside stories, movies inside movies, paintings inside paintings, matryoshka set – all reflect recursion. When you to listen to ABP News, it may happen that they switch to a field correspondent. ‘ We now take you to Anasua in front of Writers’ Building in Kolkata.’ Now Anasua may play a video clip taken earlier of ‘ Mamata’s rally’ while she comments on the happenings. You are now three levels down but keep track in your subconscious mind. The stack is tracked as each level is different from the other. There may be a loop in each track.
Ironically, something which we all have in common – the fact of being self-reflecting conscious beings leads to the rich diversity in the methods we apply to internalise evidence about all sorts of things, and in the end wind up being one of the major forces in creating distinct individuals. So much for today with Mathematics…..