Tag Archives: Fibonacci

The Fibonacci Sequence in Barcelona

Photography by Marc Adillon

Photography by Núria Casals

The picture shows the Fibonacci’s sequence monument in Port Vell in Barcelona. We can see the numbers which are part of the sequence built on the pavement and separated by a proportional distance related on the ratio between them. The Fibonacci sequence was born in a mathematical problem related to the rabbit reproduction proposed by Leonardo of Pisa in his very famous Liber abaci (1202).

Leonardo Pisano, also known as Fibonacci, was born around 1170 in the city-state of Pisa. Leonardo’s father, Guglielmo Bonacci, was a Pisan engaged in business in northern Africa, in Bugia (now Béjaïa, Algeria). This Algerian city was a very important intellectual focus during the eleventh and twelfth centuries. Therefore, it is not surprising that a businessman like Mr. Bonacci realize the possibilities that had the contact of the two cultures.

Guglielmo Bonacci put his son in a calculus positional Hindu course as well, Leonardo started to be interested about mathematics. He took profit of the frequent business trips of his father to know mathematicians of the countries they visited —Egypt, Syria, Provence, Sicily, Greece— and to make a deep study about Euclid’s Elements, that always had as a logical model of rigor and style. Therefore, it was natural that Fibonacci should have been steeped in Arabic algebraic methods, including, fortunately the Hindu-Arabic numerals and, unfortunately, the rhetorical form of expression. Fibonacci decided to write about all the knowledge that he had been collecting and that impressed him so much. He wrote it in a serie of books, the first of which was Liber Abaci (1202, reprinted 1228), Practica geometricae (1223), Liber quadratorum (1225), Flos (1225) and Epistola ad Magistrum Theodorum (1225). In Liber abaci Fibonacci explains the Arabic positional numbering system and how to read numbers, add them, multiply them… and to solve all the kind of problems which could need any trader in the Medieval Ages. The famous Fibonacci sequence is one of the problems of the chapter twelve:

How many pairs of Rabbits Are Created by One Pair in One Year

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months

Fibonacci solved the problem with the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and 377 and here one of the most famous mathematical set of numbers was introduced for the first time.

This post has been written by Marc Adillon and Núria Casals in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Pla de Miquel Taradell in Barcelona (map)


The golden number in Cosmocaixa

Photography by Pol Casellas and Eric Sandín

Photography by Pol Casellas and Eric Sandín

In the wonderfull wall full of formulas (already mentioned in this blog) that you can see in the Cosmocaixa in Barcelona, there also is the sacred equation which solution is the famous golden ratio:

Photography by Pol Casellas and Eric Sandín

Photography by Pol Casellas and Eric Sandín

Of course, one of the solutions of x2 = x + 1 is the number x = 1.6180339887498948482… (the other is -0.6180339887498948482…). At first sight it may seem a regular solution for a regular equation, but this number has revealed to the world of mathematics a whole new conception of nature and proportionality and this is the reason why it is interesting to know the history of this number and who dared to study its wonderful properties.

Since the golden ratio is a proportion between two segments, some mathematicians have assigned its origin to the ancient civilizations who created great artworks such as the Egyptian pyramids or Babylonian and Assyrian steles, even though it is thought that the presence of the ratio was not done on purpose. We can go forward on history and find the paintings and sculptures in the Greek Parthenon made by Phidias, whose name was taken by Mark Barr in 1900 in order to assign the ratio the Greek letter phi. So we can associate the first conscious appearance of the golden ratio with the Ancient Greece because of its multiple presence in geometry. Although it is usually thought that Plato worked with some theorems involving the golden ratio as Proclus said in his Commentary on Euclid’s Elements, Euclid was the first known person who studied formally such ratio, defining it as the division of a line into extreme and mean ratio. Euclid’s claim of the ratio is the third definition on his sixth book of Elements, which follows: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser”. He also described that the ratio could not be obtained as the division between two integers, referring to the golden ratio as an irrational number.

Definition VI.3  in Oliver Byrne's edition of the Elements (1847)

Definition VI.3 in Oliver Byrne’s edition of the Elements (1847)

In the 13th century, Leonardo de Pisa (also known as Fibonacci) defined his famous serie in the Liber abaci (1202) although he wasn’t aware that phi is asymptotically obtained by dividing each number in the serie by its antecedent, thus, lots of natural phenomena which follows the Fibonacci sequence in any way, are related to the golden proportion.

Another important work from the 16th century is De Divina Proportione (1509) by Luca Pacioli, where the mathematician and theologian explains why the golden ratio should be considered as “divine”, comparing properties of our number like its unicity, immeasurability, self-similarity and the fact its obtained by three segments of a line, with divine qualities as the unicity and omnipresence of God and the Holy Trinity.

Dodecahedron from De Divina Proportione attributed to Leonardo da Vinci

Dodecahedron from De Divina Proportione attributed to Leonardo da Vinci

In the Renaissance, the golden ratio was chosen as the beauty proportion in the human body and all the painters and artists used it for his great masterpieces, like Leonardo da Vinci in his Mona Lisa or his famous Vitruvian Man.

