Thursday, January 6, 2011

F(n)=F(n-1)+F(n-2), WILL YOU MARRY ME?

Is it possible to be in love with a mathematical formula? I’m kind of in love with the Fibonacci Sequence. I have been ever since I first learned about it. Which, strangely, was from a poetry textbook.
      The Fibonacci Sequence is a series of numbers that begins like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on, forever. You can probably see the pattern: After you start with 0 and 1 (by convention), each number is the sum of the previous two numbers. I won’t try to write it out in math-formula style here because I don’t know how to do subscripts on this blog (it has a bunch of capital F’s and small n’s, a bit like in the title of this piece, which is the best I can do). Many of you are no doubt already familiar with the Fibonacci Sequence and know what the formula looks like. It’s pretty popular, and not just in math circles. Like most guys, I tend to fall in love with the most popular math formula in the class.
     The sequence was unleashed in the Western world by an Italian, Leonardo of Pisa (no relation to Leonardo da Vinci), around the year 1200. Leonardo’s father was named “Bonacci”; the mathematician came to be called “Fibonacci” because that is a short-hand way of saying “son of Bonacci” in Italian. Fibonacci was a pretty sharp mathematician. For example, he was largely responsible for weaning Europeans away from Roman numerals and shifting their math over to the Hindu-Arabic numerals and the decimal system we still use today (at least those of us not living in the 0 and 1 binary world of Tron). There’s a statue of the mathematician near the Leaning Tower of Pisa. I think I kind of have a crush on Fibonacci, too. Here he is. That headgear is kind of cute.
     (The usual disclaimer: This sequence of numbers was also well known in Asian parts of the world, probably earlier than Fibonacci studied it, but we Westerners of course like to give credit to our own.)
     The Fibonacci Sequence is an eye-opener.
     First, it describes the rate at which animal populations grow. In fact, Fibonacci supposedly came up with it when he posed the question of how, in theory, rabbit populations would increase over regular periods if you begin with a single mating pair.
     Second, if you divide each number by the previous number as you go higher in the sequence, you also get closer and closer to a number (or should I call it a ratio?) called the Golden Section, which starts off 1.6180339 . . . and then goes on forever. Mathematicians call this special number Phi. Why Phi? I don’t know why Phi, says I. Phi just happens to be the ratio of the width of the Parthenon to each of its sides. That particular ratio of width to height also shows up in a zillion other pieces of architecture, including the windows of the United Nations building.
       Third, painters, especially that other Leonardo, da Vinci, have used the Golden Section, and the Golden Rectangle (the ratio of width to the side in such a rectangle yields that 1.61880339. . . number—that Phi thing) to structure their paintings.
       Fourth, and most amazingly, the sequence and Fibonacci numbers in general show up all over the natural world—and not just in the way rabbits multiply. It also shows up in the numbers of petals on many flowers, in the shape of the nautilus shell spiral, in the way many plants branch, and most dramatically in the spirals of seed heads in the dark center of the daisy flower (see below). There are, for example, exactly 34 left-oriented spirals of seedheads and exactly 55 right-oriented spirals of seedheads in the center of the daisy flower. Two Fibonacci numbers. Yeah. Like, wow.
      Finally, as other Fibonacci lovers (most way more knowledgeable, but no more ardent than I) have pointed out, humans have two hands, five fingers on each hand, each finger divided into three bones—2, 5, 3—Fibonacci numbers all. Measure your finger bones from knuckle to knuckle: In most people, the ratio of the length of the longest bone to the middle bone is pretty close to 1.6; likewise, the ratio of the middle bone to the shortest bone. You carry the Golden Section at the end of your arm.
      Mathematicians warn us not to get too carried away by all this; there are lots and lots of things in nature that have nothing to do with the Fibonacci Sequence, and some of this may be pure coincidence. But I don’t care. The Fibonacci Sequence is still so cool that I want to marry it tomorrow. I promise to buy it a rectangular wedding ring with just the right ratios.
      The very best web site devoted to the Fibonacci Sequence is here. It is created by a guy named Ron Knott at the University of Surrey in England. I’ve never met him, but I dislike Dr. Knott; I’ll bet he thinks he has the inside track on marrying the Fibonacci Sequence because his name has a Fibonacci-like eight letters and mine doesn’t. And because he has a fabulous altar of a web site devoted to it. But I love you more, Fibonacci Sequence!
      As I mentioned at the start, I first learned of the Fibonacci Sequence, not in a math class, but in a poetry book. The book was an excellent introduction to poetry called Western Wind, by John Frederick Nims. Nims begins his chapter on form, rime, meter, and structure in poetry by discussing the Fibonacci Sequence and its role in nature. His point is that nature loves form, and poetry is “natural” in this way. That seemed a bit much to me, but I thank Nims for introducing me to the Fibonacci Sequence. He can be my best man.


  1. Hi and I am sorry you hate me (this is Ron Knott)!! I never noticed the 3 and 5 letters in my name making the Fibonacci 8 before! One of the earliest references to the Fibonacci sequence seems to be to how to write Indian Sanskrit poetry in the 1100's, before Fibonacci was born.
    By the way, the formula is F(n)=F(n-1)+F(n-2) and Fibonacci never actually mentioned this, only giving the first 12 numbers (named 'Fibonacci Numbers' by a French mathematician, Edouard Lucas in the 1800s) in the solution to a problem about rabbits. Anyway, nice blog!

  2. Interesting--esp. about the Sanskrit poetry. Thanks for the correction of the formula. I've fixed it in the title now.

  3. Maybe the Phi came from Phibonacci