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Fibonacci and the sunflower

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The name Sunflower Foundation was not chosen haphazardly. It expresses the philosophy on which the MoneyMuseum is based. The sunflower stands for munificence, balance, harmony and abundance in the positive sense, but also for chance and individuality.

The sunflower is built up on a mathematical algorithm: the sunflower's flower head consists of a large number of seeds arranged in several clockwise and counterclockwise spirals. The number of spirals always corresponds to the Fibonacci series.


Fibonacci, his numbers, his geometry

Leonardo of Pisa, better known by the name Leonardo Fibonacci, lived roughly from 1170 to 1240 and is considered to be the first important European mathematician.

On journeys to Africa, Byzantium and Syria he came into contact with Arabic mathematics, which in the Christian occident was largely unknown.

In his work Liber abaci (The Book of the Abacus), which was published in 1202, he combined this knowledge with his own reflections. The book long remained unexcelled in the history of occidental mathematics and contributed to, among other things, Europe adopting the Arabic system of numbers.

Liber abaci contains a thought experiment which Fibonacci himself probably regarded as pure curiosity and did not pursue further, but which later was to become famous as the "Fibonacci sequence."

Fibonacci asked himself how many pairs of rabbits originated from a single pair in one year. To do this he assumed that none of the rabbits would die in the course of that year and that each pair of rabbits would produce exactly one more pair of both sexes per month, which in turn would be fertile from the second month after birth.

In his book Liber abaci he writes: "Because the above-mentioned pair gives birth in the first month you can double it, so that there are two pairs after one month."

So at the end of the first month – and it is here that Fibonacci begins counting – there exist two pairs of rabbits.

At the end of the second month the original pair has given birth to another pair, and the other pair became fertile. Now there are three pairs.

Of these three pairs two in the third month are now fertile and one is not yet fertile, thus at the end of the next month two more pairs of rabbits are added, so now there exist altogether five pairs.

Of these five pairs, three in turn become pregnant, so that in the fourth month there are eight pairs.

To find out how many pairs of rabbits there are, Fibonacci observed, all you have to do is in each case to add up the sum of the pairs of rabbits of the two previous months.

To begin with there is one pair of rabbits. After one month there are two pairs, after two months there is one plus two, or three pairs, after three months two plus three, or five, after four months three plus five, or eight and so on – until after eleven months 233 pairs of rabbits have resulted from the first pair.

And Fibonacci writes: "When finally the 144 pairs are added to those born in the last month, in the end there are 377 pairs. And the above-mentioned pair have finally produced that many pairs at the end of one year."

Although Fibonacci's thought-experiment is based, of course, on unrealistic assumptions, it does describe the essential features of growth processes.

While for Fibonacci his problem was thus solved, it was later discovered that the Fibonacci sequence also occurs in nature and in art – be it in the position of leaves of plants, in the spiral form of molluscs, in the structure of clouds in an area of low pressure and in paintings, the architecture of buildings and in music.

It is also possible to approach the Fibonacci numbers geometrically.

Let us assume a square whose sides measure 1. Beside it we construct a second square of the same size. We attach another square to it, which has the length 2. To this is added a square with the length of the side 3, one with a length of the side 5, one with the length of the side 8.

It is not difficult to recognise the numbers of the Fibonacci sequence.

Now we draw a quadrant in each square.

The resultant a spiral called the „Fibonacci spiral“.

It can be clearly seen in the shell of the nautilus.

The golden section and its relatives

It was the Greek mathematician Euclid who around 300 BC produced the first precise description of the golden section – he called it "division according to the outer and middle proportion" –: a length is divided into two parts in such a way that the smaller part is to the larger part in the same proportion as the larger one is to the whole.

Later, in the 15th century, the Italian mathematician and Franciscan monk Luca Pacioli took an interest in Euclid's works and devoted a whole volume to this division of lines, which he called "divina proportione" also "divine partition."

