File Pythagore chartres.jpg thumb Sculpture of Pythagoras at Chartres , France Pythagorean means of or pertaining to the ancient Ionia n mathematician, philosopher, and music theorist Pythagoras . See Philosophy Pythagoreanism is a term used for the esoteric and metaphysical beliefs purported to have been held by Pythagoras. Neo Pythagoreanism was a school of philosophy reviving Pythagorean doctrines that became prominent in the 1st and 2nd centuries AD. Pythagoreanism Vegetarianism Pythagorean diet was the name for vegetarianism prior to the nineteenth century Mathematics Pythagorean theorem Pythagorean triple Pythagorean prime Pythagorean trigonometric identity Table of Pythagoras, another name for the Multiplication table Music Pythagorean comma Pythagorean hammers Pythagorean tuning Drinkware Pythagorean cup Graphics Upsilon Symbolism Pythagorean letter Sports statistics Pythagorean expectation disambig ... more details
Pythagorean tree may refer to Tree of primitive Pythagorean triples Pythagoras tree fractal disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
Lead rewrite date November 2010 Image Perfect fifth on C.png thumb right Pythagorean perfect fifth on C audio Just perfect fifth on C.mid Play C G 3 2 1 1 3 2 . In musical tuning theory, a Pythagorean ... Explorations , p.56. ISBN 9780195144369. The frequency ratio of every Pythagorean interval is a ratio between a power of two and a power of three...confirming the Pythagorean requirements that all ... to 2 sup 2 sup 3 sup 1 sup are Pythagorean intervals. All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagorean tuning system. However, some Pythagorean intervals are also used in other tuning systems. For instance, the above mentioned Pythagorean perfect ... center 23.460 align center 0 Audio Pythagorean comma on C.mid play 12 perfect unison P1 align center 1 1 align center align center 1 1 align center 0.000 align center 0 Audio Unison on C.mid play 0 Pythagorean ... 0 Audio Pythagorean comma on C.mid play 12 minor second m2 Pythagorean limma limma , br diatonic semitone ... center 90.225 align center 100 Audio Pythagorean minor semitone on C.mid play 5 augmented unison A1 Pythagorean apotome apotome , br chromatic semitone, br major semitone align center 2187 2048 align center 3 sup 7 sup 2 sup 11 sup align center align center 113.685 align center 100 Audio Pythagorean ... 2 semiditone m3 Pythagorean minor third align center 32 27 align center 2 sup 5 sup 3 sup 3 sup align center align center 294.135 align center 300 Audio Pythagorean minor third on C.mid play 3 augmented ... 317.595 align center 300 Audio Pythagorean augmented second on C.mid play 9 diminished fourth d4 ... center 400 Audio Pythagorean diminished fourth on C.mid play 8 ditone M3 Pythagorean major third ... align center 400 Audio Pythagorean major third on C.mid play 4 perfect fourth P4 diatessaron, br sesquitertium ... center 3 sup 11 sup 2 sup 17 sup align center align center 521.505 align center 500 Audio Pythagorean ... 2 sup 9 sup align center align center 611.730 align center 600 Audio Pythagorean augmented fourth on C.mid ... more details
In algebra, a Pythagorean field is a field mathematics field in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element mrad 1 &lambda sup 2 sup for some in F . So a Pythagorean field is one closure mathematics closed under taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism , called its Pythagorean closure . Pythagorean fields can be used to construct models for some of Hilbert s axioms for geometry harv Ito 1980 loc 163 C . The analytic geometry given by F sup n sup for F a Pythagorean field satisfies many of Hilbert s axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert s axioms unless the field F has extra properties for example, if the field is also ordered then the geometry will satisfy Hilbert s ordering axioms, and if the field is also complete the geometry will satisfy Hilbert s completeness axiom. The Pythagorean closure of a non archimedean ordered field , such as the Pythagorean closure of the field of rational function s Q t in one variable over the rational numbers Q , can be used to construct non archimedean geometries that satisfy many of Hilbert s axioms but not his axiom of completeness harv Ito 1980 loc 163 D . Dehn used such a field to construct a non Legendrian geometry and a semi Euclidean geometry in which there are many lines though a point not intersecting a given line. The Witt group Witt ring of a Pythagorean field is of order 2 if the field is not Formally real field formally real , and torsion free otherwise. See also Euclidean field References Citation last1 Elman first1 Richard last2 Lam first2 T. Y. title Quadratic forms over formally real fields and pythagorean fields jstor 2373568 mr ... year 2005 volume 67 chapter Chapter VIII section 4 Pythagorean fields pages 255 264 citation first1 ... more details
