Monthly Archives: January, 2013

Pi is the guest star

Monument dedicated to Pi in Seatle
Source: Google Street View

One day by chance I discovered a Micajah Bienvenu’s sculptural work and I suppose that the picture that I had is due to the will of Destiny… I noticed that Pi was the great guest star of one of his sculptures! Reminding that discovery, I began to search in the net and I found that this Pi is or was installed at Harbor Steps in Seattle, near the corner of University St. and the 1st Avenue. However, I have not been able to discover whether the monument is still there or has been moved to another place because the author’s website says nothing about it.

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An elephant from Faro

Advertisement of a temporary exhibition
Photography by Carlos Dorce

From 2 February to 8 April 2012. there was a teporary exhibition in the Local Museum of Faro which advertisement was this curious elephant. This animal would not be too mathematical if its hat hasn’t got an infinite drawn on it. The advertisement was near the train station of faro and it was one of the last things which I could see in the capital of Algarve.

Fermat’s theorem

Another of the important equations which stands out in the main entrance of Cosmocaixa in Barcelona is Fermat’s last theorem:

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Source: Wikimedia Commons

Source: Wikimedia Commons

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 pq 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 uv co-prime such that q = 2uvy = 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 a2u2 + vp.

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 + vand 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) = zand he found that:

p = a3 – 9ab2q = 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.

Mathematical equations in Cosmocaixa

Cosmocaixa in BarcelonaPhotography by Carlos Dorce

Cosmocaixa in Barcelona
Photography by Carlos Dorce

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

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:

cosmocaixa

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 –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…

Mathematics on the walls of a school

Wall of Jovellanos school in BarcelonaPhotography by Carlos Dorce

Wall of Jovellanos school in Barcelona
Photography by Carlos Dorce

I see this picture every day when I go to work. Perhaps it’s the typical picture that you can see on the walls of a Primary school painted by the students but I think that it deserves a place in this mathematical tourism.

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There is no Zero!

Poster in the Stockmann mallPhotography by Carlos Dorce

Poster in the Stockmann mall
Photography by Carlos Dorce

One of the things which surprised me a lot in my visit to The Scandinavian countries is that they call “1” to the ground floor of a building. That fact is not a problem for the mind but when I visited the popular Stockmann mall in the centre of Helsinki and I wanted to go downstairs… The floor below the floor plant is number -1! Where is 0? There is no zero!

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Mathematical Madrid

Have you seen this adv before?

Madrid advertisement

Madrid advertisement

The translation is “The sum of everybody” and there are two letters M representing Madrid and… have you noticed that the first M is a mathematical summation? This is an example of a very mathematical advertisement. Nevertheless, I am sure that almost all the people who has read the advertisement don’t know its mathematical meaning.

Regular cubes?

Polyhedral monumentPhotography by Carlos Dorce

Polyhedral monument
Photography by Carlos Dorce

There is no doubt that this Eva Löfdahl’s monument is a set of a lot of crowed platonic cubes. I don’t know what this monument is dedicated to, but do you agree that it’s a mathematical attraction? When you are next to the monument you can see that there aren’t any cubes and everything is an optical illusion but… isn’t this perfect combination of diamonds a extraordinary geometric attraction?

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Helsinki: the World Design Capital

Metro station near KampiiPhotography by Carlos Dorce

Metro station near Kampii
Photography by Carlos Dorce

Helsinki is the World Design Capital. When you walk through the streets of its “Design District”, you can see all these mathematical figures as part of a promotional campaign. Curiously, the icosahedra, the ellipses, spheres,… are also in the metro stations!

Mathematics are in the service of advertising and good taste!

Compass and bevel in the Public University of Technology and Design

Public University of Technology and DesignPhotography by Carlos Dorce

Public University of Technology and Design
Photography by Carlos Dorce

The Public University of Technology and Design is in the centre of Saint Petersburg. It’s a very Soviet grey building which is decorated with a compass and a bevel under its windows. I suppose that both instruments represent the design but… they are mathematical instruments too, aren’t they?

Photography by Carlos Dorce

Photography by Carlos Dorce

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