Around 1600 Johannes Kepler – known for the Kepler Laws of the movements of planets – discovered the relationship between the Fibonacci numbers and the golden section.

He observed that the relationship between a number in the Fibonacci sequence and the previous number more and more closely approaches the irrational number Φ ((phi)) the longer the sequence is continued.

And Φ ((phi)) describes nothing other than the golden section.

The golden section defines a proportion which has always been perceived as especially beautiful and harmonious. In many epochs we find its application in almost all cultures throughout the world, above all in architect and art.

A rectangle in which the proportions of its sides correspond to the golden section is called a golden rectangle. Similarly, isosceles triangles in which two sides are in this proportion are termed golden triangles.

An important part is also played by the so-called golden angle Ψ ((psi)), which divides the angle of 360 degrees in the proportions of the golden section. As angles smaller than 180 degrees prove to be more handy in practice, the smaller of the resultant angles is usually called the golden angle Ψ ((psi)). It is – approximately – 137.5 degrees.

The pentagram

The pentagram is a regular, five-pointed star which is formed by the diagonals of a regular pentagon and which has been considered to be a magic sign since antiquity. The golden section appears here in an especially impressive way, as it can be found geometrically in the pentagram several times.

For each line and partial line there is a partner, which, together with it, is in proportion to the golden section.

The pentagram can also be imagined as a composition of five golden triangles. If the five intersections are connected inside it another pentagram is created there. Even when a pentagram is again drawn in its inner pentagon and so on, all the triangles contained in this drawing are golden triangles.

If an apple is cut through the middle it is found to contain a natural depiction of the pentagram in its core.

Like all members of the rose family, the apple is assigned to the female, the life-giving principle. It is thus not surprising that the pentagram is the symbol of Venus, both of the planet and the goddess.

As the symbol can be drawn in an unbroken line and at the end comes back to the beginning, it was also a sign for the cycle of life.

In the Middle Ages the pentagram, or pentacle, was used as a figure to ward off demons.

And even today it is omnipresent: the stars of numerous flags, for example those of the USA or the EU, are all pentagrams.

The symbol of Islam or the Soviet Star are also pentagrams.

 

The Fibonacci numbers in nature

What is surprising is the frequent occurrence of the golden section and the Fibonacci numbers in nature.

These structural principles reappear most conspicuously in the phyllotaxis of plants, i.e. in the arrangement of their leaves and seedcases.

In many of the more highly developed species of plants the angle between spirally shaped, consecutive leaves is on average about 137.5 degrees – the golden angle.

This arrangement of leaves is also termed the Fibonacci phyllotaxis.

As the golden angle is based on an irrational number, one leaf will never lie exactly over another. The sunlight coming from above can thus be used to the best advantage and the maximum quantity of rain that falls is passed on to the roots.

In the case of the sunflower, phyllotaxis appears in the spirally arranged seedcases on the flower's receptacle in an especially aesthetic form.

The clearly recognisable Fibonacci spirals are not formed from seeds which follow one another in the course of growth, but, rather, they result as a consequence of the fact that consecutive seeds are arranged at intervals; here the deviation from the mathematical golden angle is less than 0.01 per cent.

If we consider the number of arcs which turn counterclockwise and clockwise it will hardly be a surprise: here, too, we have consecutive Fibonacci numbers. In the outer area of sunflowers there are usually 34 and 55 spirals, in the case of larger specimens 55 and 89 or even 89 and 144. Whether the larger number of arcs turn clockwise or counterclockwise however, is left to chance.

This animation shows a constructed inflorescence with 200 seeds. Moving from the centre of germination, the seeds push outwards as growth progresses until the whole receptacle is filled out.

Seeds that follow one another emerge precisely around the golden angle separately from one another, as this ensures that the seeds are packed together in the most compact way. This, however, becomes clear if the angle between the two consecutive seeds that follow one another in the course of growth is changed.

As we can see here, 13 and 21 Fibonacci spirals emerge.

