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)

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 Padovan Sequence in Menorca

Photography by Sara Bagur

Photography by Sara Bagur

This sculpture was designed by the students and teachers (Lluís Cervera Pons and Gabriel Seguí de Vidal) of IES Pascual Calbó i Caldes to promote Maths in the population of Menorca. It represents a spiral of equilateral triangles with side lengths which follow the Padovan sequence and the “plastic number”

The plastic number is the unique real solution of the equation x3 = x + 1 which is:

plastic01

This number is approximately equal to ψ=1,324717957244746025960908854…

The other two complex solutions are:

plastic2

ψ is also the limit of the sequence ol ratios P(n+1)/P(n) where P(n) is the Padovan sequence (which is named after the architect Richard Padovan (born in 1935)). The Padovan sequence is the sequence of integers P(n) defined by the initial values P(1)=P(2)=P(3)=1 and the recurrence relation P(n)=P(n-2)+P(n-3). According to Padovan, the plastic number was discovered at the same time by Hans van der Laan.

In the Van der Laan Foundation webpage we can read:

The plastic number

[…] Through experiments with pebbles and then with building materials, he discovered a ratio he called the plastic number. The basis for the plastic number is the relationship between measures belonging to a group of measures. They increase or decrease according to the ratio four to three. The parallel in music is the ratio that relates whole and half notes to each other within an octave. The analogy between the plastic number and music goes even further: in music we can play chords, combinations of tones, with the plastic number we can compose walls and rooms and spaces that are in harmony with each other because they relate to each other as objectively as the tones within a musical key. The plastic number is not a particular measure: it disciplines the relationship between the measures we choose.

Van der Laan expressed his thoughts on the relationship between architecture and nature in this way:

If the body function of the House consists of establishing harmony between the body and its natural environment, the expression of that function will be based on the harmony between the wall that separates and the separate space. It becomes thus registration extension appreciable by the senses, both space-enclosing the separator element. That harmony will depend on the mutual proportions, who speak to the intelligence through the target language of the plastic number and that are established by the rules of the corresponding architectural ordering.

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

Location: The Padovan Sequence sculpture (map)

A solar “eight” in the Cathedral of Palma

Source: Wikimedia Commons

Source: Wikimedia Commons

Palma’s cathedral, also known as La Seu, like almost every building of this kind hide curiosities and mathematical facts that surprise due to the construction’s age. We can find trigonometry relations in a common cathedral or church, some of them even with a religious meaning behind. But in La Seu we can find two particular incidents that catch our eye. The first one happens every 2nd of February (2/2) and 11th of November (11/11), when the sunbeams go throw the east-faced rose window (known as Oculus Maior) and they are projected under the west-faced rose window, tangentially, in such a way that their centers lie on a line that is perpendicular to the ground.

Sorice: XEIX webpage

Source: XEIX webpage

It is not a coincidence the days in which the event takes place, as they are both in a similar position regarding the winter solstice. Thanks to this light  effect and the data currently available on the internet, we can easily calculate La Seu’s orientation. When this phenomenon takes place, the direction of the sunbeams coincide with the nave’s orientation. Then, we only have to set the time and figure out the exact azimuth considering this time and geographic situation. With some of the available programs, we can find out that it has a value of 122,4º and the angle of solar elevation is 10,2º (both calculus with an error smaller than 0,5º!).

Source: XEIX webpage

Source: XEIX webpage

The other incident takes place during the days near to the winter solstice. We can stare at the sunrise going through both rose windows causing a rather impressive light effect. Then again, we can find the value of the azimuth which results in 120,3º. Curiously, the bell tower, which has a square base, doesn’t have an axis of symmetry parallel to the central nave, they are out of place about 10º from each other.

Source: XEIX webpage

Source: XEIX webpage

Like many others cathedrals and churches, this one is built on an ancient mosque. In this culture, it was very important to have the ”qibla” oriented towards Mecca, specifically, towards the Ka’ba, following the precepts of the Koran. The muslim domination in Mallorca happened between 903 and 1231. In this period of time, the solution of the “qibla problem” was already known, solved by al-Khwârizmî in the 9th century. In effect, if one traces the line segment bisector of the east-faced bell tower and lengths it over the terrestrial sphere it matches the Ka’ba with an astonishing precision.

You can read more information about these two facts in this interesting article.

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

Location: Cathedral of Palma (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)

Einstein House in Bern

Photography by Carlos Dorce

Photography by Carlos Dorce

This is the house where Albert Einstein and his familiy lived from late 1903 to May 1905 and where he developed his quantic theory.

Photography by Eduard Ribas

Photography by Eduard Ribas

The house host one of the most productive career of Albert Einstein meanwhile he worked in the patent office of Bern.

Photography by Carlos Dorce

Photography by Carlos Dorce

In the house you can see the dining room and some dormitories and you can imagine Einstein’s family having lunch and Einstein discussing with his wife malena about his researches in Physics!

Photography by Carlos Dorce

Photography by Carlos Dorce

But… do you want to know something more about Eintein’s biography? Here you have one:

Childhood and youth

A man, distinguished by his desire, if possible, to efface himself and yet impelled by the unmistakable power of genius which would not allow the individual of whom it had taken possession to rest for one moment.

With these words Lord Haldane described Einstein after he had stayed at Lord Haldane’s house on his first visit to England in 1921.

