Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspiration source to Plato and platonism. One main subject that is part of pythagoreanism is musica universalis, the music of the spheres. Some Surat Shabda Yoga, Satgurus considered the music of the spheres to be a term synonymous with the Shabda or the Audible Life Stream in that tradition, becaus ...

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Pythagoreanism - Influence Pythagoreanism - Pythagorean cosmology Pythagoreanism - Influences Pythagoreanism - Reference

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Pythagoreanism, Pythagoreanism - Influence, Pythagoreanism - Pythagorean cosmology, Pythagoreanism - Influences, Pythagoreanism - Reference

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The word 'vegetarian' was coined in 1847 when the British Vegetarian Society was formed. Before this, vegetarians were known as Pythagoreans. The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans. It was called "health". ...

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Pythagoreanism, Pythagoreanism - Influence, Pythagoreanism - Pythagorean cosmology, Pythagoreanism - Influences, Pythagoreanism - Reference

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Pythagoras (approximately 569 BCE – 475 BCE, Greek: Πυθαγόρας) was an Ionian mathematician and philosopher, founder of the mysterious religious and scientific society called Pythagoreans, known best for the Pythagorean theorem which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence ...

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Pythagoras - Biography Pythagoras - Pythagoreans Pythagoras - Literary works Pythagoras - Scientific contributions

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Apollonius of Tyana (13 March 2 – 98?) was a Neo-Pythagorean philosopher and teacher of Greek origin. His teaching influenced scientific thought for centuries after his death. He is best known through the medium of the writer Philostratus, in whose biography some have seen an attempt to construct a rival to Jesus Christ. Apollonius was a vegetarian, and a disciple of Pythagoras. He is quoted as having said "For I discerned a certain sublimity in the discipline of Pythagoras, and how a certain secret wisdom enabled him to know ...

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Using a The Tetractys, also known as the decad, is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row. As a mystical symbol, it was very important to the followers of the secret worship of the Pythagoreans. * * * * * * * * * * Tetractys - Pythagorean symbol. The Tetractys symbolized the four elements - earth, air, fire, and water. The first four numbers also symbolized the harmony of the sp ...

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Tetractys - Pythagorean symbol Tetractys - Kabbalist symbol Tetractys - Tarot card reading arrangement

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A pentagram is a five-pointed star drawn with five straight strokes. In fact, the word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), a noun form of πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines". The name indicates that a pentagram is not simply a five-pointed star; the symbol must be composed of five lines. That is, it must include the interior pentagon. It is also known as a pentacle, pentalpha (as it ...

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Pentagram - Geometry
Pentagram - Some relevant trigonometric values
Pentagram - History
Pentagram - Pythagorean use
Pentagram - Christian use Pentagram - Satanic use Pentagram - Neopagan Use Pentagram - Brigate Rosse Pentagram - Flags Pentagram - In literature Pentagram - 3D

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A quincunx is the arrangement of five units in the pattern corresponding to the five-spot on dice, playing cards, or dominoes. A quincunx looks like this: The quincunx pattern originates from Pythagorean mathematical mysticism. This pattern lies at the heart of the Pythagorean tetraktys, a pyramid of ten dots. To the Pythagoreans the number five held particular significance and the quincunx pattern represented this. Sir Thomas Browne moulds his mystical discourse The Garden of Cyrus (1658) on ...

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Quincunx - Books

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Archytas (428 BC - 347 BC), was a Greek philosopher, mathematician, astronomer, statesman, strategist and commander-in-chief. Archytas was born in Tarentum, Magna Graecia (now Italy) and was the son of Mnesagoras or Histiaeus. He was taught for a while by Philolaus and he was a teacher of mathematics to Eudoxus of Cnidus. He was scientist of the Pythagorean school, famous as the intimate friend of Plato. His and Eudoxus' student was Menaechmus. Sometimes he is believ ...

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Aristoxenus of Tarentum (4th century BC) was a Greek peripatetic philosopher, and writer on music and rhythm. He was taught first by his father Spintharus, a pupil of Socrates, and later by the Pythagoreans, Lamprus of Erythrae and Xenophilus, from whom he learned the theory of music. Finally he studied under Aristotle at Athens, and was deeply annoyed, it is said, when Theophrastus was ...

