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.
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:
- 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.
- 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.
- 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!
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 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!
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)
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.
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 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!
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)
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!
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.:
There also are improvised boards graved on ceramics:
This latrunculus was also found in the ruins (1st-4th c.):
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:
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:
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.
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).
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:
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…
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.
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.
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.