Leonardo's Vitruvian Man

Leonardo’s Vitruvian Man

The golden ratio was known in the world of mathematics as the Euclidean ratio between two lines and it wasn’t until 1597 that Michael Maestlin considered it as a number and approximated the inverse number of phi, describing it as “about 0.6180340”, written in a letter sent to his pupil Johannes Kepler. Kepler, famous by his astronomical theory about planetary orbits, also talked about the golden ratio and claimed that the division of each number in the Fibonacci sequence by its precursor, will result asymptotically the phi number. He called it a “precious jewel” and compared its importance to the Pythagoras theorem.

About one century later, the Swiss naturalist and philosopher Charles Bonnet (1720-1793) found the relation between the Fibonacci sequence and the spiral phyllotaxy of plants andthe German mathematician Martin Ohm (1792-1872) gave the ratio its famous “golden” adjective. If we want to talk about artists who introduced the ratio in their paintings in the modern times, a good example would be Salvador Dalí, whose artwork is plenty of masterpieces structured by the golden ratio.

This is just a brief summary of the history behind the golden ratio, which suffices to show that the interest induced by this number over the minds of the greatest mathematicians hasn’t ceased since the Ancient Greece, and even people non-related with mathematics have used it in their own work, which shows the importance and the multiple presence of mathematics and this special number in places that one could not imagine

This post has been written by Pol Casellas and Eric Sandín in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Cosmocaixa in Barcelona (map)

The Museum of Mathematics of Catalonia (MMACA)


Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.

Mercader Palace Photography by Carlos Dorce

Mercader Palace
Photography by Carlos Dorce

The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…

Photography by Carlos Dorce

Photography by Carlos Dorce

For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.

Photography by Carlos Dorce

Photography by Carlos Dorce

Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!


Photography by Carlos Dorce

Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.

Photography by Carlos Dorce

Photography by Carlos Dorce

Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!

Photography by Carlos Dorce

Photography by Carlos Dorce

Another example: look at these three wooden pieces…


The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.

And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?

Photography by Carlos Dorce

Photography by Carlos Dorce

In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:

Source: Wikimedia Commons

Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his Zîj-i Ilkhanî (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his De Revolutionibus (1543). Al-Tûsî’s pair was very famous until the 15th century.

In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:

Photography by Carlos Dorce

Photography by Carlos Dorce

I didn’t know that it could be described only with a multiplication table! Is its equation z = xy? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.

Photography by Carlos Dorce

Photography by Carlos Dorce

Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:

Photography by Carlos Dorce

Photography by Carlos Dorce

Furthermore, one of the rooms of the palace is decoratd by a chess lover!

Photography by Carlos Dorce

Photography by Carlos Dorce

So… you must go to MMACA and enjoy Mathematics in a way ever done!


Location: MMACA in Cornellà de Llobregat (map)

Fibonacci in Pisa

Statue of FibonacciPhotography by Carlos Dorce

Statue of Fibonacci
Photography by Carlos Dorce

I didn’t only walk through Galileo Galilei’s steps in Pisa two weeks ago. It wasn’t my first time in Pisa but I hadn’t never visited the cemetery of the Piazza dei Miracoli before. I admit that the Camposanto Monumentale is a very interesting place. I could’n imagine that I would enjoy this place so much.

Camposanto monumentalePhotography by Carlos Dorce

Camposanto monumentale
Photography by Carlos Dorce

Nevertheless, I visited the cemetery because I was interested in a particular marble statue: Fibonacci is exhibited there! Leonardo da Pisa, Fibonacci (c.1170-c.1230), is one of the most famous mathematical names. We know very little about his life apart from his own biography written in his Liber abaci (1202):

After my father ‘s appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view of i ts future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the ar t very much appealed to me before all other s , and for it I realized that all i ts aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter , while on business, I pursued my study in depth and learned the give-and take of disputation. But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid’s geometric art , I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its preeminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it , as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.

The statue is one of the corners of the cloister and the inscription on the pedestal says: “A Leonardo Fibonacci. Insigne Matematico Pisano del Secolo XII (To Leonardo Fibonacci, eminent XII century Pisan mathematician)”.

The statue was planned by the marquis Cosimo Ridolfi and the baron Bettino Ricasoli from Firenze who wanted to promote the Tuscan culture among the people. Ricasoli was the prime minister of the Tuscany which had benn annexed to teh Savoy Reign in 1859 and Ricasoli was the secretary for education. So in September 23, 1859, they promoted a decree to finance a statue of Fibinacci as “the initiator of the algebraic studies in Europe” in the city of Pisa. The sculptor Giovanni Paganucci was commisioned fot that job and the statue was finished four years later and placed in the Camposanto of Pisa. In 1926, the Fascist goverment decided to place some statues of the cloister in some squares of Pisa trying to show eminent Pisan figures to the people: Fibonacci was one of them. Fibonacci was placed in front of the Ponte di Mezzo in the centre of Pisa. When in 1944 the Alied Troops arrived at Pisa, they bombed all the city and almost all the centre was destroyed. However, our statue of Fibonacci kept its position standing in the middle of the damaged city:

Fibonaci was also little damaged and nowadays he hasn’t got fingers in his hands.

After the II World War, the Camposanto was restored and Fibonacci was kept in a warehouse until he was moved to Giardino Scotto (map). In 1990’s, Fibonacci was accurately restored and was placed in the Camposanto monumentale.

Location: Camposanto monumentale (map)