Pythagoras (c. 570 - c. 495 BC) was an ancient Greek philosopher and polymath.
The exact details of Pythagoras's teachings are unclear, Aristotle writes about the teachings of the Pythagoreans, to a great extent in fact. Though, Pythagoras is not mentioned directly.
Metempsychosis
Pythagoras is thought to have either created his notion of metempsychosis or derived it from Egypt. This is the belief that all souls are immortal and that, after death, transfer to a new body. The teaching is actually referenced by Xenophanes and others. Although, nothing is known of the nature or mechanics of Pythagoras's metempsychosis.
There is an allusion in Empedocles, to Pythagoras being able to recall his past incarnations.
Diogenes Laertius also recalls another account of Pythagoras that he had recounted episodes from 4 different past lives. He could remember these episodes, supposedly, in detail.
Mysticism
There is a also this notion of the 'harmony of the spheres.' The planets and the stars move, according to this view, by mathematical equations. These movements correspond to notes that produce a symphony which cannot be heard audibly.
Porphyry says that Pythagoras taught that 7 Muses were the 7 planets singing together.
Pythagoras is also said to have practiced divination and prophecy.
Pythagoras also supposedly had a practice amongst him and his followers called 'smoke trapping.' They would enter a cave and light a fire until noxious smoke filled the room. They would inhale the smoke to induce hallucinations from oxygen deprivation, supposedly divine shapes and patterns. They would copy these shapes and patterns onto the cave walls with charcoal to be analyzed at a later point.
Numerology
Mathematics, according to Aristotle, was used by the Pythagoreans for mystical reasons, not really for practical application. They believed all things were made of numbers. The monad or the number 1, represented the origin of all things. The dyad or the number 2 represented matter. 3 was "ideal" since it had a beginning, middle and end. These are also the smallest number of points necessary to define a triangle. The triangle was a symbol of the god Apollo. 4 signifies the four elements and the four seasons. 7, the number of strings on a lyre, is also the number of known planets at the time. Apollo's birthday was also celebrated on the 7th of each month. 7 was, thus, sacred. Odd numbers were masculine while even numbers were feminine. The sum of 2 and 3 is 5, which meant marriage. 10 is the perfect number for the Pythagoreans. The Pythagoreans would never gather into groups larger than 10.
There is also the tetractys, a triangular figure constructed from 10 points in 4 rows: 1, 2, 3 and 4 points in each row. This construct was important to the secret worship of the Pythagoreans, as they regarded it with the upmost mystical importance.