Category Archives: Catalonia

The conoids in Sagrada Família Schools

Photography by Àlvar Pineda

Photography by Àlvar Pineda

It is nothing new that Antoni Gaudi’s constructions are related to mathematics. In this post, we will focus on the conoid, a ruled surface that appears in Sagrada Família Schools (Escoles Provisionals de la Sagrada Família, in Catalan), near the Basílica i Temple Expiatori de la Sagrada Família in Barcelona.

First of all, we have to explain the concept of ruled surfaces: a surface S is ruled if through every point of S there is a straight line (called ruling) that lies on S. This implies that a ruled surface has a parametric definition of the form S(t,u) = P(t) + u Q(t).

Source: Wikimedia Commons

Source: Wikimedia Commons

As you can see, the roof of the schools is one ofthese surfaces which we call conoid: we can generate it by displacing a straight line above another straight line (the axis) and above a curve (often a sinusoid). Consequently, for every point on the conoid there is a straight line that passes trough that point and intersects de axis. If all of those straight lines are perpendicular to the axis, then the conoid is called right conoid. The conoid of this post is not in Sagrada Família but on the roof and the façade of the Escoles provisionals de la Sagrada Família. Antoni Gaudí designed that building on the commission of the entity that sponsored the project of the Sagrada Família, the Associació de Devots de Sant Josep (presided over by Josep Maria Bocabella (1815-1892)), and the school was for the children of the parish and also the children of the building workers of the temple. The building was divided in three classrooms, a hall and a chapel, and was constructed with brick. Its principal promoter was Gil Parés i Vilasau (1880-1936), the first parish priest of the Sagrada Família. He was also the school’s principal until 1930 and he used the Montessori method from 1915.

The building, inaugurated on November 15, 1909, has an amazing story of destructions and reconstructions. In fact, this peculiar school was intended to be demolished because Gaudí located it occupying land reserved for the construction of the Sagrada Família’s Passion façade. However, it was dismantled and rebuilt earlier than expected as a result of the several damages during the Spanish Civil War (1936-1939). Domènec Sugrañes i Gras (1878-1938) designed the restoration that finished in 1940, but the project had few funds and for this reason, in 1943 Fransesc Quintana (1892-1966) directed another refurbishment. Many years later, in 2002, the Passion façade was going to be built, so the building of the Escoles provisionals de la Sagrada Família was dismantled again and reconstructed in the corner between Sardenya and Mallorca streets, where the picture has been taken. In this regard, we can add that the building has become a small museum. It is important to note that the fact of being surfaces generated by straight lines makes the construction of the roof and the façade easier. Besides that, the profile of the roof is highly effective to drain off waterin a rainy day.

The contrast between the simplicity of the building (it was a very cheap and quickly erected structure) and its importance in twentieth century architecture is really remarkable.

This post has been written by Àlvar Pineda in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Sagrada Família Schools (map)

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A Fractal garden in Barcelona

Photography by Àdel Alsati

Photography by Àdel Alsati

The Botanical garden of Barcelona, located in Montjuïc, has an extension of 14 hectares. It is specialized in the mediterranean climate and contains a wide range of plants from all over the world. Moreover, it is divided into the five main regions of the planet with this kind of weather, such as Chile, California, South-Africa, Australia and Southern Europe.

It was designed by the architects Carles Ferrater and Josep Lluís Canosa working in an interdisciplinary team whose two main priorities were, firstly, to distribute the plants so that they are placed together with the other ones of the same geographical region, and, in addition to that, that within every region, plants are disposed following their ecological affinities representing the different landscapes existing in those zones. Secondly, they didn’t want to do it making large earthworks.

Photography by Àdel Alsati

Photography by Àdel Alsati

They achieved the design of the park in a mathematical way since they designed the park following fractal structures: they split the land into triangles, so that every triangle contained the plants of a particular landscape, while each of the five regions was represented by a set of this triangles.