Wie wir sehen, stellen sich hier 13 und 21 Fibonacci-Spiralen ein.

The golden section in architecture and art

From time immemorial wherever people wanted to express beauty and where they tried to approach the divine ideal, we come across the golden proportion. Generally in art and in architecture, and in particular at holy sites.

Even in the pyramids of Giza, the proportions of the number Φ ((phi)) are revealed with astounding accuracy. Thus, for example, in the case of the Cheop's Pyramid the proportion of the length of the side of the pyramid to half of the pyramid's base is 356 : 220 ells, and that corresponds to 1.618 of the number Φ ((phi)).

Also, in the most famous stone monument, Stonehenge, which was built near Salisbury, England, some 3500 years ago, we again find the golden measurements.

The Parthenon, erected in Athens under Pericles around 450 BC, is one of the best-known classical buildings. At the same time it is regarded as the most beautiful and most accomplished work of ancient Greek architecture.

The proportions of the golden section are built into it in many ways and with surprising precision.

From 1940 the architect and painter Le Corbusier developed a uniform measuring tool which replaces the metric system with a scale of harmonious dimensions derived from the proportions of the human body and the golden section. In his book „The Modulor“, in which he published these ideas and which today is one of the most significant works of architectural theory, he writes:

"A person with a raised arm provides in the main points where space is displaced – foot, solar plexus, head, finger tips of the raised arm – three intervals which result in a series of golden sections named after Fibonacci.“

In art the proportions of the golden section appear in the basic structure of numerous well-known paintings, such as The Last Supper by Leonardo da Vinci or this Self-portrait by Albrecht Dürer.

An artist of modern times who conspicuously employs the golden section is, for example, the Dutch painter Piet Mondrian.

In the concourse of Zürich's Main Station there is also an example of a contemporary use of the Fibonacci numbers in the fine arts: the installation „Ovus philosophicus“ by the Italian artist Mario Mertz.

In music the golden section occurs in several forms.

Let us take a look at a piano keyboard.

The interval of an octave from C to C1 comprises 8 white and 5 black keys, together 13 keys.

The black keys are divided up into groups of 2 and 3.

All numbers and proportions from the Fibonacci series.

In the writings of the musicologist Ernö Lendvai we read that the golden section and the Fibonacci numbers reappear in the works of the composer Béla Bartok as the dominant principal of composition.

This becomes especially obvious in the "Sonata for two Pianos and Percussion", where not only the parts of the form follow the proportions of the golden section.

Bartók himself, whose favourite flower is said to have been the sunflower, has, however, never made mention of this.

Similar investigations have also been made into the works of Bach, Mozart, Schubert, Debussy and Satie.

Last but not least, the golden section is also to be found in the construction of musical instruments.

In "The New Oxford Companion to Music, Volume 2" you can read that Stradivari and Guaneri used the golden section in order, for example, to place the sound holes in their world-famous instruments at exactly the desired position.

The Fibonacci numbers and Mandelbrot's fractals

The American-French mathematician Benoît Mandelbrot is very largely responsible for the interest in fractal geometry and the chaos theory that emerged in the 1980s.

Fractals are, firstly, only mathematically defined objects which are not one, two or three dimensional, but something in between. It is impossible to imagine that clearly. Nevertheless, objects occur in nature which come close to such fractals. Structures can be found in them that repeat themselves and which are similar when considered approximately or in detail. That is why we also talk of self-similarity.

An especially beautiful example of fractal geometry in nature is the Romanesco, a green variety of cauliflower.

But it was not so much the scientific significance as the possibility of producing pictures of great aesthetic appeal using simple algorithms and the computer that no doubt contributed to the popularity of fractals. The most famous of them is probably the Mandelbrot set. In it similar, but repeatedly new and incredibly beautiful structures are revealed in the area of their edges with every enlargement.

