Pythagorians: Ideas ahead of their time in Classical Greece

When you think of great philosophers of classical Greece three names typically come to mind; Socrates, Plato, and Aristotle. While it is true that these men were very important in developing Great ideas and had a large part to do with the development of science; one man and his followers are often overlooked or understated in the importance of their idea's. This man being Pythagoras and his followers the Pythagoreans.
While very little is known about pythagoras himself and no real evidence can be given as to credit him directly with any theories or ideas. There are ideas that can be traced back to the Pythagoreans as a whole and through this are often credited to Pythagoras himself even though they may have been ideas derived from one of his many followers.

The Pythagoreans were a religious cult who's philosophy and rituals were developed by Pythagoras himself. This cult also known as the Pythagorean Brotherhood ( it is believed to have allowed woman to participate) believed in the Idea that the universe was not chaos incarnate, as was the common belief of most philosophers of the time, but was actually an ordered universe who's structure could be discovered through the use of mathematics.  This Idea would lead them to many theories that actually still have relevance today.

The most famous of these theories is the Pythagorean theorem which states, "In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)."
Is the equation form of this theorem.

Another theory developed by the Pythagoreans that is less well known is the idea that the earth not only rotated on an axis but also revolved around another celestial body. This was an idea that would be brushed aside for the next 2000 years until Copernicus citing the Pythagoreans as important predecessor to his ideas published  De revolutionibus orbium coelestium or On the Revolutions of the Heavenly Spheres in 1543(which stated that the earth was one of 9 celestial bodies revolving around the sun).

The final important theory I would like to introduce is the Pythagoreans Idea that Music was and sound was measurable by mathematics and  that "musical notes vary in accordance with the length of a vibrating string; whatever length of string a lute player starts with, if it is doubled the note always falls by exactly an octave" (still the basis of the scale in music today).

So while little is know about Pythagoras himself it can be said with certainty that the Pythagoreans as a whole have had an understated impact on both science and the world.