NOTE: Still working my way through the galleys for The Calculus Diaries, but the process made me remember this post from April 2006 about the Fibonacci sequence, an obscure mathematician named Gopala, and fun with "Fibs" -- short six-line poems based on the Fibonacci Sequence. More below, and we'll be back with original posting later this week!
"April is the cruelest month," T.S. Eliot declared in the opening lines to "The Wasteland" -- unless you're a poet or mathematician. Or both. In which case, you've got double the reason to celebrate. April is both National Poetry Month and Mathematics Awareness Month. So it seems especially appropriate that there is currently an explosion of original amateur poetry on the Internet based on the famed "Fibonacci sequence." They're called "Fibs," six-line poems whose syllables follow the Fibonacci progression. After the first two terms (1 and 1), each subsequent number is equal to the sum of the two previous numbers, so 1+1 = 2; 1+2=3; 2+3 = 5; 3+5 = 8; and so on into infinity. (Jen-Luc Piquant is relieved that the fibs mostly stop at 8 syllables instead of stretching into infinity, otherwise we'd never finish reading them -- although it might be amusing to attempt an infinitely epic version of, say, The Odyssey, in this "Fibonacci meter." Odysseus would never find his way back to Penelope.)
Apparently, more than 1000 Fibs have been written so far this month, thanks in large part to a whimsical blog posting by LA screenwriter Gregory Pincus inviting readers to submit their own six-line poems using the sequence. His original post was then linked on slashdot.org and the notion spread like a virus from there. That's the power of the Blogosphere.
Equally amazing is the fact that so many people have heard of the Fibonacci sequence, no doubt due in large part to the staggering success of Dan Brown's bestselling The Da Vinci Code, which introduced us "non-math-y" types to what was once a relatively obscure mathematical oddity -- at least to those outside rarefied academic circles. Perhaps you recall the scene in which the hero, Langdon, and the love interest, Sophie, discover a seemingly random string of numbers at a murder scene: 13-3-2-21-1-1-8-5. Because this is fiction, and therefore not even remotely realistic, Sophie quickly realizes it is the first eight numbers, in jumbled order, of the Fibonacci sequence. Divide each number in the sequence into the one that follows, and the answer will be something close to 1.618, an irrational number known as phi -- a.k.a. "the Golden ratio," which also figures prominently in Brown's book.
It's nice to see mathematics get some attention, even if it's by way of a controversial potboiler. And while we should be skeptical of some of Brown's more questionable assertions about the prevalence of the Golden Ratio in the arts, for example, Fibonacci was a real historical personage: a 13th century mathematician also known as Leonardo Pisano, or Leonardo of Pisa. He was the son of a diplomat, and therefore studied under Arabic mathematicians in North Africa and traveled widely throughout what is now modern-day Algeria, as well as Egypt, Syria, Greece, Sicily, and Provence.
Upon his return to Pisa, the now-grown Leonardo wrote the Book of Calculation (Liber Abaci) in 1200, which, among other useful elements, introduced the use of Arabic numerals to Europe, along with the base-10 system for commercial bookkeeping. (The Fibonacci moniker apparently derives from the fact that his father was nicknamed "Bonacci," so young Leonardo was dubbed "filius Bonacci" -- "son of Bonacci" -- or Fibonacci.) The Fibonacci sequence appears in the third section as the solution to a hypothetical problem on how fast a (highly idealized) population of rabbits will grow. It proved to be one of his most lasting contributions. There is still a modern journal, The Fibonacci Quarterly, devoted entirely to studying the mathematics related to this sequence.
Fibonacci deserves every bit of the attention and respect he's received of late. But it would be incorrect to assume that he "discovered" this quirky little numerical sequence that bears his name. Like Sir Isaac Newton some 400 years later, he stood on the shoulders of giants. A century before he wrote the Book of Calculation, an ancient Indian scholar named Gopala had calculated the series of meters used in some 12th century Sanskrit poetry -- a specific meter called the matra-vrttas -- in which each subsequent meter is the sum of the two preceding meters. The pattern follows the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21....
