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**

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} = *a*^{2} + *b*^{2} + 2 *a* *b* ⇒ 0 *≤ **a*^{2} + *b*^{2} – 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…

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