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Golden ratio

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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. [1]

234 relations: Acoustic scale, Acoustics, Acute and obtuse triangles, Adolf Zeising, Aesthetics, Algebraic equation, Algebraic integer, Algebraic number, Algebraic number field, Algorithm, American Mathematical Monthly, Ancient Egyptian architecture, Ancient Egyptian mathematics, Ancient Greece, Ancient Greek, Anthropometry, Arbitrary-precision arithmetic, Architect, Architecture, Arithmetic progression, Artist, Aspect ratio, Édouard Lucas, Béla Bartók, Beauty, Binary number, Bisection, Cambridge University Press, Charles Bonnet, Claude Debussy, Clockwise, Closed-form expression, Closure (mathematics), Combination tone, Compass-and-straightedge construction, Continued fraction, Convex polytope, Crystal, Cube, De divina proportione, Decagon, Decimal, Diagonal, Dihedral angle, Diophantine approximation, DNA, Dodecahedron, E (mathematical constant), Eli Maor, Elliott wave principle, ..., Equilateral triangle, Eric Temple Bell, Erica Flapan, Erik Satie, Ernő Lendvai, Euclid, Euclid's Elements, Euler's totient function, Fibonacci, Fibonacci number, Fibonacci retracement, Field (mathematics), Financial market, Franciscans, Friar, Fundamental unit (number theory), Garches, Geometric progression, Geometry, George Phillips Odom Jr., George W. Hart, Golden angle, Golden ratio base, Golden rectangle, Golden rhombus, Golden triangle (mathematics), Golden-section search, Great Mosque of Kairouan, Great Pyramid of Giza, Greek alphabet, Greek language, Group action, Gustav Fechner, Gutenberg Bible, Harold Scott MacDonald Coxeter, Heinrich Cornelius Agrippa, Heinz Bohlen, Hexadecimal, Howard Vyse, Human genome, Hurwitz's theorem (number theory), Hyperbolic geometry, Hypotenuse, Icosahedron, Ideal triangle, Inorganic compound, International Style (architecture), Intersecting chords theorem, Irrational number, Irreducible fraction, Isosceles triangle, Iterative method, Ivan Moscovich, Jan Tschichold, Jesus, Johannes Kepler, John Taylor (English publisher), Keith Devlin, Kepler triangle, La mer (Debussy), Le Corbusier, Leon Battista Alberti, Leonardo da Vinci, Liber Abaci, Limit of a sequence, Line segment, Linear fractional transformation, List of works designed with the golden ratio, Logarithmic spiral, Luca Pacioli, Lucas number, Mario Botta, Mario Livio, Martin Ohm, Mathematical coincidence, Mathematician, Mathematics, Matila Ghyka, Möbius function, Methods of computing square roots, Michael Maestlin, Minaret, Minimal polynomial (field theory), Modernism, Modular group, Modulor, Mona Lisa, Multiplicative inverse, Naqsh-e Jahan Square, Natural number, Nautilus, Nested radical, Newton's method, Numeral system, Octahedron, Optics, Organic chemistry, Origlio, Orthogonality, Parthenon, Patent, Patterns in nature, Pearl Drums, Penrose tiling, Pentagon, Pentagram, Phi, Phidias, Phyllotaxis, Phys.org, Pi, Piet Mondrian, Pisot–Vijayaraghavan number, Plastic number, Plato, Platonic solid, Polyhedron, Projective line, Proportion (architecture), Proportionality (mathematics), Ptolemy's theorem, Pyramidology, Pythagoras, Pythagorean theorem, Quadratic equation, Quadratic formula, Quasicrystal, Rate of convergence, Ratio, Rational function, Rectangle, Recurrence relation, Reflets dans l'eau, Regular polyhedron, Rhind Mathematical Papyrus, Rhombic triacontahedron, Rhombus, Right triangle, Roger Penrose, Roy Batchelor, Roy Howat, Sacred geometry, Salvador Dalí, Satellite, Scale (ratio), Section d'Or, Sequence, Series (mathematics), Silver ratio, Similarity (geometry), Skeleton, Square number, Square pyramid, Square root of 5, STARSHINE (satellite), Structure, Summation, Surface area, Symbol, Symmetric group, Symmetry, Tau, Technical analysis, Tetrahedron, The Daily Telegraph, The Economist, The New York Times, The Sacrament of the Last Supper, Theorem, Thomson problem, Timaeus (dialogue), Transcendental number, Triangle, Triangle inequality, Trigonometric functions, University of Tübingen, Visual appearance, Vitruvian Man, Vitruvius, Wolfram Demonstrations Project, Yve-Alain Bois, 1509 in literature, 3, 833 cents scale. Expand index (184 more) »