Map of the Botanical Garden

Map of the Botanical Garden

If we look at the zigzag shape of the path, and then at the trapezoidal pieces which constitute it, we can found a very good example of fractal geometry.

Photography by Àdel Alsati

Photography by Àdel Alsati

And… if we look more carefully, we’ll find it everywhere around us!

Photography by Àdel Alsati

Photography by Àdel Alsati

This post has been written by Àdel Alsati in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Botanical Garden in Barcelona (map)

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 Catalan Cane

This column measures 1 cane

This column measures 1 cane

In the corner of Santa Llúcia and Bisbe Streets, we find a corious thing that does not call the atention of any of the tourists who walk around the cathedral of Barcelona. On the romanic chapel of Santa Llúcia, erected three decades before the construction of the Cathedral, we find this semicircular column sculpted in stone on the wall which measures exactly one “destre cane”.

Photography by Robert Salla

Photography by Robert Salla

The word “cane” is engraved on the wall next to the column so everybody in the Medieval barcelona could check that this was the standard measure of longitude in the market.

A cane (from the latin qana) was an ancient unit of mesurement used on the Crown of Aragon, part of France and the north of Italy. Before the Internacional Sistem of Units it was a way to have a fixed reference of lenght. This unit was used for building specific sticks of wood that were used on the market tents to have a reference when they were selling. In Barcelona, it was equivalent to eight palms, six feet or two steps, that is about 1.55 meters, although it wasn’t exactly the same measure everywhere. For example, in Tortosa it was equivalent to 1.59 meters but the reference to the whole Catalan countries was the same as in Montpellier, equivalent to 1.99 meters. Furthermore there were the square cane which was used to measure surfaces: in Barcelona it was equivalent to 2.44 square meters, 2.42m. in Girona , 2.43m in Tarragona and in 2,45m in Mallorca. Surprisingly we see that there is less diference between the square canes than in the lenght measures. As we’ve said, it was the unit of longitude used in the markets next to the cathedral in the Middle Ages althought the Catalan “destre cane” was also used. It was equivalent to twelve palms and this is exactly the height of the column that we find next to the cathedral.

This post has been written by Ander Castillo and Robert Salla in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Carrer de la Pietat 2 (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)

A very famous magic square in Barcelona

Source: Wikimedia Commons

Source: Wikimedia Commons

If you ever visit one of the biggest Gaudí’s (1852-1926) architectural achievement in the beautiful city of Barcelona, the world-wide known and still under construction Sagrada Família, and you are passionate about maths, you might want to take a closer look at the sculpture of Judas’ betrayal; right by its side you can find embedded on the Sagrada Família’s Facade of Passion a 4×4 matrix, known as the magic square.

Magic squares are square matrices with feature integer numbers, which add up to the same amount in columns, rows and diagonals. That amount is known as the magic constant and the one concealed in Sagrada Família is the number 33. Check it out:

Source: Wikimedia Commons

Source: Wikimedia Commons

Notice that there are more combinations which add up to 33. For example, sum the red numbers and the green ones in each square:

SF3

This magic square is also included as a decoration in one of the main doors of the Passion Façade. Can you find it?

And what does it stand for? While some people argue it might have something to do with the highest degree in the Masonic lodges – and consequently relates the architect to Freemasonry – the truth is that its author is Josep Maria Subirachs (1927-2014) (Catalan sculptor famous for this design) who chose the number 33 since it’s the age at which Jesus died on the Cross. Whether or not there might be other curious legends surrounding it, it’s remarkable how maths has found room in such a masterpiece. For those who never got along with numbers, here they have a whole new and much more artistic rather than scientific perspective that might light up their face whenever they come across a magic square. Thus everyone can fully enjoy the world of maths!

This post has been written by Carles Raich in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Sagrada Família in Barcelona (map)

A catenary welcomes you to the beach in Ocata

Photography by Bernat Plandolit

Photography by Bernat Plandolit

Our daily landscape is full of images like this one. Every day we see chains that bar our way and we assume them naturally. However, which shape do these chains take? Can we identify them with known curves? These questions have thrilled some of the greatest mathematicians of all times and have led to the development of elaborated techniques valid today.

