If one were to argue for the beauty of mathematics by providing three of the simplest relevant examples, the Fibonacci sequence would unquestionably be among them. But this educational aspect of popularizing mathematics or being a subject of recreational mathematics is not the only virtue of this sequence. The mathematical properties of the Fibonacci sequence and their curious reflection in various contexts outside mathematics, including nature, have maintained over centuries an aura of mystery surrounding this mathematical concept. Where mystery is detected, scientists, mathematicians, and philosophers have an instinctive drive to solve it, and such a focused inquiry has so many times proven fertile in the history of science and mathematics by leading to important discoveries or new theories. For that reason also, the Fibonacci sequence is a virtuous concept.
We all came to know about this sequence mainly in high school, when the math teacher usually offered it as an example of a sequence defined in a recursive mode – that is, a sequence each term of which is determined as standing in a given relation with one or more of the previous terms. For the Fibonacci sequence (its name originating from Italian mathematician Leonardo Bonacci, or Leonardo of Pisa, pseudonamed Fibonacci, who lived in the 12th-13th century), the recursive rule is that each term is the sum of the previous two terms, while the first two terms are 0 and 1. According to this rule, the Fibonacci sequence can be generated as and looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… Its terms are called the Fibonacci numbers.
Now, looking at this sequence, one may fairly ask why this plain one-dimensional succession of natural numbers would be so special, where its beauty resides, and what is so mysterious about it. All these attributes can be confirmed right away one after another, if a bit of mathematical imagination is put to work.
Imagine for a moment a rectangle whose side lengths are two consecutive Fibonacci numbers in that sequence. Make them large enough, for instance 34 and 55. Now proceed to tile this rectangle successively with squares of side lengths equal to the lesser side length of the rectangle, as in the figure below:
If you draw quarter-circular arcs connecting the opposite corners of the squares in succession, you get a spiral passing through all those squares whose side lengths are successive Fibonacci numbers. The larger you make the side length of the initial rectangle as a Fibonacci number, the longer you get this spiral drawn. Well, with this drawing, the one-dimensional Fibonacci sequence has turned into a nice two-dimensional spiral. But this is not all.
The rectangles shown in Figure 1 are similar to each other – that is, the ratio of their consecutive side lengths is the same. Denoting by a and b the side lengths of the initial largest rectangle, this geometrical similarity is written as the proportion a/b = (b – a)/a. Substituting φ = b/a, the previous relation is made equivalent with the quadratic equation φ^2 – φ – 1=0, which is called the golden equation and its positive solution φ = (1 + sqrt (5))/2 = 1.61803398… is called the golden number or the golden ratio. In the geometry of curved shapes, a golden spiral is a logarithmic spiral whose growth rate is the golden number, and has the property of being self-similar – in other words, keeping the same shape when magnified in its accumulating zone. So what we have drawn above is an approximation of a golden spiral, and extending (hypothetically) our drawing infinitely both outside and inside the rectangle by the same rules – reflecting Fibonacci numbers – we would obtain a “full” golden spiral.
The golden number can be made visible not only in the golden spiral, but directly in the plain Fibonacci sequence. Mathematician Jacques Philippe Marie Binet expressed the general term of the Fibonacci sequence in the closed form Fn = [φ^n – (1 – φ)^n]/sqrt(5) , where φ is the golden number. A nice property of this sequence is closeness, in the sense that that any positive integer can be written as a sum of Fibonacci numbers each taken at most once. From this point, mathematicians developed several procedures of expressing φ, spanning over various fields of mathematics (combinatorics, number theory, topology and mathematical analysis, differential equations, and geometry), and their success connected various different concepts from this fields. However, the most striking simple representations of the golden number are these two:
On the left-hand side we have what we call a continued (infinite) fraction and on the right-hand side, a continued (infinite) radical. Of course, such numbers as those in the right-hand members above cannot even be imagined, nor computed through such representations using division and the square root; they are just symbols of more complex concepts and mathematical statements involving the limit of particular convergent sequences. What is amazing about these two different representations is that they provide the same number, the golden number, and this is definitely part of the complex beauty of mathematics. For mathematicians and philosophers of mathematics, it is not such a mystery any more that mathematical concepts come to exhibit mutual unexpected connections across the different fields of mathematics (fields using concepts of different natures, different methods, and different languages). What still remain mysterious are their connections outside mathematics, in the real world.
It is a well-known fact by now that the golden number and its spiral are visibly used by Mother Nature as a development pattern: several species of plants have been found to have their flower petals arranged and growing in a golden-spiral pattern, or their leaves distributed in the golden angle (i.e., the circular angle determined by the smallest of two circular arcs standing in the golden ratio). Mollusk shells (like the nautilus) exhibit the same spiral shape and some specific anatomic proportions in the bodies of animals and humans are equated with the golden ratio. It is perhaps due to this last fact that the Fibonacci sequence and golden number came to be seen in a mystical light.
Whatever argument may be invoked that such observations are made with a dose of illusion and self-suggestion, the fact falls within the wider belief that nature follows mathematical patterns in its evolution, and abstract mathematics effectively describes the physical reality. Such facts, although not explained rationally by any science or discipline, cannot be denied. Modern sciences, especially physics, have evolved to their current success only after describing the laws of nature in the language of mathematics and applying mathematics in any suitable context. Such universal applicability of mathematics is another kind of beauty: it is simultaneously the beauty of mathematics, the beauty of nature, and the beauty of human reason. When taking into account that the pattern of development of a plant following the golden spiral allows it to grow without changing shape – and biologists would have more to say about this in non-mathematical terms and with evolutionary explanations – the mathematical properties of the Fibonacci sequence and golden number become somehow the properties of nature. Is this something mysterious? If you have not yet decided what to answer (and don’t trouble yourself too much with this question – philosophers of science have not provided any straight answer either), compare the images showing the symbolic abstract representation of the golden number as a continued fraction or radical and the golden-spiraled plant in our image: Didn’t you feel that there is something almost indescribable that they share in common? If so, bear in mind that you have compared an abstract thing originating in a mathematician’s mind with a real live organism.
So many times in the history of science, mathematical concepts created with no particular intended application outside mathematics have come to find their application in contexts never imagined at the moment of their creation. The Fibonacci sequence and the golden ratio fall within this category, it would be an impossible task to enumerate here all of the domains of their applications. Just to mention the most relevant, these concepts have found their application in economics, sociology, architecture, art (including music), horticulture, genetics, and optics. However pragmatic the success of their application, the interest of scientists and mathematicians in the deep investigation of these concepts is still in advent – there exists a (mathematical) Fibonacci Society and a scientific journal dedicated entirely to Fibonacci sequence and the related concepts.
The mystical aura surrounding these numbers has never lost its power; it touches both scientists and ordinary people. Whatever discipline would be entitled to deal better with this matter, it cannot ignore the mathematical simplicity of the definition of the Fibonacci sequence and the idea that complex things are made of simple things. Clearly, mathematics is the natural path toward any kind of complexity, offering also the appropriate methods to deal with and investigate it.