What do fractals have to do with the Fibonacci numbers? The apple-shapes of the Mandelbrot set, which vary in size, arise in different periods of the repetition of mathematical algorithms. If we examine these apple-shapes, we make the following observation: of all the apples between one apple of period 2 and one apple of period 3, it is the apple of period 5 that is the largest. In exactly the same way, of all apples between period 5 and period 3, it is the apple of period 8 that is the largest. And of the apples between that of period 8 and that of period 5 it is the apple of period 13. All numbers from the Fibonacci series.

Fibonacci in the equity market

In the late 1920s the American mathematician Ralph Nelson Elliot developed an analysis of the equity market, which was later called the Elliot Wave Principle. Elliot examined in particular the psychological aspects of the sellers' behaviour and tried to explain movements in the market by means of patterns in crowd-psychology. His wave theory claims that share prices are guided by pre-determined cycles based on the Fibonacci sequence. According to that, during a bull market the market prices move in five upward waves and in three waves somewhat downwards again; in a bear market the pattern is reversed.

If the Elliot waves are examined from the perspective of the chaos theory different intervals can be interpreted as self-similar. The waves – five up, three down – occur not just after quite a considerable interval of time, but every day, every hour, every minute. Recent research has shown that such models, so called market fractals, may be interpreted as instruments for measuring the social and historical development of a country.

 

The quintessence of the sunflower

Let us return again to the sunflower idea as philosophy. Jürg Conzett regards the individual spirals of the sunflower as metaphors for persons. Persons with preferences, abilities and talents. Just as a sunflower grows and thrives, so, too, growth and success are experienced by people when they recognise and make use of their talents. No matter whether he or she is a dancer, an artist, a businessman or a scientist: whoever is aware of what he enjoys doing and can do well will be successful, and success will always result in new success. In the Bible it says: "For whosoever hath, to him shall be given, and he shall have abundance."

Jürg Conzett has asked himself what he needs to recognise and develop his talents. He explains his reflections with the three terms self-confidence, self-esteem and a recognition of the esteem one has for others. Self-confidence is possessed by someone who gives thought to his own aptitudes: his origins, his history, his wishes and desires, his strengths and talents, but also his weaknesses and insecurities. Anyone who has good self-esteem can be himself, contribute his abilities and exchange his views on matters. But success requires a third aspect, namely to recognise the use of one's own talents for other people – in brief: a feeling for the esteem that one has for others. This feeling requires much empathy: you have to know what moves the other person, what needs he has.

Just as in the head of a sunflower numerous spiral arcs lie side by side, so the human being is also integrated into communities, whether it be the family, whether it be a working team. A good partnership is made up of individuals who, although they have different abilities, still harmonise with one another, and all are equally respected. If the persons are too similar to one another, if there is a lack of sufficient tension, disharmony will prevail and the creative spiral will fall apart.

When people who are in harmony with one another work together, they vibrate, but do not control themselves mutually, chance plays an important part. Things occur which nobody had foreseen, but which fit in precisely. It is a kind of serendipity, of favourable providence that results from a positive constellation.

But friendly relationships between people do not stop at the working team. Success also means including the social sphere and improving interaction with it. Jürg Conzett recognises the social sphere in the spiral arcs of the sunflower rotating in opposite directions.

He considers the centre to be decisive for the creative intertwining of these two spirals: a vision, an alignment, an inner image holding everything together.

The awareness of oneself as a person, the potential that is present in partnerships and interaction with the surroundings guide the activities of the Sunflower Foundation and of the MoneyMuseum. An interest in history is for Jürg Conzett – who studied history – not an end in itself, but a means of acquiring a better understanding of himself and his fellow men, his sphere of activity. So when the MoneyMuseum deals with such subjects as economic and social history, it is not a matter of imparting dry knowledge. But the aim of the exhibition is, rather, to stimulate the visitors to become aware of the way they handle money. Completely in keeping with the sunflower idea: a philosophy of constant change, of growth and the interaction of people.