Johann Bernoulli by J.Rudolf (c.1740) Source: Wikimedia Commons

Johann Bernoulli by J.Rudolf (c.1740)
Source: Wikimedia Commons

The  main characters of this post are the Bernoulli brothers, Jakob (1655-1705) and Johann (1667-1748). They studied together Gottfried W. Leibniz’s works until they mastered it and widened the basis of what is nowadays known as calculus. One of the typical problems was the study of new curves which were created in the 17th century to prove the new differential methods. René Descartes and Pierre de Fermat invented new algebraic and geometrical methods that allowed the study of algebraic curves (those which the coordinates x and y have a polynomial relation). Descartes didn’t consider in his Géométrie curves that were not of this kind, and he called “mechanical curves” to all those ones that are not “algebraic”. In order to study them, he developed non algebraic techniques that allowed the analysis of any kind of curves, either algebraic or mechanical. These mechanical curves had already been introduced a lot of time ago, and were used to solve the three classical problems: squaring the circle, angle trisection and doubling the cube.

When developing calculus, Leibniz’s objective was to develop this general method that Descartes asked for. When Bernoulli brothers started to study curves and its mechanical problems associated, calculus would become its principal solving tool. For instance, in 1690 Jakob solved in Acta Eruditorum, a new problem proposed by Leibniz. In this document, he proved that the problem was equivalent to solve a differential equation and the power of the new technique was also shown.

During the 17th century, mathematicians often proposed problems to the scientific community. The first challenge that Jakob exposed was to find the shape that takes a perfectly flexible and homogeneous chain under the exclusive action of its weight and it is fixed by its ends. This was an old problem that hadn’t been solved yet. As we can see in the picture, the shape taken by the chain is very similar to a parabola and, owing to the fact that it is a well-known curve for centuries, it comes easy to think that it is indeed a parabola (for example, Galileo Galilei thought that he had solved the problem with the parabola). However, in 1646 Christiaan Huygens (1629-1695) was capable to refute it using physical arguments, despite not being able to determine the correct solution.

Christiaan Huygens by C.Netscher (1671) Source: Wikimedia Commons

Christiaan Huygens by C.Netscher (1671)
Source: Wikimedia Commons

When Jakob aunched the challenge, Huygens was already 60 years old and successed in finding the curve geometrically, while Johann Bernoulli and Leibniz used the new differential calculus. All of them reached the same result, and Huygens named the curve “catenary”, derived from the latin word “catena” for chain.

Hence the first picture shows a catenary which we observe without paying much attention although all its history. Nowadays, we know it can be described using the hyperbolic cosine, although any of all these great mathematicians couldn’t notice it, as the exponential function had not been introduced yet. Then… How did they do it? Using the natural geometric propreties of the curve so Huygens, Johann Bernoulli and Leibniz could construct it with high precision only usiny geometry. Jakob didn’t know the answer when he proposed the problem and neither found the solution by himself later. So Johann felt proud of himself for surpassing his brother, who had been his tutor (Jakob, who was autodidactic, introduced his younger brother to the world of mathematics while he was studying medicine). What is more, the catenary problem was one of the focusses of their rivalry.

Jakob and Johann were also interested in the resolution of another fmous problem: the braquistochrone. Now, Johann proposed the challenge of finding the trajectory of a particle which travels from one given point to another in the less possible time under the exclusive effect of gravity. In the deadline, only Leibniz had come up with a solution which was sent by letter to Johann with a request of giving more time in order to receive more answers. Johann himself had a solution and in this additional period Jakob, l’Hôpital and an anonymous english author. The answer was a cycloid, a well-known curve since the 1st half of the 17th century. Jakob’s solution was general, developing a tool that was the start of the variational calculus. Johann gave a more imaginative solution based on Fermat’s principle of maximums ans minimums and Snell’s law of refraction of light: he considered a light beam across a medium that changed its refraction index continuously. Given this diference between their minds, Johann enforced his believe that he was better due to his originality and brightness, in contrast with his brother, who was less creative and worked more generally.

