# Bonanza

### Formulae that changed the world.

*«Geometry holds two great treasures: one is the Pythagorean theorem, the other is the division according to the extreme and the mean ratio [i.e. according to the golden ratio]. The former we can compare to a bushel of gold, the latter we may call a precious jewel.»*

Johannes Kepler (1571–1630)

Right triangle, square, and regular pentagon are the main actors of our Bonanza. For they are connected with mathematics rich in relations – and a long history with serious consequences on top of that. It was not for nothing that the Pythagorean theorem appeared on a stamp from Nicaragua in the 1971 series “The 10 formulae that changed the face of the Earth”. James A. Garfield, who later became the 20th President of the United States, even found an algebraic proof for this famous theorem.

The integers 1, 2, 3,… and the harmonic intervals created from them, such as the octave (1:2), the fifth (2:3) or the fourth (3:4), made the Greek philosophers assume that “the whole world is harmony and numbers”. That is, until a groundbreaking discovery on the natures of the numbers square root of 5 or square root of 2 destroyed this conception of the world. The Ancient Greeks were shaken by this finding, but not put off. They turned away from number theory and went to find the beauty of geometry – its golden properties.