Refimprove date November 2007 A Pythagorean prime is prime number of the form 4 n 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle . The first few Pythagorean primes are 5 number 5 , 13 number 13 , 17 number 17 , 29 number 29 , 37 number 37 , 41 number 41 , 53 number 53 , 61 number 61 , 73 number 73 , 89 number 89 , 97 number 97 , 101 number 101 , 109 number 109 , 113 number 113 , OEIS id A002144 . Fermat s theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely up to order , and that no other primes can be represented this way, aside from 2 1 sup 2 sup 1 sup 2 sup . Thus these primes and 2 occur as norms of Gaussian integers , while other primes do not. The law of quadratic reciprocity says that if p and q are distinct odd primes, at least one of which is Pythagorean, then p is a quadratic residue mod q if and only if q is a quadratic residue mod p by contrast, if neither p nor q is Pythagorean, then p is a quadratic residue mod q if and only if q is not a quadratic residue mod p . &minus 1 is a quadratic residue mod p if and only if p is a Pythagorean prime or 2 . In the field Z p with p a Pythagorean prime, the polynomial x 2 1 has two solutions. Category Classes of prime numbers Prime number classes state collapsed numtheory stub da Pythagor iske primtal es N mero primo pitag rico sl Pitagorejsko pra tevilo zh ... more details
Image Pythagorean comma difference A1 m2 .PNG thumb right 400px Pythagorean comma PC defined in Pythagorean ... direction from to C Music to D Music b . Image Pythagorean comma on C.png thumb right Pythagorean comma on C. audio Pythagorean comma on C.mid Play . The note depicted as lower on the staff B ... tuning , the Pythagorean comma or ditonic comma ref not to be confused with the syntonic comma ... in Pythagorean tuning between two enharmonic enharmonically equivalent notes such as C and B Music audio Pythagorean comma on C.mid Play , or D Music b and C Music . ref Apel, Willi 1969 . Harvard Dictionary of Music , p.188. ISBN 978 0 674 37501 7. ...the difference between the two semitones of the Pythagorean ... temperament s often refer to tempering is the Pythagorean comma. ref Coyne, Richard 2010 . The Tuning of Place Sociable Spaces and Pervasive Digital Media , p.45. ISBN 978 0 262 01391 8. ref The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean .... ref i.e., between a chromatic and a diatonic semitone , as determined in Pythagorean tuning , or the difference ... between three Pythagorean ditone s and one octave this is the reason why the Pythagorean comma is also called ditonic comma . The diminished second , in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides therefore with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma e.g. from C Music to D Music b , equal to about &minus 23.46 cents. Derivation As described in the introduction, the Pythagorean comma may be derived in multiple ways Difference between two enharmonic enharmonically equivalent notes in a Pythagorean scale, such as C and B Music audio Pythagorean comma on C.mid Play , or D Music b and C Music see Pythagorean comma Circle of fifths and enharmonic change below . Difference between Pythagorean apotome and Pythagorean limma . Difference between twelve just perfect fifth s and seven Perfect octave octaves ... more details
A Pythagoras Pythagorean quadruple is a tuple of integer s a , b , c and d , such that d 0 and math a 2 b 2 c 2 d 2 math , and is often denoted math a,b,c,d math . Geometrically, a Pythagorean quadruple math a,b,c,d math defines a cuboid with side Norm mathematics Euclidean norm lengths a , b , and c , whose space diagonal has integer length d . Pythagorean quadruples are thus also called Pythagorean Boxes . ref R.A. Beauregard and E. R. Suryanarayan, Pythagorean boxes , Math. Magazine 74 2001 , 222 227. ref Parametrization of primitive quadruples The Set mathematics set of all primitive Pythagorean quadruples, i.e., those for which gcd a , b , c 1, where gcd denotes the greatest common divisor of a , b , and c , is parametrized by, ref R.D. Carmichael, Diophantine Analysis , New York John Wiley & Sons, 1915. ref ref L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics , in Villat Henri , ed., Conf rence g n rale, Comptes rendus du Congr s international des math maticiens, Strasbourg, Toulouse, 1921, pp. 41 56 reprint Nendeln Liechtenstein Kraus Reprint Limited, 1967 Collected Works 2, pp. 579 594. ref ref R. Spira, The diophantine equation math x 2 y 2 z 2 m 2 math , Amer. Math. Monthly 69 1962 , 360 365. ref math a m 2 n 2 p 2 q 2, , math ... negative integer s and gcd m , n , p , q 1 and m n p q 1 mod 2 . Thus, all primitive Pythagorean ... Pythagorean quadruple math a,b,c,d math Parametrization parametrized by math m,n,p,q math corresponds ..., Letter to the Editor , Amer. Math. Monthly 94 1987 , 757 758. ref Pythagorean quadruples with small ... Alternate parametrization All Pythagorean quadruples including non primitives, and with repetition ... a and b as follows ref Wac aw Sierpi ski Sierpi ski, Wac aw , Pythagorean Triangles , Dover, 2003 ... be an even factor of math a 2 b 2 math . No such method exists if both a and b are odd. See also Pythagorean ... title Pythagorean Quadruple MathWorld urlname LebesgueIdentity title Lebesgue s Identity ... more details
