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)

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)

The “Taula” in Torretrencada

Photography by Laura Barber

Photography by Laura Barber

The name of the Talaiotic culture comes from conic towers built with stones, probably used as a dwelling, watch towers and defense towers. These tables (“Taules” in Catalan) consist in a vertical rectangular stone and another one placed horizontally on its top, so the name of the table is given by the form of “T”. But… why these old monuments are mathematic? The front of most of them is oriented to the south! This orientation is related to the possible use as calendar in this former culture. The construction of the first monuments in Balearic islands dates from the end of the 2nd millenium BC to the beginnings of the 1st millenium BC. At this moment, these monuments began to proliferate on Mallorca and Menorca (there are 31 only in this small Mediterranean island!) appearing in isolated fashion as a territorial boundary stone.

The tables served as sanctuaries next to other monuments and all of them were built in almost the same latitude (and longitude?). For example, Sa naveta des Turons (latitude = 39.99º and longitude = 3.93º), Torretrencada (latitude = 40.003º and longitude = 3.89º) and Torre d’en Gaumès (latitude = 39.93º and longitude = 4.12º) seems to be aligned!

Naveta des Tudons Source: Wikimedia Commons

Naveta des Tudons
Source: Wikimedia Commons

In 1996, Vicente Ibáñez Orts published his hypothesis on the Table explaining that their design was very well computed and not the result of chance. Regarding Torretrencada, it seems that the monument was built from some mathematical computation indicating that Talaiotic men had a system of writing numbers and a deep knowledge of arithmetic and geometry)

This post has been written by Laura Barber and Anabel Luís in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Torretrencada (map)

Kukulkan’s temple in Chichen Itza

Kulkunkan's temple Photography by Roberto Lara

Kukunkan’s temple
Photography by Roberto Lara

Today, the ancient Mayans are particularly famous by their incredible calendar. In fact, Mayans made a really powerful calendar inspired by astronomical events, as they really were essentially farmers and very superstitious. This is the reason why they didn’t have an unique counting system in their calendar, that is, they had ‘sub-calendars’ which different periods as reference. For example, they had a holy calendar (called Tzolkin), which had 260 days, and also a civil solar calendar (called Haab) with 365 days (it’s not clear what was the motivation for the Tzolkin). Tzolkin means “division of days” was probably based on the 224-day Venus sidereal period although there are some hypothesis which defend that it is related with the human gestation period. The Haab calendar consisted in 18 months of 20 days each plus an additional period of five days at the end of the year. It was first used around 500 BC. Mayans were so religious and these astronomical calendars were exposed in their most important buildings like the World-wide famous Temple of Kukulkan (“Feathered serpent”) in the archeological site of Chichen Itza. The temple was founded around 525 AD although the current building was completed between the 9th and the 12th centuries. The pyramid has four sides, each one with 91 steps, which adds up to 364 steps. If we count the last platform as a step we get 365 steps, which is equal to the days we find in the Haab calendar.

Photography by Roberto Lara

Photography by Roberto Lara

But the most famous thing about the Kukulkan’s temple is the descent of Kukulkan: during the autumn and spring equinoxes the late afternoon Sun strikes off the northwest corner of the pyramid and casts a series of triangular shadows against the northwest balustrade, creating the illusion of a feathered serpent ‘crawling’ down the pyramid. We should remark that the balustrade and corners of the pyramid are perfectly aligned, which makes us admire even more the work that Mayans had on the building:

The Feathered Serpent in the Spring Equinox Source: Wikimedia Commons

The Feathered Serpent in the Spring Equinox
Source: Wikimedia Commons

The pyramid also shows us that Mayans had some knowledge about acoustics. If you stand in front of any of the four stairway and clap your hands, the pyramid reflects the sound in such way that you hear the sing of a quetzal, a bird from the jungle. It’s fascinating! Isn’t it? Moreover, the shaman was known as ‘the man with the great voice’, because when people met for a ritual, he didn’t have to speak loudly, as everybody could hear him perfectly.

From all these facts, we can easily conclude that mathematics in the ancient Mayan world wasn’t only a help for agriculture but a tool through which the leaders could control the population. In fact, in the picture below we can see the ruins of a Mayan school. Only those from the upper class had access to the education, and we can see from the building they truly wanted to keep it as a privilege!

Photography by Roberto Lara

Photography by Roberto Lara

The hole at the right of the picture was made by an adventurer who thought gold was hiding inside it and used dynamite to enter the building.

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

Location: Chichen Itzá (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)

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