Albert Einstein at the age of 3. Source: Wikimedia Commons

Albert Einstein at the age of 3.
Source: Wikimedia Commons

Einstein has become, with no doubt, one of the most well known scientists in history. He was born in March 14, 1879 in Ulm, in the German Empire. In a Jewish family. His parents Hermann Einstein and Pauline Koch. Albert was the first of two sons: he had a sister, Maria, -or Maja, as she was always called- to whom Einstein felt very close. At his early years, Einstein had a great devotion for music, specially Mozart and Beethoven sonatas that he used to play with his mother. At his 12 birthday, he was given a book which he later referred as “the holy geometry book”: it was a book on Euclidean geometry, “the clarity and certainty of its contents made an indescribable impression on me”. It is true that Einstein was slow to speak, but the widespread belief that he was also a bad student is a myth, probably because the first bibliographers that wrote about him didn’t know that in Germany 1 is the maximum grade while 6 is the worst one, and in Switzerland is the opposite way. He actually was one of the best ones. There is a story that Einstein himself would occasionally tell quite amused from when he went at the Gymnasium in Munich. A teacher once said to him that he would be much happier if Einstein was not in his class. Einstein replied to him that he had done nothing wrong and the teacher said “Yes, that is true. But you sit there in the back row and smile, and that violates the feeling of respect which a teacher needs from his class”.

Switzerland: Bern and Zurich

Photography by Albert Ribas

Photography by Albert Ribas

When he was 16, Hermann’s business didn’t go so well and Einstein’s parents moved to Pavia, Italy, while he stayed in Munich to finish his courses. Albert felt alone and depressed in Munich, so he decided to leave before he had finished them and study by his own for passing the exam for the admission at the ETH in Zurich. He did it very well on sciences and mathematics, although he failed the general exam. Then he went to a school in Arau , in the German speaking part of Switzerland. That school made a great impression on him as he wrote shortly before his death:

This school has left an indelible impression on me because of its liberal spirit and the unaected toughness of the teachers, who in no way relied on external
authority.

In that year, 1896, Einstein successfully obtained the Matura, gave up the German citizenship, and finally enrolled at the ETH. During his years in Zurich he liked to go at a Kaeehaus to talk with friends. He spent happy hours with the distinguished historian Alfred Stern and his family, and started a life-long friendship with Michele Angelo Besso, a young engineer whom Einstein called “the best sounding board in Europe” for his scientic ideas. And it was also in this first year in the ETH when Einstein met Mileva Maric, a Serbian classmate -and the only woman in a group of six students- whom Einstein fell in love with, and would later become his wife. In 1900, Einstein passed the exams together with three other students, who immediately found a position as assistant at ETH. Mileva was unable to pass and, although Einstein did pass, he was jobless. After some more tries to find an university position, he worked as a teacher in Winterthur, Schahausen, and finally moved to Berna where he spend the most creative years of his life.

He moved to Berna thanks to Marcel Grossman, a classmate in the ETH which afterwards would develop a principal role in the mathematics behind general relativity, whose father recommended Einstein to Friedrich Haller, the director of the federal patent office in Berna, back in 1900. Finally, Einstin applied for a vacant in the patent office and he was settled there in February 1902. Firstly Einstein worked as provate teacher of mathematics and physics and hence he met Maurice Solovine, a student of philosophy who read the advertisement where Einstein offered private lessons and contacted him because he was tired of the great abstraction of philosophy and he wanted to learn more about physics. Instead of that, they began to meet on a regular basis to discuss their shared interests in physics and philosophy. Soon Konrad Habicht, a good friend of Solovine, joined them. They called themselves the Akademie Olympia, and although sometimes a friend would join them, the Akademy remain basically among this trio.

Source: Wikimedia Commons

Source: Wikimedia Commons

Einstein and Mileva married on January 6, 1903 and hired his residence at Kramgasse 49, second door, in the autumn (they had a secret daughter since 1902). On May 1904, they had a son, Hans Albert Einstein. That same year Einstein got a permanent job at the patent office.

Photography by Carlos Dorce

Photography by Carlos Dorce

In 1905 Einstein widened the horizons of physics in such a short time as no one had done before or since. This period is often referred as the Annus Mirabilis. In March 18 he completed a paper on the Photoelectric effect On a Heuristic Viewpoint Concerning the Production and Transformation of Light which let him to win the Noble prize and in May 11 he finished an article on Brownian motion On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat which gained him the PhD degree from the University of Zurich. apparently this idea. On June 30 Einstein sent his third paper that year On the Electrodynamics of Moving Bodies which was known later as Einstein’s special theory of relativity. Finally on September 27 Einstein sent to Annalen der Physik a fourth paper Does the Inertia of a Body Depend Upon Its Energy Content? in which Einstein developed an argument for probably the most famous equation in Physics: E=mc2. In 1906 Einstein was promoted to “2nd class technical expert” and at the end of 1907 Einstein made the first attempts to apply the laws of gravitation to the Special Theory of Relativity, which would eventually become the General Theory of Relativity. During all these years he did not have easy access to a complete set of scientic reference materials, although he regularly read and contribute reviews to Annalen der Physik. In addition to that, he often met with scientic colleagues such as Michele Besso or the members of the Akademie Olympia, and the most important colleague Einstein had: Mileva, his wife. In his own words:

How lucky I am to have found a creature who is my equal, who is as strong and independent as I am myself.

Also:

I’ve got an extremely lucky idea that will make it possible to apply our theory of molecular forces to gases as well.

Eventually , in 1909 Einstein resigned from his job at the patent office and accepted a position at the Zurich University, where began another phase of his life: his academic career. At the end of his life, Einstein wrote that the greatest thing Marcel Grossman did for him was to recommend him to the patent office.

This post has been written by Pau de Jorge and Eduard Ribas in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Kramgasse 49, Bern (map)

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