A third controversial curve was the tautochrone which is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. This property was studied by Huygens and he applied it to the construccion of pendulum clocks.

While we can see catenaries in a park to prevent children from danger or in the entrance to this Catalan beach, we luckily do not see cycloid slides. Thus, children will not descend in the minimum time but will reach the floor safe and sound!

But… who was the misterious english author? For Johann Bernoulli the answer to this question did not involve any mistery: the solution carried inside Isaac Newton’s genius signature -to whom we apologise for having mentioned Leibniz as the calculus inventor-, and he expressed that in the famous sentence:

I recognize the lion by his paw.

Isaac Newton by G.Kneller (1702) Source: Wikimedia Commons

Isaac Newton by G.Kneller (1702)
Source: Wikimedia Commons

Here we see one more example of a catenary in our daily life:

Photography by Bernat Plandolit

Photography by Bernat Plandolit

You have more information in this older post.

This post has been written by Bernat Plandolit and Víctor de la Torre in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Ocata beach (map)

A masonic geometric symbol in Barcelona

Photography by Paula Arrebola and Abel Hernadez

Photography by Paula Arrebola and Abel Hernadez

There are a lot of hidden enigmas and misteries in the modern buildings of Barcelona and some of these details are unknown by almost everyone. For example, on the facade at 11 Portaferrissa Street in the Old Town (Barri Gòtic), we can see the sculpture of two little boys above the main door on the top of a pile of bricks between them and the boy on the right holds some kind of square, whereas the one on the left holds a compass and a paddle. At first sight, it may seem another figure of the architectural style of the 19th Century. Nevertheless, it is a masonic sign indeed. The square and the compass were some of the most remarkable symbols of Freemasonry since both appear in the Masonry emblem.

Source. Wikimedia Commons

Source: Wikimedia Commons

In ancient times, the compass symbolized the Heavens inasmuch as it was used to study the starry Heavens, while the square represented the Earth because it was used to measure it. Nowadays both might have some philosophical and ethic connotations such as boundaries, so as to keep the equilibrium, and morality respectively. So, according to the Barcelona Historical Archive, the construction was made in the late 19th Century since there is a file with a works license of 1867, projected by Domingo Sitjas. It is interesting to notice that, in the original plans of the project, there is no sign of this sculpture. Due to Masonry persecution, everything related to it had to be kept secret. So, logically, the sculpture doesn’t appear in the plans!

Freemasonry describes itself as a beautiful system of morality, veiled in allegory and illustrated by symbols. Traditionally, masons are fond of architecture and are dedicated to the seven Liberal Arts: Grammar, Rhetoric, Arithmetic, Logic, Music, Geometry and Astronomy. Therefore, since Arithmetic, Geometry and Logic belong to mathematics, there is a link between Masonry and mathematics somehow. For instance, number 1 is represented by a point, which has no dimensions and turns to be the generator of any imaginable figure. According to the Masons, one is the arithmetic symbol of the Unity as well as the point is the geometric image of the Being. On top of that, if there are two points, they can be joined with a compass considering the straight segment connecting them like the one-dimensional projection of the geometrical link. It is important to note that the symbol of the Unit is the generator of duality, ternary, etc. using the compass. In addition to that, Geometry is the basis on which the Masonic superstructure stands and is considered by them as the mother of Science. However, Arithmetic has an important role in Masonry philosophybecause under Freemasons’ point of view, each of the four fundamental operations corresponds to a present value in their lives. For example, the sum is related to adding knowledge to our cognizance. In regard to such importance of the Unity as the genesis of the rest, it should be noted that the quaternary arises just like the ternary does. The last one would be represented in the form of a square. Mathematically, this generating method from the Unit would be considered as N = 1 + n (0<n<9 integer number), where N is associated with geometric figures representing values ​​or facts of their daily life. It must be kept in mind that this is a cycle whose culmination is the following expression: 9+1=10=1+0=1. Let’s look at some examples of the corresponding symbolism to certain geometrical figures identified, as we have seen, with numbers. Remember that the “construction” of these is given by using a compass and a square like the ones in the image:

