[Music] 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 accident was largely [Music] unknown in his work Libra abachi 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 accidental mathematics and contributed to among other things Europe adopting the Arabic system of numbers Liber abachi 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 [Music] 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 furtle from the second month after birth in his book Liber AI he writes because the above mentioned pair gives birth in the first month you can double it so 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 [Music] 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 all together 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 [Music] months to begin with there is one pair of rabbits after 1 month there are two pairs after 2 months there is 1 + 2 or three pairs after 3 months 2 + 3 or five after 4 months 3 + 5 or 8 and so on until after 11 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 air have finally produced that many pairs at the end of one [Music] 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 and in art be it in the position of leaves and plants in the spiral form of Mullis 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 one beside it we construct a second square of the same size we attach another Square to it which has the length two to this is added a square with the length of the side three one with the length of the side five one with the length of the side eight it is not difficult to recognize the numbers of the Fibonacci sequence now we draw a quadrant in each Square the result is a spiral called the Fibonacci spiral it can be clearly seen in the Shell of the Nautilus [Music] [Music] it was the Greek mathematician uid 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 large 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 Puli took an interest in uet's works and devoted a whole volume to this division of lines which he called Divina proport or Divine partition [Music] around 1600 Johannes Kepler known for the Kepler laws of the movements of planets discovered the relationship between the Fibonacci numbers and the golden [Music] 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 fi the longer the sequence is [Music] continued and F describes nothing other than the golden section the golden section defines a proportion which has always been perceived as especially beautiful and [Music] harmonious in many epics we find its application in almost all cultures throughout the world above all in architecture and [Music] art a rectangle in which the proportions of its sides correspond to the golden section is called a golden rectangle similarly isoceles 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 Sai which divides the angle of 360° in the proportions of the golden section as angles smaller than 180° Pro proved to be more handy in practice the smaller of the resultant angles is usually called the Golden angle saigh it is approximately 137.5 de [Music] 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 inter 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 [Music] core like all members of the Rose family the Apple is assigned to the female the lifegiving principle it is thus not surprising that the pentagram is the symbol of Venus both of the planet and the [Music] 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 [Music] [Applause] [Music] [Music] what is surprising is the frequent occurrence of the golden section and the Fibonacci numbers in [Music] nature these structural principles reappear most conspicuously in the filot taxis of plants I.E in the arrangement of their leaves and Seed cases in many of the more highly developed species of plants the angle between spiral shaped consecutive leaves is on average about 137.5 de the golden angle this arrangement of leaves is also termed the Fibonacci filot taxis as the go 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 onto the [Music] roots in the case of the sunflower filot taxis appears in the spirally arranged seed cases on the flower's receptacle in an especially aesthetic form [Music] the clearly recognizable 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% [Music] 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 center 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 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 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 [Music] [Applause] [Music] 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 SES even in the Pyramids of Giza the proportions of the number fi are revealed with astounding accuracy thus for example in the case of the Chop's pyramid the proportion of the length of the side of the pyramid to half of the pyramid's base is 356 to 220 L's and that corresponds to 1.618 of the number [Music] five also in the most famous stone monument Stone Hedge which was built near Salsbury England some 3,500 years ago we again find the golden measurements the Parthenon erected in Athens under paricles 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 luier 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 modular 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 fingertips 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 Al Dura an artist of modern times who conspicuously employs the golden section is for example the Dutch painter Pete mandrian [Music] in The Concourse of zurich'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 MS [Music] [Music] 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 eight white and five Black Keys together 13 Keys The Black Keys are divided up into groups of two and [Music] three all numbers and proportions from the Fibonacci series in the writings of the musicologist ano Leni we read that the golden section and the Fibonacci numbers reappear in the works of the composer Bella BTO as the dominant principle 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 Bok himself whose favorite flower is said to have been the sunflower has however never made mention of [Music] this similar investigations have also been made into the works of Bak Mozart shubber de and Sati 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 stradavar and gueri used the golden section in order for example to place the sound holes in their world famous instruments at exactly the desired position [Music] [Applause] [Music] the American French mathematician benoir Mandel brro is very largely responsible for the interest in fractal geometry and the chaos theory that emerged in the 19 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 a 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 colorflower 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 mandal Bro 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 mandal Bro 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 the 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 [Music] [Applause] [Music] 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 sell's behavior 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 predetermined 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 bare Market the pattern is reversed if the Elliot waves are examined from the perspective of the chaos theory different intervals can be interpreted as selfsimilar the waves five up three down occur not just after quite a considerable 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