original research date January 2009 Image Pythagorean.svg thumb The Pythagorean theorem nowrap a sup 2 sup b sup 2 sup c sup 2 sup File Theorem.gif thumb Animation demonstrating the simplest case of the Pythagorean Triple 3 sup 2 sup 4 sup 2 sup 5 sup 2 sup . A Pythagorean triple ... 3, 4, 5 . If nowrap a , b , c is a Pythagorean triple, then so is ka , kb , kc for any positive integer k . A primitive Pythagorean triple PPT is one in which a , b and c are pairwise coprime . A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle . The name is derived from the Pythagorean theorem , stating that every right triangle has side lengths satisfying the formula nowrap a sup 2 sup b sup 2 sup c sup 2 sup thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non integer sides do not form Pythagorean ..., but 1, 1, 2 is not a Pythagorean triple because 2 is not an integer. Moreover, 1 and 2 do not have ... of a primitive Pythagorean triple, one in which the terms are relatively prime. Image Pythagorean triple scatterplot2.png thumb A scatter plot of the legs a , b of the Pythagorean triples with c ... clearly. There are 16 primitive Pythagorean triples with nowrap c 100 style margin auto cellspacing ... plot. Additionally these are all the primitive Pythagorean triples with nowrap 100 c 300 style ... Pythagorean triples Image PrimitivePythagoreanTriplesRev08.svg thumb 300px alt Primitive Pythagorean triples shown as triangles on a graph The primitive Pythagorean triples. The odd leg a is plotted ... of constant m n and of constant m n in Euclid s formula. Image Pythagorean Triples from Grapher.png ... Pythagorean triples given an arbitrary pair of positive integers m and n with nowrap m n . The formula states that the integers math a m 2 n 2 , , b 2mn , , c m 2 n 2 math form a Pythagorean ... and n are coprime. ref Mitchell, Douglas W., An alternative characterisation of all primitive Pythagorean ... more details
Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team ... s actual and Pythagorean winning percentage can be used to evaluate how lucky that team was by examining ... to the Pythagorean theorem . The basic formula is math mathrm Win frac text runs scored 2 text ... pyth Baseball reference.com What is pythagorean winning percentage? Bot generated title ref The updated ... Prospectus Articles Revisiting the Pythagorean Theorem Bot generated title ref developed by Clay ... Prospectus Articles Revisiting the Pythagorean Theorem Bot generated title ref Less well known but equally ... scored and allowed into the pythagorean formula, one can generate second order wins, the number of wins .... In 2003, Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean ... of James Pythagorean Formula Long url http groups.google.com group rec.puzzles browse thread thread ... Magazine 20 2007 , no. 1, 40 48 pages 9698 title A Derivation of the Pythagorean Won Loss Formula ... journal first Steven J last Miller title A Derivation of the Pythagorean Won Loss Formula in Baseball .... By comparing the Giants expected wins as calculated by the Pythagorean theory with an exponent of 1.83 ... to adapt James Pythagorean expectation to professional basketball while a researcher at STATS, Inc ... . math Daryl s Modified Pythagorean Theorem was first published in http morey.org pythbook.gif STATS ... James Pythagorean theory to professional basketball. The result was similar. Another noted basketball statistician, John Hollinger , uses a similar Pythagorean formula except with 16.5 as the exponent ... stat website and publisher Football Outsiders , where it is known as Pythagorean projection . The 2011 ... Football League NFL in Pythagorean wins, while only seven were one by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins ... participants from 2005 2008, it reasserted itself in 2009 and 2010. Furthermore, t he Pythagorean projection ... more details
Ecphantus or Ecphantos lang grc is a shadowy Greek pre Socratic philosopher. He may not have actually existed. ref Some scholars have argued that Hicetas and Ecphantus, both of Syracuse, were not historical figures at all but rather characters in dialogues written by Heraclides of Pontus . From the Stanford Encyclopedia of Philosophy , article http plato.stanford.edu entries pythagoreanism Pythagoreanism . ref He is identified as a Pythagoreanism Pythagorean of the 4th century BCE, and as a supporter of the heliocentric theory . Described as from Syracuse, Sicily Syracuse , this may or may not be the same figure as the attested Ecphantus of Croton . Notes references External links fr icon http www.cosmovisions.com Ecphante.htm DEFAULTSORT Ecphantus The Pythagorean Category 4th century BC Greek people Category 4th century BC philosophers Category Ancient Syracusians Category Ancient Greek philosophers Category Pythagoreans of Magna Graecia Category Ancient Greek astronomers Philosopher stub de Ekphantos el it Ecfanto di Siracusa pt Ecfanto, o Pitag rico ru sk Ekfantos zo Syrak z sl Ekfant ... more details
