Correspondence with Martin Gardner
on Fibonacci Kolams
© Copyright 2011
Prof. S. Naranan
Martin Gardner is an iconic polymath with extraordinary learning, spanning Philosophy, Metaphysics, Science (Mathematics and Physics), Recreational Mathematics, games, puzzles, magic tricks and art. He is best known for his series of columns “Mathematical Games” that appeared in Scientific American for 25 years during 1956-1981. There are 330 columns, reprinted in 15 volumes, in several editions with updates. They define “Mathematical Recreations”, a genre of Mathematics he created all by himself.
Gardner gave up writing in Scientific American, perhaps because it left little time for writing about his other variegated interests. He was active and publishing books until his death in 2010 at age 95. In all he wrote more than 100 books, the last one in 2009 to mark his 95th birthday. A list of his writings – columns, articles, reviews and books – runs to a few hundred pages.
It is said that Gardner never used computers and used only a typewriter for all his writing and correspondence with a large network of his readers and admirers. Long before the present-day Internet ‘blog’, Gardner innovated exchange of letters with hundreds of his readers consisting of mathematicians, computer scientists and programmers and puzzle-lovers to make his columns interactive. Based on feedback from readers, he published updates on solutions to the Mathematics problems in his columns and new editions of his books. It is reported that he spent a whole day every week on correspondence, which he typed standing in front of an old typewriter mounted on a pedestal.
Martin Gardner’s mathematical education was only up to high school. He got a B.A. in Philosophy from the University of Chicago in 1936. He dropped out of graduate study in the first year. Some believe, this lacuna may have had an advantage. Gardner is reported to have said that if something was not clear to him, it will most probably be unclear to many of his readers. His intellectual integrity made him check out many facts with “original sources” and double check all he wrote. Perhaps that accounts for the extraordinary clarity of his writing, reflecting similar clarity in his thinking.
The Scientific American columns spawned an ever-growing community of laymen-turned-mathematicians. At the same time thousands of mathematicians turned puzzle-lovers. There is a subtle distinction between Mathematical Recreations and Recreational Mathematics. In Mathematical Recreations, the emphasis in on the puzzles (games, magic tricks), based on mathematical principles. Most of them are Gardner’s creations. In Recreational Mathematics, key mathematical ideas – some very abstract – are unraveled by lucid exposition to lay readers. Gardner excelled in both. In early 1970’s, Donald E. Knuth, published the book “Fundamental Algorithms”. It was the first of six volumes planned, covering algorithms for computation, meant mostly for undergraduates in Computer Science. A large part of the book was devoted to problems. Gardner discovered that many of the problems could be adapted for Recreational Mathematics. Many readers turned avid computer programmers.
Gardner revealed many new important advances in Computer Science for the first time in his Scientific American columns. Here are a few examples. Gardner introduced the “Game of Life” of the famous mathematician John Conway, a game with applications in ‘cellular automata’. In 1977, Ronald Rivest of the Massachusetts Institute of Technology (M.I.T), Cambridge (U.S.A) – one of the famous trio ‘Rivest, Shamir and Adelman’– approached Gardner offering to reveal for the first time to the public, the path-breaking ‘RSA’ algorithm based on trap-door ciphers. It made possible secret and private communication on a public channel like the Internet. Such a move was strongly resisted by the National Security Agency (NSA) and the Central Intelligence Agency (CIA) of the U.S. Govt. But Gardner could not refuse Rivest’s offer and published the RSA algorithm in the August 1977 issue of Scientific American. Although brief, the revealed algorithm could be easily implemented in Fortran code by anyone with knowledge of elementary Number Theory. This was perhaps the ‘greatest coup’ in Gardner’s lifetime. Gardner popularized ‘Fractals’ discovered by Benoit Mandelbrot. Fractals had links with the beautiful Mandelbrot set, order and chaos in complex systems (non-linear dynamics). They were highlighted in a column in 1983. Fractals, perhaps more than any other concept, brings out the strong links between Art and Mathematics with the aid of powerful computers.