Acoustic scale

In music, the acoustic scale, overtone scale, Lydian dominant scale, or Lydian 7 scale, is a seven-note synthetic scale.

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Acoustics is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound.

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Acute and obtuse triangles

An acute triangle is a triangle with all three angles acute (less than 90°).

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Adolf Zeising

Adolf Zeising (24 September 181027 April 1876) was a German psychologist, whose main interests were mathematics and philosophy.

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Aesthetics (also spelled esthetics) is a branch of philosophy that explores the nature of art, beauty, and taste, with the creation and appreciation of beauty.

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Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

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Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

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Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

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Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

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In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Ancient Egyptian architecture

Ancient Egyptian architecture is the architecture of one of the most influential civilizations throughout history, which developed a vast array of diverse structures and great architectural monuments along the Nile, including pyramids and temples.

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Ancient Egyptian mathematics

Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. 300 BC, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt.

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Ancient Greece

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).

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Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

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Anthropometry (from Greek ἄνθρωπος anthropos, "human", and μέτρον metron, "measure") refers to the measurement of the human individual.

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Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system.

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An architect is a person who plans, designs, and reviews the construction of buildings.

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Architecture is both the process and the product of planning, designing, and constructing buildings or any other structures.

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Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

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An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating an art.

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Aspect ratio

The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions.

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Édouard Lucas

François Édouard Anatole Lucas (4 April 1842 – 3 October 1891) was a French mathematician.

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Béla Bartók

Béla Viktor János Bartók (25 March 1881 – 26 September 1945) was a Hungarian composer, pianist and an ethnomusicologist.

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Beauty is a characteristic of an animal, idea, object, person or place that provides a perceptual experience of pleasure or satisfaction.

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Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

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In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Charles Bonnet

Charles Bonnet (13 March 1720 – 20 May 1793), Genevan naturalist and philosophical writer, was born at Geneva, of a French family driven into the region by the religious persecution in the 16th century.

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Claude Debussy

Achille-Claude Debussy (22 August 1862 – 25 March 1918) was a French composer.

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Two-dimensional rotation can occur in two possible directions.

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Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.

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Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

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Combination tone

A combination tone (also called resultant or subjective tone"", Britannica.com. Accessed September 2015.) is a psychoacoustic phenomenon of an additional tone or tones that are artificially perceived when two real tones are sounded at the same time.

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Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

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Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

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Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

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A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.

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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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De divina proportione

De divina proportione (On the Divine Proportion) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, composed around 1498 in Milan and first printed in 1509.

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In geometry, a decagon is a ten-sided polygon or 10-gon.

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The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers.

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In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.

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Dihedral angle

A dihedral angle is the angle between two intersecting planes.

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Diophantine approximation

In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers.

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Deoxyribonucleic acid (DNA) is a thread-like chain of nucleotides carrying the genetic instructions used in the growth, development, functioning and reproduction of all known living organisms and many viruses.

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In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

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Eli Maor

Eli Maor, an Israel-born historian of mathematics, is the author of several books about the history of mathematics.