File Distorted truncated square tiling.png thumb A Pythagorean tiling File Jacob Ochtervelt Street Musicians ... Ochtervelt , 1665. As observed by Nelsen ref name rbn the floor tiles in this painting are set in the Pythagorean tiling In geometry , the Pythagorean tiling or two squares tessellation is a tessellation ... Tile Inc. date August 2008 . ref Topology and symmetry The Pythagorean tiling is the unique tiling ... b39h2mar.ps.gz volume 39 year 1998 . ref Topologically, the Pythagorean tiling has the same structure ... page 171 publisher W. H. Freeman title Tilings and Patterns year 1987 . ref The smaller squares in the Pythagorean ... tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate ... symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic group cyclic ... triangles and regular hexagons, and one more, the Pythagorean tiling. ref harvtxt Gr nbaum Shephard 1987 , pp. 73 74. ref Pythagorean theorem and dissections File Pythagorean dissections.svg ... and by Henry Perigal right This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth century Arabic mathematicians Al Nayrizi ..., the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple 5,12,13 . By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate ..., overlaying two Pythagorean tilings may be used to generate a six piece dissection of two unequal ... Aperiodic tilings File Pythagorean tiling section.svg thumb 240px An aperiodic sequence generated from tilings by two squares whose side lengths form the golden ratio Although the Pythagorean tiling ... forms a Pythagorean tiling with two squares for which the ratio between the two side lengths is an irrational ... in the Pythagorean tiling meet edge to edge, ref name mms but this fact does not violate Keller s conjecture ... more details
ALLTYPES PPA98,M1 online Struck regards the magico religious performative power of the Pythagorean symbol as exceeding the methodology methodological confines of conventional semiotics . ref The Pythagorean symbols Walter Burkert featured Androcydes in his Textual criticism Stemmatics stemma of Pythagorean ... Abstain from beans see Kenneth Sylvan Guthrie, The Pythagorean Sourcebook and Library Red Wheel Weiser ... &num 100&as brr 3&as pt ALLTYPES online. ref or the quasi monasticism monastic rule governing the Pythagorean ... Pythagorean dietary practice . ref Robert McQueen Grant, Early Christians and Animals Routledge ... and philosophical modes of discourse, in that he cited passages from Hesiod in interpreting the Pythagorean ... Androcydes who was physician to Alexander the Great . ref Gillian Clark, Iamblichus On the Pythagorean ...&pg PA64&dq ANdrokydes&lr &num 100&as brr 3&as pt ALLTYPES PPA64,M1 online. ref Pythagorean dietary ... Androcydes Category Pythagoreans Category Pythagorean philosophy Category Ancient philosophers ... more details
the toilet. Common occurrences File Jewishmug1.jpg thumb upright A Pythagorean cup sold in Crete File Pythagoras cup 2.jpg thumb upright A Pythagorean cup sold in Samos The Pythagorean cup is credited ... rock to serve as an aqueduct bringing water to the city. Hero of Alexandria c. 10 70 AD used Pythagorean ... more details
Image Pythagorean diatonic scale on C.png thumb 330px Pythagorean diatonic scale on C audio Pythagorean ... intonation. Image Pythagorean major chord on C.png thumb Pythagorean tonic major chord on C audio Pythagorean major chord on C.mid Play compare audio C major triad.mid Play equal tempered and audio Just major triad on C.mid Play just . Pythagorean tuning lang el is a system of musical ... consonant . Attributed to Pythagoras sixth century BC , the Pythagorean system would appear to be ideal ... 2011 . The Physics of Music and Color , p.362. ISBN 978 1 4614 0556 6. ref The Pythagorean scale ... of fifths generated above gives seven notes, a diatonic major scale on C in Pythagorean tuning, shown in notation on the top right. Method Pythagorean tuning is based on a stack of intervals called perfect ... see Pythagorean comma . Thus, A music b and G music , when brought into the basic octave, will not coincide ...&pg PA82&dq pythagorean interval&hl en&ei aNrlTpK7BorWiALk1eyZBg&sa X&oi book result&ct result&resnum 10&ved 0CF0Q6AEwCQ v onepage&q pythagorean 20interval&f false Transactions of the Asiatic ..., as the above table indicates, in Pythagorean tuning they have different ratios with respect to D ... one quarter of a semitone, is known as a Pythagorean comma . To get around this problem, Pythagorean ... as a wolf interval . In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ..., the position of the wolf fifth can be changed. For example, a C based Pythagorean tuning would ... flat the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible ... ratios in D based symmetric Pythagorean tuning powers for large numbers .PNG frame right Frequency ratio of the 144 intervals in D based Pythagorean tuning. Interval music Quality Interval names are given ... Interval ratios in D based symmetric Pythagorean tuning powers of 2 26 3 .PNG here and http commons.wikimedia.org wiki File Interval ratios in D based symmetric Pythagorean tuning.PNG here . Image ... more details
