The new advertising campaign in the city of Barcelona is so numerical. The city is divided in ten districts and the first ten numbers are the stars of the new posters which are in all the streets. Enjoy them!
1. The former Barcino which is the focus of the city where the Roman ruins are together with the Mediaeval Gothic architecture, was the origin of the first district: Ciutat Vella (Old Town). Their streets and squares are an open book to the history of Barcelona.
2. Eixample is the Modernist district! At the end of the 19th century, Barcelona was expanded in this rectangular net with their new wide streets which nowadays are the focus of the urban movement of the city.
3. Sants-Montjuïc offers a long walk from the mountain to the sea pointing our attention in the history of the comercial Barcelona.
4. Between Diagonal, full of shops and malls, and the noise of the Camp Nou, Les Corts is a Barcelona full of gardens where you can be lost in this residential town.
5. Old Sarrià-Sant Gervasi is the most residential district of Barcelona. It’s like a quiet town rounded by gardens and museums which ends in Tibidabo mountain.
6. Gracia is the opened town to all the cultures, the urban artists, the music, the theatre ans the cinema although the people whi lives here are proud of it and its past.
7. Horta-Guinardó is the great balcony in the city. It’s full of water and the old houses and “masias” and the gardens are the treasure and the image of this richness.
8. This different and wide district called Nou Barris (Nine Towns) is the focus of a lot of gardens and green zones to walk and enjoy its cultural life which is independent form the rest of the city.
9. sant Andreu is like its neighbours: characters, fight and a deep respect for its traditions. From the Tres Tombs to the Esclat and other parties, tradition is the most important thing in this old town.
10. Beside the sea, ubiquitous chimneys welcome the Sant Martí district recalling its industrial past. An industry that has led to innovation and new technology in the district and has become the engine of the new Barcelona.
This little “Giralda” built in the Catalan town called L’Arboç was projected by Joan Roquer i Marí after a trip in Andalusia. He loved Andalusian architecture and decided to copy this Spanish style in his home. The Giralda was built between 1877 and 1889 designed by Roquer although measures half the height of the real Giralda of Seville. If you visit it you will also find a replica of the Court of the Lions of the Alhambra of Granada.
The decoration of the building was designed from several photographies taken by Roquer in Seville and therefore we have a lot of mosaics which can be admire in this mathematical post:
Location: La Giralda in L’Arboç (map)
On May 2, 2014, I told you that I vidited the MMACA with some of my students and we also noticed this sundial walking from the underground station (Gavarra) to Mercader Palace. We were in a hurry so we could’n stop to analise the shadow of the gnomon but this picture must be the first step for coming back in the not too distant future.
Some weeks ago I had to stop to put gas in my car and Destiny led me to a petrol station next to Sidamon (a small village near Lleida). I took the opportunity to have a drink in the bar and… what was that? There is a big sundial in the roundabaut next to the petrol station!
All the people who lives in Sidamon (less than 700 people!) see this sundial all the days of their lifes. Why don’t they paint it? So it will bright in the middle of this big plain!
One thing more… the coffee in the bar wasn’t nice.
Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.
The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…
For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.
Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!
Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.
Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!
Another example: look at these three wooden pieces…
The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:
There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.
And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?
In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:
Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his Zîj-i Ilkhanî (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his De Revolutionibus (1543). Al-Tûsî’s pair was very famous until the 15th century.
In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:
I didn’t know that it could be described only with a multiplication table! Is its equation z = xy? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.
Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:
Furthermore, one of the rooms of the palace is decoratd by a chess lover!
So… you must go to MMACA and enjoy Mathematics in a way ever done!
