THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN
The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model or describe an amazing variety of phenomena, in mathematics and science, art and nature. The mathematical ideas the Fibonacci sequence leads to, such as the golden ratio, spirals and self- similar curves, have long been appreciated for their charm and beauty, but no one can really explain why they are echoed so clearly in the world of art and nature.
The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a young man in his twenties, a member of an important trading family of Pisa. In his travels throughout the Middle East, he was captivated by the mathematical ideas that had come west from India through the Arabic countries. When he returned to Pisa he published these ideas in a book on mathematics called Liber Abaci, which became a landmark in Europe. Leonardo, who has since come to be known as Fibonacci, became the most celebrated mathematician of the Middle Ages. His book was a discourse on mathematical methods in commerce, but is now remembered mainly for two contributions, one obviously important at the time and one seemingly insignificant.
The important one: he brought to the attention of Europe the Hindu system for writing numbers. European tradesmen and scholars were still clinging to the use of the old Roman numerals; modern mathematics would have been impossible without this change to the Hindu system, which we call now Arabic notation, since it came west through Arabic lands.
The other: hidden away in a list of brain-teasers , Fibonacci posed the following question:
If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?This apparently innocent little question has as an answer a certain sequence of numbers, known now as the Fibonacci sequence, which has turned out to be one of the most interesting ever written down. It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. Its method of development has led to far-reaching applications in mathematics and computer science.
But even more fascinating is the surprising appearance of Fibonacci numbers, and their relative ratios, in arenas far removed from the logical structure of mathematics: in Nature and in Art, in classical theories of beauty and proportion.
Consider an elementary example of geometric growth - asexual reproduction, like that of the amoeba. Each organism splits into two after an interval of maturation time characteristic of the species. This interval varies randomly but within a certain range according to external conditions, like temperature, availability of nutrients and so on. We can imagine a simplified model where, under perfect conditions, all amoebae split after the same time period of growth.
So, one amoebas becomes two, two become 4, then 8, 16, 32, and so on.
We get a doubling sequence. Notice the recursive formula:
Now in the Fibonacci rabbit situation, there is a lag factor; each pair requires some time to mature. So we are assuming
Now let the computer draw a few more lines:
The pattern we see here is that each cohort or generation remains as part of the next, and in addition, each grown-up pair contributes a baby pair. The number of such baby pairs matches the total number of pairs in the previous generation. Symbolically
So we have a recursive formula where each generation is defined in terms of the previous two generations. Using this approach, we can successively calculate fn for as many generations as we like.
So this sequence of numbers 1,1,2,3,5,8,13,21,... and the recursive way of constructing it ad infinitum, is the solution to the Fibonacci puzzle. But what Fibonacci could not have foreseen was the myriad of applications that these numbers and this method would eventually have. His idea was more fertile than his rabbits. Just in terms of pure mathematics - number theory, geometry and so on - the scope of his idea was so great that an entire professional journal has been devoted to it - the Fibonacci Quarterly.
Now let's look at another reasonably natural situation where the same sequence "mysteriously" pops up. Go back 350 years to 17th century France. Blaise Pascal is a young Frenchman, scholar who is torn between his enjoyment of geometry and mathematics and his love for religion and theology. In one of his more worldly moments he is consulted by a friend, a professional gambler, the Chevalier de Mé ré , Antoine Gombaud. The Chevalier asks Pascal some questions about plays at dice and cards, and about the proper division of the stakes in an unfinished game. Pascal's response is to invent an entirely new branch of mathematics, the theory of probability. This theory has grown over the years into a vital 20th century tool for science and social science. Pascal's work leans heavily on a collection of numbers now called Pascal's Triangle, and represented like this:
This configuration has many interesting and important properties:
Next, notice what happens when we add up the numbers in each row - we get our doubling sequence.
Now for visual convenience draw the triangle left-justified. Add up the numbers on the various diagonals ...
and we get 1, 1, 2, 3, 5, 8, 13, . . . the Fibonacci sequence!
Fibonacci could not have known about this connection between his rabbits and probability theory - the theory didn't exist until 400 years later.
What is really interesting about the Fibonacci sequence is that its pattern of growth in some mysterious way matches the forces controlling growth in a large variety of natural dynamical systems. Quite analogous to the reproduction of rabbits, let us consider the family tree of a bee - so we look at ancestors rather than descendants. In a simplified reproductive model, a male bee hatches from an unfertilized egg and so he has only one parent, whereas a female hatches from a fertilized egg, and has two parents. Here is the family tree of a typical male bee:
Notice that this looks like the bunny chart, but moving backwards in time. The male ancestors in each generation form a Fibonacci sequence, as do the female ancestors, as does the total. You can see from the tree that bee society is female dominated.
The most famous and beautiful examples of the occurrence of the Fibonacci sequence in nature are found in a variety of trees and flowers, generally asociated with some kind of spiral structure. For instance, leaves on the stem of a flower or a branch of a tree often grow in a helical pattern, spiraling aroung the branch as new leaves form further out. Picture this: You have a branch in your hand. Focus your attention on a given leaf and start counting around and outwards. Count the leaves, and also count the number of turns around the branch, until you return to a position matching the original leaf but further along the branch. Both numbers will be Fibonacci numbers.