pp move small yes See also Pythagorean trigonometric identity File Pythagorean.svg thumb The Pythagorean ... on the hypotenuse c . Trigonometry In mathematics , the Pythagorean theorem or Pythagoras theorem ... as an equation relating the lengths of the sides a , b and c , often called the Pythagorean ... author Judith D. Sally, Paul Sally page 63 chapter Chapter 3 Pythagorean triples url http books.google.com ... represent the lengths of the other two sides. The Pythagorean theorem is named after the Greeks ... Method of Generating Pythagorean Triples author Abdulrahman A. Abdulaziz year 2010 eprint ... at all, but n dimensional solids. The Pythagorean theorem has attracted interest outside ... two sides, the Pythagorean theorem can be expressed as the Pythagorean equation math a 2 b 2 ... c 2 a 2 . , math The Pythagorean equation relates the sides of a right triangle in a simple way, so ... of cosines reduces to the Pythagorean equation. Proofs This theorem may have more known proofs than any other the law of quadratic reciprocity being another contender for that distinction the book The Pythagorean ... times e c times d c times d e c 2 , , math which, tidying up, is the Pythagorean theorem math a 2 b ... Illustration to Euclid s proof of the Pythagorean theorem.svg thumb Proof in Euclid s Elements In outline ... See for example Mike May S.J., http www.slu.edu classes maymk GeoGebra Pythagoras.html Pythagorean ... s proof of the Pythagorean theorem2.svg thumb Illustration including the new lines File Illustration to Euclid s proof of the Pythagorean theorem3.PNG thumb Showing the two congruent triangles of half ... the knot.org pythagoras index.shtml 10 title Pythagorean Theorem, proof number 10 author Alexander ... File Pythagorean theorem rearrangement.svg thumb Proof using an elaborate rearrangement Algebraic ... www.cut the knot.org pythagoras index.shtml 3 title Cut the knot.org Pythagorean theorem and its many ... pythagoras index.shtml 4 title Cut the knot.org Pythagorean theorem and its many proofs, Proof ... more details
In mathematics , Pythagorean addition is the following binary operation on the real number s math a oplus b sqrt a 2 b 2 . math The name recalls the Pythagorean theorem , which states that the length of the hypotenuse of a right triangle is nowrap a b , where a and b are the lengths of the other sides. This operation provides a simple notation and terminology when the summands are complicated for example, the energy momentum relation in physics becomes math E mc 2 oplus pc. math Properties The operation is associative and commutative, and math sqrt x 1 2 x 2 2 cdots x n 2 x 1 oplus x 2 oplus cdots oplus x n math . This is enough to form the real numbers into a commutative semigroup . However, is not a Group mathematics group operation for the following reasons. The only element which could potentially act as an identity element is 0, since an identity e must satisfy e e e . This yields the equation math sqrt 2 e e math , but if e is nonzero then math sqrt 2 1 math is a contradiction, so e could only be zero. Unfortunately 0 does not work as an identity element after all, since 0 1 1. This does indicate, however, that if the operation is restricted to nonnegative real numbers, then 0 does act as an identity. Consequently the operation acting on the nonnegative real numbers forms a commutative monoid . See also Euclidean distance Hypot function Further reading expand further date February 2012 cite journal author Moler, Cleve and Donald Morrison title Replacing Square Roots by Pythagorean Sums journal IBM Journal of Research and Development volume 27 issue 6 pages 577 581 year 1983 url http www.research.ibm.com journal rd 276 ibmrd2706P.pdf doi 10.1147 rd.276.0577 Try also http citeseerx.ist.psu.edu viewdoc download?doi 10.1.1.90.5651&rep rep1&type pdf here . cite journal first Augustin A. last Dubrulle title A Class of Numerical Methods for the Computation of Pythagorean Sums journal IBM Journal of Research and Development volume 27 issue 6 pages ... more details
fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics ... books.google.co.uk books?id 8vw9z mnzH8C&pg PA517&lpg PA517&dq pythagorean hammers chisels&source ... v onepage&q pythagorean 20hammers 20chisels&f false ref As Aristotle was later to write, the Pythagoreans ... reflist DEFAULTSORT Pythagorean Hammers Category Acoustics Category Music theory Category Ancient Greek science Category Pythagorean philosophy Category Mathematics of music Link GA de de Pythagoras ... more details
Image MathematicalMeans.svg thumb right A geometric construction of the Quadratic mean and the Pythagorean means of two numbers a and b . Harmonic mean denoted by H , Geometric by G , Arithmetic by A and Quadratic mean also known as Root mean square denoted by Q . In mathematics, the three classical Pythagorean means are the arithmetic mean A , the geometric mean G , and the harmonic mean H . They are defined by math A x 1, ldots, x n frac 1 n x 1 cdots x n math math G x 1, ldots, x n sqrt n x 1 cdots x n math math H x 1, ldots, x n frac n frac 1 x 1 cdots frac 1 x n math Each mean has the following properties Value preservation math M x,x, ldots,x x math First order homogeneous function homogeneity math M bx 1, ldots, bx n b M x 1, ldots, x n math Invariance under exchange math M ldots, x i, ldots, x j, ldots M ldots, x j, ldots, x i, ldots math for any math i math and math j math . Averaging math min x 1, ldots,x n leq M x 1, ldots,x n leq max x 1, ldots,x n math These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians Thomas Heath, History of Ancient Greek Mathematics because of their importance in geometry and music. There is an ordering to these means if all of the math x i math are positive , along with the quadratic mean math Q sqrt frac x 1 2 x 2 2 cdots x n 2 n math math min leq H leq G leq A leq Q leq max math with equality holding if and only if the math x i math are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized mean s. This inequality sequence can be proved for the math n 2 math case for the numbers a and b using a sequence of right triangle s x , y , z with hypotenuse z and the Pythagorean theorem , which states that math x 2 y 2 z 2 math and implies that math z x math and math z y math . The right triangles are ref Kung ... links http mathworld.wolfram.com PythagoreanMeans.html Pythagorean means on MathWorld Category Means ... more details
Eurytus lang el , an eminent Pythagoreanism Pythagorean philosopher, lived c. 400 BC, who Iamblichus in one passage ref Iamblichus, de Vit. Pyth. 28 ref describes as a native of Croton , while in another, ref Iamblichus, de Vit. Pyth. 36 ref he enumerates him among the Taranto Tarentine Pythagoreans. He was a disciple of Philolaus , and Diogenes La rtius ref Diogenes La rtius iii. 6, viii. 46 ref mentions him among the teachers of Plato , though this statement is very doubtful. It is uncertain whether Eurytus was the author of any work, unless we suppose that the fragment in Stobaeus , ref Stobaeus, Phys. Ecl. i. ref which is there ascribed to one Eurytus, belongs to this Eurytus. Aristotle , Metaphysics Aristotle Metaphysics 1092b mentions Eurytus, speaking about points as limits of spatial magnitude It was in this sense that Eurytus determined the number of anything for he computed the number of a man or that of a horse or of any living thing by outlining its shape with pebbles, as one would number the sides of a triangle or a square, ref Aristotle, Metaphysics , 1092b. translated by Richard Hope, p. 314, Columbia University Press, 2008. ref Notes reflist SmithDGRBM Persondata Metadata see Wikipedia Persondata . NAME Eurytus ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Eurytus Category Ancient Greek philosophers Category Pythagoreans of Magna Graecia Category Ancient Crotonians Category 4th century BC philosophers Category 4th century BC deaths ca Eurit de Eurytos Philosoph it Eurito pitagorico fi Eurytos filosofi roa tara Eurito, pitagoriche uk ... more details
Unreferenced date April 2008 Image Non Pythagorean Scale Comparison.jpg 300px right thumb Comparison of Non Pythagorean pitches red audio Non Pythagorean scale on C.mid Play to equal temperament 12 tone equal tempered pitches blue audio ChromaticScaleUpDown.ogg Play . Note that, while the equal tempered pitches increase exponentially, the pitches found lower on the Non Pythagorean scale have frequencies that are farther apart while pitches found higher on the scale are closer together. The Non Pythagorean scale is a musical scale first conceived and developed by Robert Schneider of The Apples in Stereo . It is based on natural logarithms . The term Non Pythagorean is a reference to Ancient Greeks Greek philosopher Pythagoras in respect to his work with the chromatic scale . The scale was introduced in 2007 with the release of New Magnetic Wonder , the sixth studio album by The Apples in Stereo. The album featured two brief compositions using the scale. Enhanced CD versions of the album included a third composition as well as a variety of information from Schneider concerning the scale. Audio files and instructions are also on the disc, enabling the listener to prepare a MIDI keyboard to play in the Non Pythagorean scale. Though use of the scale is still sparse, Schneider encourages those interested to experiment with it and continue development of a possible musical theory. External links http www.applesinstereo.com pythagorean.php The Non Pythagorean scale at the official Apples in Stereo website http popculturewilleatitself.com music 2007 02 11 brief explanation of differences between pythagorean and non pythagorean scales A brief explanation of the differences between Pythagorean and Non Pythagorean scales at Pop Culture Will Eat Itself The Apples in Stereo Category Musical scales music stub ... more details
