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Philosophical system based on the teachings of Pythagoras
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans
Pythagoreanism
Greek philosopher (c. 570 – c. 495 BC)
(1455–1522) synthesized Pythagoreanism with Christian theology and Jewish Kabbalah, arguing that Kabbalah and Pythagoreanism were both inspired by Mosaic
Pythagoras
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Topics referred to by the same term
mathematician, philosopher, and music theorist Pythagoras, may refer to: Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by
Pythagorean
Method of tuning a musical instrument
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which
Pythagorean_tuning
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Sports formula
Pythagorean expectation is a sports analytics formula devised by Bill James to estimate the percentage of games a baseball team "should" have won based
Pythagorean_expectation
Set of ancient Greek and Hellenistic religious beliefs
even be considered a single entity, termed "Orphico-Pythagoreanism." The belief that Pythagoreanism was a subset or direct descendant of Orphic religion
Orphism
Length of a line segment
Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient
Euclidean_distance
School of Hellenistic and Roman philosophy
Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic and Ancient Roman philosophy which revived Pythagorean doctrines. Neopythagoreanism
Neopythagoreanism
Small interval between musical notes
Pythagorean comma (531441:524288) on C In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher
Pythagorean_comma
Classical averages studied in ancient Greece
Press. p. 163. ISBN 1139444077. Huffman, Carl (2014). A History of Pythagoreanism. Cambridge University Press. p. 168. ISBN 978-1139915984. Virginia Tech
Pythagorean_means
Field in which every sum of two squares is a square
a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has a Pythagoras number equal to 1. A Pythagorean extension
Pythagorean_field
Cup with a central siphon drain
A Pythagorean cup (also known as a Pythagoras cup, greedy cup, cup of justice, anti greedy goblet or Tantalus cup) is a practical joke device in the form
Pythagorean_cup
Type of spline curve
In mathematics, a Pythagorean hodograph curve or PH curve is a curve defined by a polynomial parametric equation for which the speed (the derivative of
Pythagorean_hodograph_curve
Ancient Greek astronomer, philosopher and mathematician
of Pontus." From the Stanford Encyclopedia of Philosophy, article "Pythagoreanism". Eusebius, Praeparatio evangelica, Book 15, chapter 58, section 3,
Ecphantus_the_Pythagorean
Hypotenuse of right triangle from its sides
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its
Pythagorean_addition
Tiling by squares of two sizes
A Pythagorean tiling or two squares tessellation is a tiling of the Euclidean plane by squares of two different sizes, in which each square touches four
Pythagorean_tiling
Transmigration of the soul
from early Pythagoreanism is a retrojection, possibly of Nietzschean ideas about the opposition of the Apollonian (associated with Pythagoreanism) and the
Metempsychosis
other Pythagoreans, but accepted death when they reached a bean field because Pythagoreanism forbade touching beans. Ocellus Lucanus was a Pythagorean, but
List of pre-Socratic philosophers
List_of_pre-Socratic_philosophers
Musical interval
In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or
Pythagorean_interval
Natural number
underlies the tetractys, a triangular figure of ten points important in Pythagoreanism. A ten sided polygon is called a decagon. Mathematics portal Weisstein
10
Symbol of ten points laid in four rows
As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary
Tetractys
Hypothesized astronomical system
occasionally mixes up Platonism with Pythagoreanism. According to Eudemus, a pupil of Aristotle, the early Pythagoreans were the first to find the order of
Pythagorean astronomical system
Pythagorean_astronomical_system
Country in South Asia
(c. 700–400 BCE) contain the earliest extant verbal expression of the Pythagorean theorem (although very likely it had been known to the Old Babylonians
India
Basic musical interval
functions as a chromatic semitone in a Pythagorean tuning. The Pythagorean limma and Pythagorean apotome are only a Pythagorean comma apart, and may be considered
Semitone