Finally, I must talk about these two cuneiform tablets where we can see Mesopotamian figures. There are two ancient Sumerian administrative documents (offerings to a temple and a palace) from the Vorderasiastisches Museum of Berlin. They are dated in c. 2300 BC and we can distinguish the Archaic Mesopotamian numerals in them. In the first one (on the left), we can see some Mesopotamian numerals in the upper left corner:
The Archaic Mesopotamian System of Numeration consisted in the following symbols:
From left to right, the values of the respectively symbols are 1, 10, 60, 600, 3.600 and 36.000. So we can see in the cuneiform tablet the numbers 3 x 600 = 1.800, 1 x 10 = 10 and 5 x 1 = 5. The Archaic numeration wasn’t a positional system so each number was constructed from the iteration of the different ciphers. However, the Sumerians adopted a system in which they could subtract ciphers in spite of adding them. For example, in our tablet, the number 1 x 10 + 5 x 1 = 15 is located inside an angle. This angle represents the operation “minus” so the number which we read in the tablet is 1.800- 15 = 1.785.
In the second tablet we can see the number 11 in the lower left corner.
I have found two more interesting pictures checking my photos of the Caixaforum exhibition. The first one is the plane of a house (c. 2000 BC) from the Vorderasiatisches Museum of Berlin:
The second one is the plane of a sanctuary or a private house (c. 200 BC) from the Musée du Louvre:
From November 30, 2012 to February 24, 2013, we can enjoy this wonderful exhibition in the Caixaforum of Barcelona:
Another 2,500 years would go by before the first dolmens and menhirs were built in Europe, and Egypt was not yet a uni_ed state ruled by the Pharaohs. But, in what is now southern Iraq, a people had built a great city with 40,000 inhabitants. Perhaps the first city in history, this was the capital of a kind of “empire”, with colonies as far-flung as southern Turkey. Its name was Uruk.
The first monumental architecture; the first territorial planning; the first writing in history, perhaps even predating Egypt; the first accounting. All this came about in Uruk in around 3500 BC.
The exhibition showed about 400 pieces from the most important museums of the World and it was possible to contact a lost civilization and all its characteristics:
It seems they spoke Sumerian, a tongue with no links to any other known language, past or present. After the fall of this great state in around 2900 BC, a number of independent city-states sprang up along the southern banks of the Tigris and Euphrates rivers and in the marshlands of the delta. Five hundred years later, these cities were united, firstly, under the Akkadian Empire, whose capital, Akkad, may have occupie what is now Baghdad. Short-lived, Akkad was replaced by a second empire, that of Ur III, whose capital was the city of Ur. Replacing Akkadian, Sumerian once more became the language of this empire.
I enjoyed it a lot. Furthermore, the mathematical objects were also exhibited and I could see different mathematical cuneiform tablets. First of all, I could find Bonaventura Ubach’s suitcase:
Ubach (1879-1960) was a Catalan priest who was an orientalist interested for the Bible. He traveled to the lands of the former Mesopotamia and wrote Dietari d’un viatge per les regions d’Iraq (1922-1923).
The first exhibited cuneiform tablet is about a contract of sale of lands:
One shar of house and orchard/Shamash-nâsir’s house/From Shamash-nâsir/Tâbîya/buys/The full price/A silver shekel and a half/ will be paid/ He won’t claim in the future/ In the name of [¿?]
Different planes of fields were also exhibited:
A round tablet with measurements of the Third Dynasty of Ur (2100-2000 BC) from the Musée royaux d’Art et d’Histoire of Brussels:
A map of an arable bounded land (c. XXth. century BC) from the Musée du Louvre:
Undoubtedly, it was a really wonderful exhibition!
Another of the important equations which stands out in the main entrance of Cosmocaixa in Barcelona is Fermat’s last theorem:
Fermat was reading Diophantus’ Arithmetic about the pythagorean triples x2 + y2 = z2 in 1637 when he noticed immediately that n = 2 was the only case which satisfies the equation xn + yn = zn. Then, he wrote in the margin of his edition of Diophantus’ work:
it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain
Probably, Fermat was only able to prove the new theorem for n = 4 using his proof by infinite descent: Fermat proved that there are not three numbers x, y, z such that x4 + y4 = z2 . First of all, we can suppose that x, y, z are co-prime from (x2)2 + (y2)2 = z2. If <x,y,z> = d, then we can divide all the equation by d2 and we’ll have another equation (x2)2 + (y2)2 = z2 with <x,y,z> = 1.