file Pythagorean.tree.svg 500px thumb The tree of primitive Pythagorean triples. In mathematics , a Pythagorean ... divisor . The set of all primitive Pythagorean triples has the structure of a rooted Tree structure ... multiplied on the right by a column vector whose components form a Pythagorean triple, then the result is another column vector whose components are a different Pythagorean triple. If the initial triple is primitive, then so is the one that results. Thus each primitive Pythagorean triple has three children . All primitive Pythagorean triples are descended in this way from the triple 3 ... . This tree also appeared in papers of A. Hall in 1970 ref A. Hall, Genealogy of Pythagorean Triads .... Kanga in 1990. ref Kanga, A. R., The family tree of Pythagorean triples, Bulletin of the Institute ... primitive Pythagorean triples It can be shown Induction mathematics inductively that the tree contains primitive Pythagorean triples and nothing else by showing that starting from a primitive Pythagorean ... Pythagorean and primitive. Preservation of the Pythagorean property If any of the above matrices, say A , is applied to a triple a , b , c sup T sup having the Pythagorean property a sup 2 sup b sup ... is also Pythagorean. This can be seen by writing out each of d , e , and f as the sum of three terms ... Pythagorean triple a , b , c sup T sup to obtain another triple d , e , f sup T sup , we have d , e ... of every primitive Pythagorean triple exactly once To show that the tree contains every primitive Pythagorean triple, but no more than once, it suffices to show that for any such triple ... to an arbitrary primitive Pythagorean triple d , e , f , noting that by the above reasoning primitivity and the Pythagorean property are retained, and noting that for any triple larger than 3, 4, 5 ... generation. ref name Price cite arXiv last Price first H. Lee eprint 0809.4324 title The Pythagorean ... 21, 20, 29 respectively. If either of A or C is applied repeatedly from any Pythagorean triple ... more details
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem ... also Trigonometric functions List of trigonometric identities Mathematically, the Pythagorean identity ... sup 2 sup . This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric ... identity follows from the Pythagorean theorem . Proofs and their relationships to the Pythagorean ... hypotenuse frac a c . math The Pythagorean identity follows by squaring both definitions above, and adding ... mathrm hypotenuse 2 math which by the Pythagorean theorem is equal to 1. Note, however, that this definition ... math 1 tan 2 theta sec 2 theta , math and math 1 cot 2 theta csc 2 theta , math are also called Pythagorean ... the Pythagorean theorem. The angle opposite the leg of length 1 this angle can be labeled ... from the Pythagorean theorem. Tabulation of derivations Another way of thinking about the other ... by dividing each element of the original Pythagorean Identity by a common divisor. class wikitable sortable ..., from the equation for the unit circle math cos 2 theta sin 2 theta 1 , math the Pythagorean identity ... varies as the angle changes quadrant. Because the x and y axes are perpendicular, this Pythagorean identity is actually equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1 which is in turn equivalent to the full Pythagorean theorem by applying a similar triangles argument ... sin 2 x cos 2 x 1 , math which is the Pythagorean trigonometric identity. The Pythagorean theorem is not closely related to the Pythagorean identity when the trigonometric functions are defined in this way ... is a constant so z 1 for all x , so the Pythagorean identity is established. A similar proof can ... has no direct connection with Euclid s demonstration of the Pythagorean theorem. See also Pythagorean theorem Trigonometric identity Unit circle Power series Differential equation In line notes and references references External links http sympl.org book examples interactive plots pythagorean ... more details
