2019.05.24; The Numbers of Life
Published by DB,
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.
Science Notes / Winter 1994 / Science Communication Program / University of California, Santa Cruz