school of the Sextii, which was founded by Quintus Sextius and combined Pythagoreanism with Stoicism. Sotion was the teacher of Seneca the Younger, who "sat
Sotion_(Pythagorean)
Topics referred to by the same term
Pythagorean tree may refer to: Tree of primitive Pythagorean triples Pythagoras tree (fractal) This disambiguation page lists articles associated with
Pythagorean_tree
Prime number congruent to 1 mod 4
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum
Pythagorean_prime
Any of the five regular polyhedra
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are
Platonic_solid
Mystical properties of numbers
graffiti dating back to the 6th century BCE. Aristotle wrote that the Pythagorean tradition, founded in the 6th century by Pythagoras of Samos, practiced
Numerology
Legend about the discovery of musical tuning
ISBN 9780521623711. Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375. ISBN 9780674539181. Cited in Christensen 2002, p.143. Markus
Pythagorean_hammers
the Pythagorean symbol as exceeding the methodological confines of conventional semiotics. Walter Burkert, Lore and Science in Ancient Pythagoreanism, translated
Androcydes_(Pythagorean)
Four integers where the sum of the squares of three equals the square of the fourth
A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only
Pythagorean_quadruple
Enchantress-goddess in Greek mythology
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Circe
Ancient philosophical concept
of music. The theory, originating in ancient Greece, was a tenet of Pythagoreanism, and was later developed by 16th-century astronomer Johannes Kepler
Musica_universalis
Musical interval encompassing three half steps
a semiditone (or Pythagorean minor third) is the interval 32:27 (approximately 294.13 cents). It is the minor third in Pythagorean tuning. It arises
Minor_third
Ancient Greek philosopher (c. 428-347 BC)
to southern Italy to study with Archytas and other Pythagoreans. The influence of these Pythagoreans appears to have been significant. According to R.
Plato
5th-century BC Greek philosopher
Stobaeus, Phys. Ecl. i. Reviel Netz,The problem of Pythagorean mathematics in A History of Pythagoreanism, ed C. Huffman, Cambridge UP, 2014: " Pseudo-Alexander
Eurytus_(Pythagorean)
Study of general and fundamental questions
Platonism Academy Middle Neo Peripatetic school Pre-Socratic Ionians Pythagoreans Eleatics Atomists Sophists School of the Sextii Second Sophistic Hindu
Philosophy
Most beautiful woman in Greek mythology
Greek word Σελήνη (Selēnē), meaning "Moon, goddess of the moon". One Pythagorean source claimed that Helen had originally come from a colony on the moon
Helen_of_Troy
Concept of rebirth in different physical form
systematized Socrates' thought with concepts he took directly from Pythagoreanism or Orphism. Recent scholars have come to see that Plato has multiple
Reincarnation
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
Netz, Reviel (2014), "The problem of Pythagorean mathematics", in Huffman, Carl A. (ed.), A History of Pythagoreanism, Cambridge University Press, pp. 167–184
Ancient_Greek_mathematics
Ancient Greek pythagorean philosopher
Philosophy. London: Search Press, 1946. Guthrie, W.K.C. "Pythagoras and Pythagoreanism," in Encyclopedia of Philosophy. Vol. 7. Edited by Paul Edwards. NY:
Damo_(philosopher)
Dialogue by Plato on the immortality of the soul
impiety. Many of the key characters in the dialogue are associated with Pythagoreanism, a religious and philosophical doctrine that flourished early in the
Phaedo
Ancient Greek god
hymns to Apollo, the swans would join the chant in unison. Among the Pythagoreans, the study of mathematics and music were connected to the worship of
Apollo
Mathematical proof by James Garfield
Garfield's proof of the Pythagorean theorem is an original proof of the Pythagorean theorem discovered by James A. Garfield, the 20th president of the
Garfield's proof of the Pythagorean theorem
Garfield's_proof_of_the_Pythagorean_theorem
Ancient Greek deity and herald of the gods
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Hermes
Relation between the side lengths and altitude of a right triangle
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows: Let
Inverse_Pythagorean_theorem
Ancient Greek goddess
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Artemis
Ancient Greek goddess
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Athena
Mathematical tree of integer right triangles
primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean triple is
Tree of primitive Pythagorean triples
