Tuesday, April 01, 2014

Pythagorean Legends, Pythagorean Triples

The Pythagoreans were a secret society in ancient Greece, a mystical brotherhood dedicated to mathematical study. Their motto was "All is number." Esoteric groups such as the Freemasons and Rosicrucians claim descent from the Pythagorean Brotherhood.

The central assumption of the Pythagorean brotherhood was that the entire universe could be explained by careful application of whole numbers or their ratios (i.e. fractions, or rational numbers). Legend has it that when a member of the cult of Pythagoras proved that even something as simple as the diagonal of a square could not be measured using whole numbers or their ratios, he was summarily executed to avoid insulting prevailing Pythagorean doctrine.

Nevertheless, Pythagorean theories have been applied over the centuries to such diverse problems such as building the great pyramids of Egypt, establishing musical scales, and mapping the orbits of planets.

The good ol' Pythagorean Theorem is one of the most fundamental formulas in all of mathematics. Falsely attributed to Pythagoras of Samos, this famous rule defines the relationship between the lengths of the sides of all right triangles:

a^2 + b^2 = c^2.

Pythagorean Triples are trios of whole numbers that conform to the requirements of the Pythagorean Theorem, and therefore measure the three sides of a right triangle: the two "legs" forming the right angle, and the "hypotenuse" (not "hippopotamus") opposite the right angle.

Here's a fun and simple way to generate Pythagorean Triples from any given natural number (indeed, all Pythagorean Triples can be generated this way):

1. Pick a natural number larger than two. Call this number n.

2. If n is even: let y be 1 and let x be half of n; otherwise (if n is odd) first subtract 1 from n, then let y be half of this new number and let x be one more than y.

3. The following equations will generate Pythagorean Triples for any such x, y:

a = x^2 – y^2

b = 2xy

c = x^2 + y^2

