Pythagoras (572-497 BC) himself left no written recored of his work. In fact, little is known of Pythagoras's childhood, although as a child he spent much his early years on the island of Samos. During those years on Samos, Pythagoras' teacher was a man named Pherekydes. It is thought that Pythagoras visited Miletus when he was between 18 and 20 years old where he was introducing to the concepts of mathematical ideas by the philosophers Thales and his pupil Anaximander. Thales, although quite old at the time, contributed to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Pythagoras then traveled to Egypt where he learned geometry, supplementing what he had already been taught from Thales and Anaximander.
After his return from Egypt, Pythagoras formed a school on Samos. Iamblichus writes in the third century AD that: - "Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics..."
After a time, Pythagoras left Samos and went to Croton on Italy's southern coast. It was where Pythagoras founded a philosophical and religious school that attracted a following of likeminded students, known as mathematikoi. The mathematikoi practiced communalism and had no personal possessions and, like Pythagoras himself, were vegetarians. They were taught and obeyed strict rules, one of the strictest was that of secrecy. The Greek philosopher, Proclus, who lived around 450 AD wrote, "Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational numbers and the construction of the cosmic figures." Some have proposed the Greek word should be interpreted to mean proportional or proportion, It may be inferred with regard to Doric temples, at least, the two terms are indistinguishable from each other.
The issue with irrational numbers centered around finding the length of a square’s hypotenuse. Pythagoras learned the length of the hypotenuse cannot be determined mathematically. The problem was, there were no fractional parts small enough to accurately divide the length without a portion remaining. For Pythagoras, he formulated a set of geometric shapes that represent the irrational numbers which he used in defining the footprints of Doric temples with 6 x 11, 6 x 12, 6 x 13, 6 x 14 and 6 x 15 columns. As for the code of secrecy, it is said Hippasus was expelled from the school because he published doctrines of Pythagoras on the use of irrational or incommensurable numbers and was subsequently drowned at sea because of it.