Now, from the solution of the Pythagorean triples, there are p, q such that x2 = 2pq, y2 = p2 – q2 and z = p2 + q2.
We can observe that y2 = p2 – q2 implies that p2 = q2 – q2 is another Pythagorean triple so there are u, v co-prime such that q = 2uv, y = u2 – v2 and p = u2 + v2.
So, x2 = 2pq = 2(u2 + v2)(2uv) = 4(uv)(u2 + v2). Since <p,q> = 1, we know that <uv,u2 + v2> = 1 and since their product by 4 is a perfect square, they must be perfect squares too. So there exists a < p such that a2= u2 + v2 = p.
Finally, if we had (x2)2 + (y2)2 = z2 , we have obtain p and q co-prime with z = p2 + q2. Then it is possible to obtain another pair u and v co-prime with p = u2 + v2 and p < z, u < p and v < q. So we can iterate this algorithmic procedure so we’ll obtain a set of pairs of natural numbers each of them lower than the previous and this fact is false, since we’d have an infinite decreasing succession of natural numbers without end!
Nobody accepts that Fermat could have known any other case apart of n = 4! In 1753, Leonhard Euler wrote a letter to Christian Goldbach telling him that he had proved the case n = 3 but the demonstration that he published in his Algebra (1770) was wrong. He tried to find integers p, q, z such that (p2 + 3q2) = z3 and he found that:
p = a3 – 9ab2, q = 3(a2b – b3) ⇒ p2 + 3q2 = (a2 + 3b2)3
He worked with numbers of the form a + b√-3 to find two new numbers a and b less than p and q such that p2 + 3q2 = cube and then he applied the method of infinite descent. Unluckily, he made some mistakes working with the new complex numbers a + b√-3.
This story is so exciting but, almost quoting Fermat, this post is too short to contain everything. If you are interested in, you must read Simon Singh’s Fermat last theorem which is as interesting as the theorem itself.
When you enter the Cosmocaixa building in Barcelona (the museum of science), you notice that the main door is surrounded by a very big panels full of Mathematical, Chemical and Physical equations:
The reason of this decoration is:
Not everything imaginable happens in reality. Objects and facts have inviolable restrictions: the laws of nature. Its knowledge helps to fulfill an old dream of living beings: to anticipate the uncertainty. The laws are written with mathematical equations, a relationship between relevant quantities (mass, energy, charge …) and their changes over time and space. An equation is rises its fundamental law as greater is its scope. A great equation of a law is like a poem: a concentrate of intelligibility and beauty.
Curiously, one equation that stands out is:
The demonstration of this formula is very easy. If we suppose a, b > 0, then we can square both terms of the inequality to obtain:
4 a b ≤ (a + b)2 = a2 + b2 + 2 a b ⇒ 0 ≤ a2 + b2 – 2 a b ⇒ 0 ≤ (a – b)2
which is obviously right.
The geometric and arithmetic means between two numbers was known in time of Pythagoras (and also the harmonic mean: a, b and c are in harmonic progression if a(b-c) = c(a-b)). Archytas (c.430 BC-c.360 BC) defined the three means in his On Music . Iamblicus (IIIrd c.) relates that Pythagoras knew them and Archytas and Hippasus adopted the name of harmonic mean instead of subcontrary mean. Heath (A History of Greek Mathematics, 1981) says that:
Nichomachus [of Gerasa] too says that the name ‘harmonic mean’ was adopted in accordance with the view of Philolaus about the ‘geometrical harmony’, a name applied to the cube because it has 12 edges, 8 angles and 6 faces, and 8 is the mean between 12 and 6 according to the theory of harmonics.
The arithmetic mean of 6 and 12 is 9 and their geometrical mean is 8,4852…