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An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m > n, then is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but ...

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Pythagorean triple, Pythagorean triple - Generating Pythagorean triples, Pythagorean triple - Properties of Pythagorean triples, Pythagorean triple - Some relationships, Pythagorean triple - Unit circle relationships, Pythagorean triple - A special case: the Platonic sequence, Pythagorean triple - Generalizations, Pythagorean triple - Other Formulas for Generating Triples

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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the British Isles shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b,  ...

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An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m > n, then is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but ...

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Pythagorean triple, Pythagorean triple - Generating Pythagorean triples, Pythagorean triple - A special case: the Platonic sequence

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The properties of primitive pythagorean triples include: Exactly one of a,b is odd, c is odd. exactly one of a,b is divisible by 3 exactly one of a,b is divisible by 4 exactly one of a,b,c is divisible by 5 exactly one of a,b,(a+b),(a-b) is divisible by 7 at most one of a,b is a square c is an odd number Every integer >2 is part of a pythagorean triple The area (Area=1/2*a*b) is not an integer For any Pythagorean triple, the product of the two nonhypotenuse legs is always ...

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Pythagorean triple, Pythagorean triple - Generating Pythagorean triples, Pythagorean triple - Properties of Pythagorean triples, Pythagorean triple - Some relationships, Pythagorean triple - Unit circle relationships, Pythagorean triple - A special case: the Platonic sequence, Pythagorean triple - Generalizations, Pythagorean triple - Other Formulas for Generating Triples

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This theorem may have a greater variety of known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains over 250 different proofs. James Garfield, who later became a President of the United States, devised an original proof of the Pythagorean theorem in 1876. The external links below provide a sampling of the many different proofs of the Pythagorean theorem. Pythagorean theorem - Geometrical proof. Like many of the proofs of the Pythagorean theorem, this one is based on the proporti ...

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A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 is called a Pythagorean quadruple. A generalization of the concept of Pythagorean triples is the search for triples of positive integers a, b, and c, such that an + bn = cn, for some n strictly greater than 2. Pierre de ...

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Pythagorean triple, Pythagorean triple - Generating Pythagorean triples, Pythagorean triple - Properties of Pythagorean triples, Pythagorean triple - Some relationships, Pythagorean triple - Unit circle relationships, Pythagorean triple - A special case: the Platonic sequence, Pythagorean triple - Generalizations, Pythagorean triple - Other Formulas for Generating Triples

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Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a merchant from Tyre). As a young man, he left his native city for Croton in Southern Italy, to escape the tyrannical government of Polycrates. Many writers credit him with visiting the sages of Egypt and Babylon before going west; such travels feature in the biographies of many Greek sages. Upon his migration from Samos to Croton, Pythagoras established a secret religious society very similar to (and possibly ...

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Pythagoras' followers were commonly called "Pythagoreans." For the most part we remember them as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements. The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed ascetic, dietary and moral rules which they believed would enable their soul to ...

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Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training. In astronomy, the Pythagoreans were well aware of the periodic numerical relations of the planets, moon, and sun. The celestial spheres of the planets were thought to prod ...

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Pythagoras, Pythagoras - Biography, Pythagoras - Pythagoreans, Pythagoras - Literary works, Pythagoras - Scientific contributions

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The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to π / 2; this violates the Euclidean Pythagorean theorem because . This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean t ...

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Pythagorean theorem, Pythagorean theorem - History, Pythagorean theorem - Proofs, Pythagorean theorem - Geometrical proof, Pythagorean theorem - A visual proof, Pythagorean theorem - Converse of the theorem, Pythagorean theorem - Algebraic Proof, Pythagorean theorem - Pythagorean triples, Pythagorean theorem - Generalizations, Pythagorean theorem - The Pythagorean theorem in non-Euclidean geometry, Pythagorean theorem - Other facts, Pythagorean theorem - Notes

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