There's precious little information available about Gopala; he's the shortest of stubs on Wikipedia, and even the exhaustive collection of historical biographies of mathematicians throughout the ages compiled by the good folks at St. Andrews College in England contains very little information. (Ironically, that same database reports that Fibonacci's work in number theory was in turn largely unknown during the Middle Ages; it was "rediscovered" 300 years later by the mathematician Maurolico.) There's only a wee bit more information about Gopala's fellow Indian scholar, Hemchandra, who studied the same sequence 20 years later while looking at the various possible ways of exactly bin-packing certain items of two given lengths. Hemchandra authored a proliferation of textbooks on science and Indian philosophy, as well as an epic poem and several Sanskrit grammars.
The latter is a fascinating field of study on its own, and doesn't lack for significance in both the arts and science. "Sanskrit" means "perfected," "refined," or "polished," and it may very well be the oldest language in the world, dating back to 2000 BC, although its formal rules weren't recorded until about 400 BC. But it's hardly a dead language: it's still spoken by hundreds of millions of people. Goethe borrowed from the Sanskrit tradition when writing portions of Faust. I opened this post by quoting "The Wasteland." Eliot was a student of Indian philosophy and ended his masterpiece with the Sanskrit lines, "Shantih Shantih Shantih." (A college pal of Jen-Luc Piquant's, known affectionately as "Red," once wrote a parody of this famous poem, declaring, "Eliot is the cruelest poet/Breeding art out of dead legend...." Even more fun can be had with Sanskrit via an online interactive game, "The Trials of Vajra.")
Henry David Thoreau read the Bhagavad Gita. So did physicist J. Robert Oppenheimer, the former head of the Manhattan Project, who famously quoted that timeless work after the successful Trinity Test ushered in the nuclear age in 1945: "I am become Death, the shatterer of worlds." In the early days of the periodic table, scientists used Sanskrit prefixes to refer to as-yet-undiscovered elements. NASA researchers are considering the possibility of adapting Sanskrit as a possible computer language because its structure leaves little room for error -- there are some 3959 rules to define the basic elements of sentence structure, consonants, nouns and verbs, all laid out like a mathematical function -- and bears some similarity to modern programming languages.
I bring up Gopala, Hemchandra, and the grand Sanskrit tradition mostly because (a) I find it historically fascinating, and (b) we tend to be far too Western-centric in American culture, and ignorant of the rich diversity and flourishing culture of other ethnic traditions. I'm no exception: I would never have heard of Gopala had I not read Mario Livio's The Golden Ratio. nor is that an isolated instance. The Independent recently ran a fascinating article detailing the many ways in which forgotten Islamic inventors changed the world, inspired by a new exhibit featured at the Science Museum in Manchester in March and still making the rounds of Merry Olde England. It seems we honor certain historical breakthroughs and inventions out of sequence, like the jumbled Fibonacci numbers in the opening pages of The Da Vinci Code.
Leaving aside the more obvious contributions to algebra, architecture, and cryptology, Muslim scholars assumed the Earth was round some 500 years before Galileo realized it. One of the earliest pinhole cameras was invented by a 10th-century Muslim scientist named Al-Hazen (or Ibn al-Haitham). Around 800, Muslim scientist Jabir ibn Hayyan invented the process of distillation (separating liquids through differences in their boiling points), technically making him one of the founders of modern chemistry. An ingenious Muslim engineer called al-Jazari invented a rudimentary crank shaft to raise water for irrigation, and also invented a prototype of the combination lock. And while I'm a big fan of Eilmer of Malmesbury, an 11th century English Benedictine monk who fashioned a pair of makeshift wings and jumped off the roof of Malmesbury Abbey, he wasn't the first to attempt such a stunt. In 852, a Muslim poet, astronomer, musician and engineer named Abbas ibn Firnas jumped from the minaret of the Grand Mosque in Cordoba with nothing more than a loose cloak fitted with wooden struts. It fortunately served as a parachute, so he escaped with minimal injuries. But really, what is it about early aviation attempts and jumping off the roofs of religious structures? It lends a whole new meaning to the term "leap of faith."
Jen-Luc Piquant largely eschews poetics in favor of the highly diverting -- for those with a mathematical bent -- Fibonacci Puzzles Page. But she was nonetheless inspired by Mr. Pincus, his poetical followers, and the forgotten fathers of Fibonacci to pen the following lines:
and friend Hemchandra,
not just Fibonacci's sequence.