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…