Tree_of_primitive_Pythagorean_triples
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Lists of Greek mythological figures
Lists_of_Greek_mythological_figures
President of the United States in 1881
Garfield's aptitude for mathematics extended to his own proof of the Pythagorean theorem, published in 1876, and his advocacy of using statistics to inform
James_A._Garfield
Body of myths originating in ancient Greece
Peripteros Philosophy Cynicism Epicureanism Neoplatonism Peripateticism Pythagoreanism Stoicism Events Founding of Rome Natale di Roma Rape of the Sabine Women
Greek_mythology
Natural number
often representing God or the universe in monotheistic traditions. The Pythagoreans considered the numbers to be plural and therefore did not classify 1
1
Ancient Greek god of the sea, earthquakes, and horses
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Poseidon
Greek philosopher (c. 470 – c. 385 BC)
mixes up Platonism with Pythagoreanism. Alcmaeon of Croton Apeiron Nicomachus Parmenides Protrepticus (Aristotle) Pythagoreans "The most likely date for
Philolaus
Polygonal curve made from right triangles
geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles
Spiral_of_Theodorus
theorist. Pythagoreanism – the system of philosophy of Pythagoras and his followers Neopythagoreanism – a later philosophical system Pythagorean cup – a
List of things named after Pythagoras
List_of_things_named_after_Pythagoras
Besides Euclid's formula, many other formulas for generating Pythagorean triples have been developed. Euclid's, Pythagoras' and Plato's formulas for calculating
Formulas for generating Pythagorean triples
Formulas_for_generating_Pythagorean_triples
Measurement of a string instrument
The scale length of a string instrument is the maximum vibrating length of the strings that produce sound, and determines the range of tones that string
Scale length (string instruments)
Scale_length_(string_instruments)
2nd-century collection of maxims
is Quintus Sextius, a Roman philosopher who combined Stoicism with Pythagoreanism, and who lived in the 1st century BC. The work is similar to the sayings
Sentences_of_Sextus
Philosophical concept of a most basic substance, or supreme being
to a most basic or original substance. As originally conceived by the Pythagoreans, the Monad is therefore Supreme Being, divinity, or the totality of all
Monad_(philosophy)
Divine hero in Greek mythology
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Heracles
Greek god and personification of the Sun
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Helios
Greek goddess of spring and the queen of the underworld
Peter (1995). Ancient Philosophy, Mystery, and Magic: Empedocles and Pythagorean Tradition. Oxford University Press. Kourouniōtēs, Kōnstantinos; Mylonas
Persephone
German-born theoretical physicist (1879–1955)
made such rapid progress that he discovered an original proof of the Pythagorean theorem before his thirteenth birthday. A family tutor, Max Talmud, said
Albert_Einstein
Triangle containing a 90-degree angle
integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. The relations between the
Right_triangle
Musical scale set of twelve pitches
decimal): where 256⁄243 is a diatonic semitone (Pythagorean limma) and 2187⁄2048 is a chromatic semitone (Pythagorean apotome). In 5-limit just intonation the
Chromatic_scale
Twentieth letter in the Greek alphabet
writer Persius wrote in Satire III: and the letter which spreads out into Pythagorean branches has pointed out to you the steep path which rises on the right
Upsilon
Right triangle with a feature making calculations on the triangle easier
know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles". The following are all the Pythagorean triple
Special_right_triangle
Result in Fourier analysis
sum of squares of the amplitudes). Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity
Parseval's_identity
Unique positive real number which when multiplied by itself gives 2
across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The
Square_root_of_2
Educational methodology
Pythagorean Method of Memorization (PYMOM), also known as Triangular Movement Cycle (TMC), is a game-based, educational methodology or associative-learning
Pythagorean Method of Memorization
Pythagorean_Method_of_Memorization
Set of integers, the lengths of the sides of a triangle with a 60° angle
Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle
Eisenstein_triple
Greek mythological figure in the Iliad