Perhaps because of his university education in philosophy, Gardner had an abiding interest in metaphysics and religion. Yet he was a champion of the scientific method – “defending reason, attacking fraud” – and a founder of the modern skeptic movement.
Gardner’s seminal role in propagating Mathematics and Mathematics Education is recognized by a commemorative book “The Colossal Book of Mathematics” published in 2002. Note that the compilation stressed Mathematics rather than the puzzles. This is remarkable for a mathematician who probably never studied calculus.What was Gardner’s first publication? Gardner was the editor of a children’s magazine “Humpty Dumpty” and he designed ‘paper-folding puzzles’ for the magazine. They caught the attention of Scientific American, and led to the first article by Gardner in the magazine: ‘Hexaflexagons’. Soon after, began the regular columns on MG (Mathematical Games!) by MG starting from January 1957 that entertained the readers for 25 years. The rest is history. Today we remain grateful not only to Martin Gardner for the bounty he created in Recreational Mathematics but also to Scientific American for popularizing it.
I had the luxury of a personal subscription to Scientific American from 1957 for several years. It was a gift from my brother-in-law Dr. S. Ramachandran who lived in the U.S. When the monthly magazine arrived, I would skip its contents to go to Gardner’s column. So, I was hooked to it from very early days. Around early 1960’s, I wrote a letter to Martin Gardner after reading his column on Mathematical Billiards (Bouncing balls in polygons and polyhedrons). I pointed out a similarity between one of the billiard ball trajectories and a ‘kolam’ design. Kolam is a closed curve meandering through the gaps in a rectangular grid of dots, a folk-art popular in South India. Gardner replied immediately, saying he will comment upon it in a future updated version, when it happens.
I found the August 1977 article on trap-door ciphers and the revolutionary RSA algorithm, mentioned earlier, fascinating because of its connection to Number Theory and Prime Numbers. The RSA algorithm for secure communication is based on the fact that while it is easy to create large prime numbers (say 100 digits), it is very difficult to factorize a large number (say 200 digits) into its two large prime factors (each 100 digit). To me it was incredible that such an algorithm could be invented in cryptography. I was spurred to write the RSA code with the help of a programmer colleague. After nearly 35 years, the RSA code remains the workhorse of secure communication on the Internet. The article on Fractals in 1983 led me to a life-long interest in non-linear dynamics, order and chaos in complex systems. In short, Martin Gardner was responsible for broadening my interests beyond Physics and Astronomy, to Mathematics, Statistics, Number Theory, Cryptography, Computer Science, Complex systems, and Quantitative Linguistics.
In 2007, I wrote an article in two parts titled “Kolam Designs Based on Fibonacci Numbers”. I found a new modular approach to build kolam designs of any arbitrary size and shape using some unique properties of Fibonacci Numbers.
It was my brother S. Srinivasan (“Seenu”) of Austin, TX, who suggested the idea to write to Martin Gardner about my kolam work, especially because I had a brief correspondence with him about 50 years ago! The hard copy of the article had to be sent by post. Seenu, after some effort, succeeded in finding the postal address. The subsequent events are described in Seenu’s words:
I acknowledged Gardner’s letter and posted it on 1 May 2010. I wrote:
Looking back at the sequence of events, I recall that for nearly three years I hesitated to write to Martin Gardner. On 16 April 2010, one day before I completed 80 years, I thought I should do something worthwhile. On the spur of the moment I wrote the letter. I was unprepared for the laudatory reply so promptly written by Gardner. I am grateful to my brother Seenu for being instrumental in getting my kolam work into the hands of Gardner. My letters of 16 April and 1 May and Gardner’s letter of 22 April 2010 are reproduced below as items 1, 2 and 3.
Sadly Martin Gardner passed away on May 22, 2010 at age 95, within weeks of reviewing and commenting on my paper.
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