Besides Euclid s formula, many other formulas for generating Pythagorean triple s have been developed ... triples have been described here main Pythagorean triple The methods below appear in various sources ... yields Pythagorean triples not all of them primitive for any given value, under the conditions given, but it does not yield all valid Pythagorean triples, or even all primitive Pythagorean triples ... of the fractions are the odd numbers, math 3, text 5, text 7, text 9, math etc. To calculate a Pythagorean ... to himself the following method for generating Pythagorean triples. To find integer solutions ... and that s and t are factors of math tfrac r 2 2 math . All Pythagorean triples may be found ... of Pythagorean triples starting from a sub 3 sub , b sub 3 sub , c sub 3 sub 4, 3, 5 via math ... Fibonacci triangles Fibonacci triangles . II. A Pythagorean triple can be generated using any ... is a Pythagorean triple. ref Horadam, A. F., Fibonacci number triples , American Mathematical Monthly ..., Frank R. and Price, H. Lee title Heron s formula, Descartes circles, and Pythagorean triangles ... q , q are not coprime , the same procedure leads to a non primitive triple. Pythagorean triples and Descartes circle equation This method of generating primitive Pythagorean triples also provides ... sub 0 sub 554. A Ternary Tree Generating All Primitive Pythagorean Triples Each primitive Pythagorean ... and primitive Pythagorean triple. In this section we shall use the Fibonacci Box in place of the primitive triple it represents. An infinite ternary tree containing all primitive Pythagorean ... H. Lee last Price title The Pythagorean Tree A New Species year 2008 pages 14 eprint 0809.4324 class ... 129 139 language Swedish postscript . ref See also Tree of primitive Pythagorean triples . Generating ... leg of a Pythagorean triple. ref J. L. Poet and D. L. Vestal, Jr. 2005 . Curious Consequences of a Miscopied ... 4 252 405 477 21 6 Pythagorean triples by use of matrices and linear transformations main Tree of primitive ... more details
File Pythagorean.svg thumb Visual representation of the Pythagorean theorem. Under the proposal the shape seen here would be drawn at vast size on the Siberian tundra using pine trees and fields of wheat Gauss Pythagorean right triangle proposal is a proposal attributed to Carl Friedrich Gauss for a method to signal extraterrestrial beings by constructing an immense right triangle and three squares on the surface of the earth. The shapes would be a symbolical representation of the Pythagorean theorem , large enough to be seen from the Moon or Mars . Although credited in numerous sources as originating with Gauss, with exact details of the proposal set out, the specificity of detail, and even whether Gauss made the proposal, has been called into question. Many of the earliest sources for forms of the proposal do not actually name Gauss as the originator, instead crediting a German Astronomer or using other nonspecific descriptors, and in some cases naming a different author entirely. The details of the proposal also change significantly upon different retellings. Nevertheless, Gauss writings reveal a belief and interest in finding a method to contact extraterrestrial life , and that he did, at the least, propose using amplified light using a Heliotrope instrument Heliotrope , his own 1818 invention, to signal supposed inhabitants of the Moon. The proposal Carl Friedrich Gauss is credited with an 1820 proposal ref cite journal author Glenn Garelik coauthors J. Madeleine Nash and Richard Woodbury title Space Onward to Mars journal Time Magazine url http www.time.com time magazine article 0,9171,967915,00.html date July 18, 1988 page 50 format subscription required volume 132 issue 3 ref for a method to signal extraterrestrial beings in the form of drawing an immense right triangle and three squares on the surface of the earth, intended as a symbolical representation of the Pythagorean ... that this geometric figure, invoking the Pythagorean theorem through the squares on the outside ... more details
Cronius lang el 2nd century was a celebrated Neopythagorean philosopher. ref Porphyry, Vita Plotini , 20 Eusebius , Church History Eusebius Church History vi. 19. ref He was probably a contemporary of Numenius of Apamea , who lived in the 2nd century, and he is often spoken of along with him. ref Porphyry, De Antr. Nymph. 10 ref Nemesius ref Nemesius, de Anima , 2. ref mentions a work of his On Reincarnation , lang el , and Origen is said to have diligently studied the works of Cronius. ref Suda , Origenes . ref Porphyry philosopher Porphyry also states that he endeavoured to explain the fables of the Homer ic poems in a philosophical manner. This is all we know about Cronius, although he appears to have been very distinguished among the Neopythagoreans. Notes reflist External links http www.sacred texts.com cla plotenn enn001.htm Life of Plotinus by Porphyry SmithDGRBM Category 2nd century philosophers Category Neoplatonists Category Neo Pythagoreans Category Roman era philosophers ca Croni de Kronios ru ... more details
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