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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
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
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
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
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
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
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
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
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
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
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
Basic musical interval
and functions as a chromatic semitone. The Pythagorean limma and Pythagorean apotome are only a Pythagorean comma apart, and may be considered enharmonic
Semitone
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
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
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
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
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
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
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
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
Five-pointed star polygon
a depiction on a vase possibly dating back to the 7th century BCE. Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol
Pentagram
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)
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
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
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
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
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
the Pythagorean symbol as exceeding the methodological confines of conventional semiotics. Walter Burkert, Lore and Science in Ancient Pythagoreanism, translated
Androcydes_(Pythagorean)
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
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
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
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
Musical interval
prefer 45/32 to a small-number interval of about the same width. In the Pythagorean ratio 81/64 both numbers are multiples of 3 or under, yet because of
Tritone
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
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
American philosopher (born 1951)
known for his works on ancient Greek philosophy. Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician King, Cambridge University Press, Cambridge
Carl_A._Huffman
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
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
Relationship among tones of the chromatic scale
pitch class, but instead overshoots it by a small interval known as the Pythagorean comma. This leads to tuning issues when transposing or modulating, and
Circle_of_fifths
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
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, the Supreme Being, divinity, or the totality
Monad_(philosophy)
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
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
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
Longest side of a right-angled triangle, the side opposite of the right angle
legs or catheti. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse
Hypotenuse
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
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
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
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
Musical interval
41-ET, 53-ET, and 72-ET. Conversely, in twelve-tone equal temperament, Pythagorean tuning, and meantone temperament (including 19-ET and 31-ET) all major
Major_second
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
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
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
Mathematical papyrus
problems on the Papyrus may suggest that the ancient Egyptians knew the Pythagorean theorem. The Berlin Papyrus 6619 is an ancient Egyptian papyrus document
Berlin_Papyrus_6619
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
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
Philosophical origins and foundation of Western civilization
rationalism into Pythagoreanism, that he was simply a rationalist whose successors are responsible for the mysticism in Pythagoreanism, or that he was
Ancient_Greek_philosophy
Generalization of Pythagorean theorem
\beta .\end{aligned}}} The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if γ {\displaystyle
Law_of_cosines
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)
Concept of perpetual recurrence of time
that "nothing was entirely new". Eudemus of Rhodes also references this Pythagorean doctrine in his commentary on Aristotle's Physics. In a fragment preserved
Eternal_return
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
N-th root of the arithmetic mean of the given numbers raised to the power n
for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). If p is a non-zero
Generalized_mean
Musical interval
third, and is instead the same width as the septimal major third. The Pythagorean diminished fourth (F♭, 8192:6561 = 384.36 cents), also known as the schismatic
Diminished_fourth
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
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
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
Chess problems Leaper that makes moves 5 units in length. Due to the Pythagorean theorem, it has twelve possible directions. Also named Root-25-Leaper
List_of_fairy_chess_pieces
Relation between the sides of a convex quadrilateral and its diagonals
can be seen as generalisation of the Pythagorean theorem. Because of the latter the restatement of the Pythagorean theorem in terms of quadrilaterals is
Euler's_quadrilateral_theorem
Natural number
sequence. 265 is the 7th number to be the hypotenuse for two separate Pythagorean Triples. The other two values would be 23 and 264 or 96 and 247. 265
265_(number)
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
Greek Neopythagorean philosopher (c.15–100)
2nd century the satirist Lucian of Samosata was a sharp critic of Neo-Pythagoreanism. After AD 180 he wrote a pamphlet wherein he attacked Alexander of Abonoteichus
Apollonius_of_Tyana
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)
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
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
American football website
in Pythagorean wins, while only seven were won by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins
Football_Outsiders
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
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
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
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
Shamanic practice
soul ascends as high as from which it came (cf. Metempsychosis). The Pythagoreans adopted many of the Orphic teachings into their own philosophical tradition
Soul_flight
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
1st century BC Greco-Egyptian philosopher
Platonism. He attempted to reconstruct Plato's philosophy in terms of Pythagoreanism. Little is known about Eudorus' life. Chronologically, he lived in the
Eudorus_of_Alexandria
Early Christian and Jewish religious systems
called æons. According to Hippolytus, this view was inspired by the Pythagoreans, who called the first thing that came into existence the Monad, which
Gnosticism
Ancient Greek philosopher (c. 470–399 BC)
Anaximander Anaximenes Heraclitean Heraclitus Cratylus Diogenes Italian Pythagorean Pythagoras Hippasus Philolaus Archytas Lamiskos Alcmaeon Brontinus Theano
Socrates
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
Brazilian philosopher
Philosophical work Era 20th-century philosophy Region Western philosophy School Pythagoreanism Neoplatonism Thomism Scotism Mutualism Main interests Dialectic Logic
Mário_Ferreira_dos_Santos
Head of the Catholic Church from 257 to 258
him with the contemporary author Xystus, who was a Greek student of Pythagoreanism. Sixtus II restored the relations with the African and Eastern churches
Pope_Sixtus_II
Historical region of West Asia
Plimpton 322 tablet, created around 1900–1600 BC, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to
Mesopotamia
2024 documentary about the religion and animal rights
(Seventh-day Adventist Church) Hinduism Sattvic Ahimsa Islam Jainism Judaism Pythagoreanism Rastafari Sikhism Taoism Food and drink List of vegetarian and vegan
Christspiracy
War goddess, wife of Mars in Roman mythology
Peripteros Philosophy Cynicism Epicureanism Neoplatonism Peripateticism Pythagoreanism Stoicism Events Founding of Rome Natale di Roma Rape of the Sabine Women
Nerio
Idea for signaling extraterrestrial beings from Earth
Gauss's Pythagorean right triangle proposal is an idea attributed to Carl Friedrich Gauss for a method to signal extraterrestrial beings by constructing
Gauss's Pythagorean right triangle proposal
Gauss's_Pythagorean_right_triangle_proposal
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