  1. The triangle, which represents number 3, is the geometric figure par excellence. The rest of figures can be represented as a set of triangles. It is known among Masons as Radiant Delta and each of its vertices represents space, time and energy. The union of these vertices turns out to be the force that gives structure to the universe, which is the Great Architect of the Universe (G), i.e., God.
  2. The square, which represents number 4, symbolizes at the same time two squares or the union of two triangles, that is, harmony and balance. If these two squares are within a circle, then they represent the harmony between the Earth and the Sacred World.
  3. The 5-pointed star (or pentagrammon) represents number 5 and symbolizes the man and life, what is masculine and what is feminine, and the union represents androgyny.

It’s imperative to comment that, although there may be no direct link between Pythagoras and Freemasonry, the teachings of Pythagoras have greatly influenced its structure and its teachings: numbers (especially numbers from 1 to 10) are symbols representing the philosophical universal organization and the only way to reach God. Furthermore, every degree of the Masonic initiation corresponds to one number.

Finally, it is necessary to remark that this is just a selection of the whole mathematics they studied, since the 47th Euclid problem and the golden proportion should be also mentioned.

In conclusion, the sculpture emblems a fraternal community full of mysteries which has always been passionate about mathematics. One last mathematical fact: if you come in the building, you will see 7 steps followed by 7 steps and 14 steps more!

Photography by Paula Arrebola and Abel Hernandez

Photography by Paula Arrebola and Abel Hernandez

This post has been written by Paula Arrebola and Abel Hernandez in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location Casa Domingo Sitjas (map)

Roman gambling in MUHBA

Photography by Carlos Dorce

Photography by Carlos Dorce

MUHBA (Museu d’Història de Barcelona) is one of the most interesting museums in Barcelona. Located in Plaça del Rei, it involves a journey through an area stretching over 4000 m2 under the actual square which reveal the Roman’s ruban structure of the city. The remains allow the visitor to take a look at the commercial life of the city and its craft production centres and the everyday life of Barcelona’s first Christian citizens.

The main focus of the exhibition is the Roman ruins through which you can explore the life of the citizens of the former Barcino. There is a lot of information about Roman life and… of course, gambling was very important for our ancestors. For example, look at these bone dice (1st-3rd centuries) and terra sigillata globets (1st-2nd c.) found in the ruins! One of them is a weighing one for the most cheating players!

Photography by Carlos Dorce

Photography by Carlos Dorce

Although gambling was prohibited by law, Romans played a lot and traps were so common among them. When the lusoria tabula was not available, it was improvised by stripes on the ground or on stones, as we can see in this board from the 1st-4th c.:

Photography by Carlos Dorce

Photography by Carlos Dorce

There also are improvised boards graved on ceramics:

Photography by Carlos Dorce

Photography by Carlos Dorce

This latrunculus was also found in the ruins (1st-4th c.):

Photography by Carlos Dorce

Photography by Carlos Dorce

The latrunculus was a very popular game derived from the Greek Petteia to which Homer quotes in his works. Varro (1st c. BC) wa sthe first Roman author who mentions this game.

Another popular game was the traditional coin flopping (navia aut caput) which was played with these coins:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, I must talk about the tali (knucklebones) of the first photography. They probably were the most popular game in the Roman Empire and we have a lot of witnesses of their use until the 19th century. For example, you can notice the knucklebones in this 18th century painting:

Girl playing knucklebones. Jean-Baptiste-Siméon Chardin (1734)

Location: MUHBA in Barcelona (map)

The Dalí Theatre-Museum

Source: Wikimedia Commons

The Dalí Theatre-Museum, opened in 1974, is the largest surrealistic object in the World. It was built on the ruins of the ancient theater of Figueres and hosts the most important collection of Dalí’s pictures and sculptures.

