My essay on the Fibonacci numbers published in the «Washington Post»
By Nik Walter
As a boy I loved geography. I spent countless hours under a dim light bent over unfolded maps from all over my native Switzerland. I knew every valley, every mountain and every town. I was determined to become a cartographer, until my high school mathematics teacher forever changed that plan.
Not that he lessened my love for maps. But he aroused my curiosity for the obscure world of numbers and equations. With his help, I started to recognize that abstract mathematical figures can have an inherent beauty. More important, I realized that mathematical beauty exists not only in mere numbers--it is also an intrinsic feature of the living world. It was hard for me to grasp at that time--and somehow still is today--that the structures of plants and animals alike seem to obey mathematical laws. Yet, when I was about 16, one such law, the "numbers of life" or Fibonacci sequence, awakened my interest in biology--an interest that carried me all the way through a Ph.D. in molecular genetics.
The pattern of the "numbers of life" is elegantly simple. In the Fibonacci sequence, every number (after the first two) is the sum of the two preceding numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, and so on. This looks like a simple pattern, yet it determines the shape of a mollusk's shell and a parrot's beak, or the sprouting of leaves from the stem of any plant--a revelation as surprising to me, at 16, as it probably was to Leonardo Pisano--later known as Fibonacci--almost 800 years ago. Pisano, the first great mathematician of medieval Europe, discovered these magical numbers by analyzing the birth rate of rabbits.
He wrote in the Book of the Abacus, in 1202: "Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, to find out how many pairs will be born in the course of one year, it being assumed that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their own birth." When Fibonacci checked after one month, he found one adult pair and one juvenile pair. After two months, the count was one adult pair (the original) and two juvenile pairs. After three months, there were two adult pairs and three juvenile pairs. One month later, the count was three and five, then five and eight, eight and 13, 13 and 21, and so forth. Rabbits helped Fibonacci to discover one of the great marvels of nature.
It wouldn't be a marvel, though, if these numbers were found only in the growth of a rabbit population. Interestingly enough, the "numbers of life" appear throughout biology. Botany offers countless examples. The leaves of many plant species sprout in well-defined geometrical arrays spirally from the stem. In willows, roses, and many other plants, consecutive leaves follow each other by an average angle of 144*. Therefore, five leaves account for 720* or two complete circles. In other words, the periodicity consists of two windings and five leaves. Other plants show widely varying periodicities that are nevertheless consistent with the numbers of life. In cabbage, asters or hawkweeds, for example, eight leaves complete a period after three circles. In the cones of spruce and fir trees, 21 scales turn eight times for one period. The cones of pines, in contrast, use 34 scales in 13 windings.
Yet, Fibonacci numbers appear not only in the leaves and cones of plants, but also in flower blossoms. Pick some random flowers and count their colored petals. On average, daisies will have 21, 34, 55 or 89 petals, chrysanthemums 21, and some senecio species either 13 or 21 petals. Although exceptions to the Fibonacci rule are not difficult to find, the "numbers of life" occur so frequently in nature that they cannot be explained by chance. There must be a general law of symmetry, aesthetics and beauty.
In fact, such a law seems to govern the Fibonacci numbers. The ratio between one number and its predecessor in the series approaches 1.6180 as the numbers increase (5/3=1.667, 8/5=1.600, 13/8=1.625, 21/13=1.615, 34/21=1.619, 55/34=1.618). This magical ratio turns out to be a universal measure of beauty, which the Greeks called the "golden section" or "divine proportion." Most of the ancient Greek temples, including the Parthenon in Athens, obey this law of divine proportion. They are exactly 1.618 times as long as they are wide. Long before the Greeks, the ancient Egyptians had already built the pyramids along the same rules. A pyramid's base length is 1.618 times its height. And many artists, too, including Leonardo da Vinci, have used the divine proportion to structure their paintings and sculptures.
Returning to the living world, let's go one step further. Draw a "golden" rectangle with a width-length ratio of 1.6180. Then, draw a square in one end of this rectangle and you end up with a smaller golden rectangle in the space left. Next, place a square into that smaller rectangle, following the same rules, and you produce yet another, smaller golden rectangle. Theoretically, this can be done infinitely. After you've nested about ten rectangles within the original rectangle, try drawing a curved line connecting the centers of all the squares. You'll be surprised to find that the line forms a perfect spiral.
This "golden spiral" defines the shapes and structures of many features of living organisms. The claws of a lion, the horns of a ram, the tusks of an elephant, the beak of a parrot and the shell of a snail all obey the rules of the golden spiral. Such perfect shapes appeal to us through an irresistible combination of order and beauty. Yet, the golden spiral appears unexpectedly in many non-living things, too--in the shape of a breaking wave or the structure of a galaxy, for example.
This enmeshing of mathematical laws and the natural world awakened my love for biology and shaped my scientific career. After graduating from high school, pondering whether to enroll in geography--my old love for maps had not vanished--or biology classes at the university, I chose biology rather spontaneously on the last day of enrollment.
Throughout the time I spent at the university, the "numbers of life" accompanied my scientific career. For example, I remember very well one field trip to Marettimo, a small Mediterranean island west of Sicily. A group of about 25 undergraduate students, we set out to Marettimo in spring 1984 to learn about the local flora. On our daily hikes across the rugged island, we detected all kinds of gorses, ericas, holm-oaks and orchids, as well as many other wonderful plants. But the most memorable experience was when I realized how many different blossoms have either 5, thirteen or twenty-one petals, all of them "numbers of life."
Later, during my graduate studies in molecular genetics, I had another encounter with the Fibonacci sequence. Trying to find out more about the molecular mechanism of how the nervous system forms during the development of an organism, I chose the tiny fruit fly Drosophila melanogaster as my object of study. Looking at anesthesized fruit flies under a stereo-microscope, I always admired their perfect shape. One day, I realized why fruit flies look so beautiful. In fact, the segmentation of their bodies matches the law of beauty: A fruit fly has two or three head segments (hardly visible), three thoracic segments (where the legs and wings are attached) and eight abdominal segments (with no legs), all of them Fibonacci numbers. That makes 13 segments in total, just another "number of life."
Aware of this universal law of beauty, I have tried since high school to find the Fibonacci numbers wherever they might be hidden--in cones, blossoms, or even in fruit flies. And I always think, with a little smile, what an irony it actually is, that I had been imprinted by my high school math teacher to delve into the biology of beauty and the mathematics of aesthetics.
published in Science Notes Winter 1994 / Washington Post, Feb 8, 1995