Project Gutenberg Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism. Translated by Minar, Edwin L., Jr. Harvard University Press. pp. 138–141
Euphorbus
Relation between sine and cosine
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
Third planet from the Sun
3402/tellusa.v26i1-2.9731. Kahn, Charles H. (2001). Pythagoras and the Pythagoreans: A Brief History. Indianapolis, IN and Cambridge, England: Hackett Publishing
Earth
5th-century BC Pythagorean philosopher
50–51. Springer. Walter Burkert (1972), Lore and Science in Ancient Pythagoreanism, page 459. Harvard University Press. Plato, Theaetetus, 147d ff. Thomas
Hippasus
6th-century BCE Greek mythographer and proto-philosopher
the Ancient Greek religion, which influenced the metempsychosis of Pythagoreanism, and the theogonies of Orphism. Various legends and miracles were ascribed
Pherecydes_of_Syros
Dios was an ancient Greek philosopher of uncertain date. Certain Pythagorean writings on aesthetics are ascribed to him. Two fragments found in Stobaeus
Dios_(philosopher)
Group of Vedic Sanskrit texts
including an approximation of the square root of 2 and the statement of the Pythagorean theorem. Baudhayana's Śrauta sūtras related to performing Vedic sacrifices
Baudhayana_sutras
Ethnoreligious group of the Levant
Christianity, Gnosticism, Neoplatonism, Zoroastrianism, Manichaeism, and Pythagoreanism.[page needed] This has led to the development of a distinct and secretive
Druze
Texts belonging to the Śrauta ritual
statements of the Pythagorean theorem, both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples. In
Shulba_Sutras
Mathematical treatise by Euclid
elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common
Euclid's_Elements
Lifestyle of frugality and abstinence
Greek philosophical schools (Epicureanism, Cynicism, Stoicism, and Pythagoreanism), Indian religions (Buddhism, Hinduism, Jainism), Abrahamic religions
Asceticism
Dissonant musical interval
A common example is the abnormally small fifth that results from the Pythagorean comma. Particularly out of tune intervals produce an unpleasant effect
Wolf_interval
Hypothetical planet on the other side of the Sun from Earth
"Central Fire" was developed in the 5th century BC and attributed to the Pythagorean philosopher Philolaus. Philolaus' universe moved "the earth from the
Counter-Earth
Field of knowledge
and Latin. Traditionally, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant
Mathematics
Nereid of Greek mythology
Games Nemean Games Pythian Games Isthmian Games Philosophy Schools Pythagoreanism Neopythagoreanism Platonism Middle Platonism Neoplatonism Philosophers
Thetis
Roman philosopher and writer
their esoteric Pythagoreanism, into which Nigidius incorporated Stoic elements. Jerome called him Pythagoricus et magus, a "Pythagorean and mage," and
Nigidius_Figulus
Roman-era word square with a Latin palindrome
Several German academics have written on the links of the square to Pythagoreanism and Stoicism, including philologist Hildebrecht Hommel [de], historian
Sator_Square
1st-century-BC Roman poet
Peripteros Philosophy Cynicism Epicureanism Neoplatonism Peripateticism Pythagoreanism Stoicism Events Founding of Rome Natale di Roma Rape of the Sabine Women
Virgil
resurgence of interest in Pythagoreanism about the first century BCE. Neo-Pythagoreans such as Numenius soon began claiming that Pythagorean doctrines were symbolically
Allegorical interpretations of Plato
Allegorical_interpretations_of_Plato
Sets whose elements have degrees of membership
_{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1} , which is reminiscent of the Pythagorean theorem. Pythagorean fuzzy sets can be applicable to real life applications in which
Fuzzy_set
Natural number
the union of the physical (number 4) with the spiritual (number 3). In Pythagorean numerology the number 7 means spirituality. The number seven had mystical
7
Numerical symbol
Gerbert of Aurillac marked triples of columns with an arc (called a "Pythagorean arc"), when using his Hindu–Arabic numeral-based abacus in the 10th century
Decimal_separator
Natural number
Greek goddess Hecate and the Triple Goddess of Wicca. Pythagoras and the Pythagorean school highlighted that the number 3, which they called triad, is the
3
Natural number
seconds and 8 minutes past 8 p.m. (local time) on 8 August 2008. In Pythagorean numerology the number 8 represents victory, prosperity and overcoming
8
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