Salvador Felipe Jacinto Dalí i Domènech, Marquis of Púbol (11 May 1904 – 23 January 1989) was born in Figueres. Although his principal mean of expression was the painting, he also made inroads in different fields such as cinema, photography, sculpture, fashion, jewellery and theatre, in collaboration with a wide range of artists in different media. His wife and muse, Gala Dalí was one of the essential characters in his biography. His public appearances never failed to impress and his ambiguous relationship with Francisco Franco’s regime made of this multifaceted character an icon of the 20th century and more than an artist. During his life he lived in Madrid, Paris and Catalonia and for this reason he was influenced by other important artists. He died in Barcelona and was buried in his own museum against his desire.

Dalí’s tomb. Source: Wikimedia Commons

Why did I say that he is more than an artist? If you visit the Dalí’s Theatre-Museum in Figueres, you will see his art based on mathematics and physical laws. Dalí’s relationship with science began in his teens when he started reading scientific articles and this passion for science was preserved all his life. In the museum you can find a great reflection of that passion. Furthermore, the painter’s library contains hundreds of books with notes about various scientific topics: physics, quantum mechanics, life’s origin, evolution and mathematics. In addition to that, he was subscribed to several scientific journals to be informed about the new scientific advances.

To show this relation between Mathematics and his masterpieces, I will explain three artworks which are exhibited in the museum from a mathematical point of view. The first one is Leda Atomica (1949). He created it from studying Luca Pacioli’s De Divina Proportione (Milan, 1509) Dalí made different computations for three months with the help of Matila Ghyka (1881-1965). Ghyka wrote some mathematical treatises related with the golden number like Le nombre d’or: Rites et rythmes pythagoriciens dans le development de la civilisation occidentale (1931), The Geometry of Art and Life (1946) or A Practical Handbook of Geometry and Design (1952).

Matila Ghyka Source: Wikimedia Commons

Matila Ghyka
Source: Wikimedia Commons

The painting synthesizes centuries of tradition of Pythagorean symbolic Mathematics. It is a watermark based on the golden ratio, but making the viewer not appreciate it at first glance. In 1947’s sketch, it can be noticed the geometric accuracy of the analysis done by Dalí based on the Pythagorean mystic staff, which is a five-pointed star drawn with five straight strokes:

Source: Wikimedia Commons

Source: Wikimedia Commons

You can see that Gala, in the centre of the painting, is enclosed in a regular pentagon and her proportions are according the golden ratio. The picture depicts Leda, the mythological queen of Sparta, with a swan suspended behind her left. There also are a book, a set square, two stepping tools and a floating egg. Dalí himself described the picture in the following way:

Dalí shows us the hierarchized libidinous emotion, suspended and as though hanging in midair, in accordance with the modern ‘nothing touches’ theory of intra-atomic physics. Leda does not touch the swan; Leda does not touch the pedestal; the pedestal does not touch the base; the base does not touch the sea; the sea does not touch the shore…

File:Leda atomica.jpg

Leda atomica (1949)( Source: Wikimedia Commons

Another mathematical example is Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors from 1973. This is a stereoscopic work which is an example of the experiments conducted by him during the seventies. Dalí wished to reach the third dimension through stereoscopy and to achieve the effect of depth.

Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors (1973)  Photography by Roger Pijoan Català

Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors (1973)
Photography by Roger Pijoan Català

The last example is Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln (1975). In this case, Dalí used the double image techinque for creating akind of illusion which is very common in his work.

Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln  Photography by Roger Pijoan Català

Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln
Photography by Roger Pijoan Català

 

So, Dalí was more mathematician than one can imagine.

This post has been written by Sara Puig Cabruja in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

More information about Dalí’s scientific motivation: Salvador Dalí and Science and Salvador Dalí and Science. Beyond a mere curiosity.

Location: Dalí Theatre and Museum in Figueres (map)