For example, for a pear tree there will be 8 leaves and 3 turns. Here are some more examples:
|Branches of the Fibonacci Family|
You can take a walk in a park and find this pattern on plants and bushes quite easily.
Many flowers offer a beautiful confirmation of the Fibonacci mystique. A daisy has a central core consisting of tiny florets arranged in opposing spirals. There are usually 21 going to the left and 34 to the right. A mountain aster may have 13 spirals to the left and 21 to the right. Sunflowers are the most spectacular example, typically having 55 spirals one way and 89 in the other; or, in the finest varieties, 89 and 144.
Pine cones are also constructed in a spiral fashion, small ones having commonly with 8 spirals one way and 13 the other. The most interesting is the pineapple - built from adjacent hexagons, three kinds of spirals appear in three dimensions. There are 8 to the right, 13 to the left, and 21 vertically - a Fibonacci triple.
Why should this be? Why has Mother Nature found an evolutionary advantage in arranging plant structures in spiral shapes exhibiting the Fibonacci sequence?
We have no certain answer. In 1875, a mathematician named Wiesner provided a mathematical demonstration that the helical arrangement of leaves on a branch in Fibonacci proportions was an efficient way to gather a maximum amount of sunlight with a few leaves - he claimed, the best way. But recently, a Cornell University botanist named Karl Niklas decided to test this hypothesis in his laboratory; he discovered that almost any reasonable arrangement of leaves has the same sunlight-gathering capability. So we are still in the dark about light.
But if we think in terms of natural growth patterns I think we can begin to understand the presence of spirals and the connection between spirals and the Fibonacci sequence.
Spirals arise from a property of growth called self-similarity or scaling - the tendency to grow in size but to maintain the same shape. Not all organisms grow in this self-similar manner. We have seen that adult people, for example, are not just scaled up babies: babies have larger heads, shorter legs, and a longer torso relative to their size. But if we look for example at the shell of the chambered nautilus we see a differnet growth pattern. As the nautilus outgrows each chamber, it builds new chambers for itself, always the same shape - if you imagine a very long-lived nautilus, its shell would spiral around and around, growing ever larger but always looking exactly the same at every scale.
Here is where Fibonacci comes in - we can build a squarish sort of nautilus by starting with a square of size 1 and successively building on new rooms whose sizes correspond to the Fibonacci sequence:
Running through the centers of the squares in order with a smooth curve we obtain the nautilus spiral = the sunflower spiral.
This is a special spiral, a self-similar curve which keeps its shape at all scales (if you imagine it spiraling out forever). It is called equiangular because a radial line from the center makes always the same angle to the curve. This curve was known to Archimedes of ancient Greece, the greatest geometer of ancient times, and maybe of all time.
We should really think of this curve as spiraling inward forever as well as outward. It is hard to draw; you can visualize water swirling around a tiny drainhole, being drawn in closer as it spirals but never falling in. This effect is illustrated by another classical brain-teaser:
Four bugs are standing at the four corners of a square. They are hungry (or lonely) and at the same moment they each see the bug at the next corner over and start crawling toward it. What happens?The picture tells the story. As they crawl towards each other they spiral into the center, always forming an ever smaller square, turning around and around forever. Yet they reach each other! This is not a paradox because the length of this spiral is finite. They trace out the same equiangular spiral.
Now since all these spirals are self-similar they look the same at every scale - the scale does not matter. What matters is the proportion - these spirals have a fixed proportion determining their shape. It turns out that this proportion is the same as the proportions generated by successive entries in the Fibonacci sequence: 5:3, 8:5,13:8, and so on. Here is the calculation:
As we go further out in the sequence, the proportions of adjacent terms begins to approach a fixed limiting value of 1.618034 . . . This is a very famous ratio with a long and honored history; the Golden Mean of Euclid and Aristotle, the divine proportion of Leonardo daVinci, considered the most beautiful and important of quantities. This number has more tantalizing properties than you can imagine.
By simple calculation, we see that if we subtract 1 we get .618 . . which is its reciprocal. If we add 1 we get 2.618 . . . which is its square.
Using the traditional name for this number, the Greek letter f ("phi") we can write symbolically:
Solving this quadratic equation we obtain
Here are some other strange but fascinating expressions that can be derived:
, an infinite cascade of square roots.
, an infinite cascade of fractions.
Using this golden ratio as a foundation, we can build an explicit formula for the Fibonacci numbers:
Formula for the Fibonacci numbers:
But the Greeks had a more visual point of view about the golden mean. They asked: what is the most natural and well-proportioned way to divide a line into 2 pieces? They called this a section. The Greeks felt strongly that the ideal should match the proportion between the parts with that of the parts to the whole. This results in a proportion of exactly f.
Forming a rectangle with the sections of the line as sides results in a visually pleasing shape that was the basis of their art and architecture. This esthetic was adopted by the great Renaissance artists in their painting, and is still with us today.
Translations of this page include:
A Ukrainian translation provided by Anna Matesh, a student at Kyiv International University.
A Dutch translation posted on Johanne Teerink's blog.
A Finnish translation posted on Elsa Jansson's blog.