Sat. Feb 16th, 2019

# From the sacred triangle to the Pythagorean theorem | Science

Long before Pythagoras (or one of his disciples) demonstrated his famous theorem, the Babylonians, the Indians and the Egyptians knew-and used effectively-the properties of the triangle of sides 3, 4 and 5, which was considered sacred. The most remarkable thing about this triangle is that the angle opposite the greater side is straight, and it is not necessary to point out the importance of the right angle in all types of measurements and constructions. In ancient Egypt, the triangle of proportions 3-4-5 most used in architecture and surveying was the one with equal sides at 15, 20 and 25 cubits respectively (about 7.5, 10 and 12.5 meters), called "isiac triangle" in honor to the goddess Isis, which was already used in the construction of the pyramid of Khafre, in the XXVI century a. C. But were the Pythagoreans who, two thousand years later, demonstrated the theorem and gave it its well-known canonical expression:

"In every right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse."

The right angle is a fundamental element of our physical environment, to the point that the great architect Le Corbusier called it "our pact of solidarity with nature" and dedicated an extensive -and intense- poem to him; Here is a fragment:

Erect on the terrestrial plane / of understandable things, / contrasted with nature a pact / solidarity: it is the right angle. / Standing upright before the sea, / there you are on your legs

From the moment they stood on their hind legs, the first hominids had to become fully aware of the horizontal-vertical binomial, or what is the same, of the perpendicularity.

It is probable that the first demonstrations of the Pythagorean theorem were geometric, since the Pythagoreans saw it as a relation of the squares built on the three sides of the right triangle rather than as an algebraic equation. One of the most elegant demonstrations is the one that illustrates the figure (which is also represented in some very ancient Chinese documents).

The square on the center and the one on the right are the same. The one on the right is formed by the square on the side equal to the hypotenuse and four triangles equal to the original; that of the center, by two squares of equal sides, respectively, to both legs and four triangles equal to the original; therefore, the area of ​​the largest square is equal to the sum of the areas of the two smaller ones, or what is the same, atwo + btwo = ctwo.

The proof that we frequently find in current geometry books (attributed to Pythagoras himself) is based on the similarity of triangles.

The triangles ABC and ACH are similar because both are rectangles and have in common the angle A, and therefore their sides are proportional, then b '/ b = b / c, where btwo = cb '. They are also similar ABC and BCH, because they have in common the angle B, then a '/ a = a / c, from where atwo = ca '. Adding both equalities: atwo + btwo = ca '+ cb' = c (a '+ b') = ctwo.

### From Pythagoras to Fermat

The Pythagorean formula totwo + btwo = ctwo invites us to ask ourselves what happens if we generalize it to other exponents and turn it into an + bn = cn, where n is a whole number. Well, in 1637 Pierre de Fermat he concluded that for n greater than 2 there are not three natural numbers (integers and positives) a, b, c such that this equality is met. Fermat wrote in the margin of a book that he had found an "admirable" proof of this theorem; but such a demonstration was never found, and experts assume that Fermat was wrong ... or wanted to play a joke on the mathematical community. In fact, the theorem remained in a state of conjecture for three and a half centuries, until, after numerous attempts, it was demonstrated by Andrew Wiles in 1995.

"I have found a really admirable demonstration, but it does not fit in the meager margin of this book," Fermat wrote. In this, too, time proved him right: Wiles' demonstration would be difficult to fit into the margins of a book, since it occupies about 100 pages.

Master forms is a section by Carlo Frabetti dedicated to explaining the main formulas of mathematics and physics, their origin, evolution and precise meaning.

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.