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Elliott wave principle

The Elliott wave principle is a form of technical analysis that finance traders use to analyze financial market cycles and forecast market trends by identifying extremes in investor psychology, highs and lows in prices, and other collective factors.

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Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

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Eric Temple Bell

Eric Temple Bell (February 7, 1883 – December 21, 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life.

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Erica Flapan

Erica Flapan (born August 14, 1956) is an American mathematician, the Lingurn H. Burkhead Professor of Mathematics at Pomona College.

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Erik Satie

Éric Alfred Leslie Satie (17 May 18661 July 1925), who signed his name Erik Satie after 1884, was a French composer and pianist.

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Ernő Lendvai

Ernő Lendvai (February 6, 1925 – January 31, 1993) was one of the first music theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartók's music.

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Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

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Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

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Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

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Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".

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Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

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Fibonacci retracement

In finance, Fibonacci retracement is a method of technical analysis for determining support and resistance levels.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Financial market

A financial market is a market in which people trade financial securities and derivatives such as futures and options at low transaction costs.

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The Franciscans are a group of related mendicant religious orders within the Catholic Church, founded in 1209 by Saint Francis of Assisi.

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A friar is a brother member of one of the mendicant orders founded since the twelfth or thirteenth century; the term distinguishes the mendicants' itinerant apostolic character, exercised broadly under the jurisdiction of a superior general, from the older monastic orders' allegiance to a single monastery formalized by their vow of stability.

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Fundamental unit (number theory)

In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic).

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Garches is a commune in the western suburbs of Paris, France.

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Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

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Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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George Phillips Odom Jr.

George Phillips Odom Jr. (born 1941) is an American artist and amateur geometer, who is primarily known for his work on the golden ratio (\Phi).

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George W. Hart

George William Hart (born 1955) is an American geometer who expresses himself both artistically and academically.

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Golden angle

In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc.

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Golden ratio base

Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number ≈ 1.61803399 symbolized by the Greek letter φ) as its base.

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Golden rectangle

In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.

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Golden rhombus

In geometry, a golden rhombus is a rhombus whose diagonals are in the ratio \frac.

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Golden triangle (mathematics)

A golden triangle, also known as the sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the distinct side: Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons.

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Golden-section search

The golden-section search is a technique for finding the extremum (minimum or maximum) of a strictly unimodal function by successively narrowing the range of values inside which the extremum is known to exist.

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Great Mosque of Kairouan

The Great Mosque of Kairouan (جامع القيروان الأكبر), also known as the Mosque of Uqba (جامع عقبة بن نافع), is a mosque in Tunisia, situated in the UNESCO World Heritage town of Kairouan.

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Great Pyramid of Giza

The Great Pyramid of Giza (also known as the Pyramid of Khufu or the Pyramid of Cheops) is the oldest and largest of the three pyramids in the Giza pyramid complex bordering what is now El Giza, Egypt.

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Greek alphabet

The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.

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Greek language

Greek (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean and the Black Sea.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Gustav Fechner

Gustav Theodor Fechner (19 April 1801 – 18 November 1887), was a German philosopher, physicist and experimental psychologist.

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Gutenberg Bible

The Gutenberg Bible (also known as the 42-line Bible, the Mazarin Bible or the B42) was the first major book printed using mass-produced movable metal type in Europe.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Heinrich Cornelius Agrippa

Heinrich Cornelius Agrippa von Nettesheim (14 September 1486 – 18 February 1535) was a German polymath, physician, legal scholar, soldier, theologian, and occult writer.

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Heinz Bohlen

Heinz P. Bohlen (26 June 1935 – 2 February 2016)"", Bohlen-Pierce-Conference.org.

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In mathematics and computing, hexadecimal (also base, or hex) is a positional numeral system with a radix, or base, of 16.

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Howard Vyse

Major General Richard William Howard Vyse (25 July 1784 – 8 June 1853) was a British soldier, anthropologist and Egyptologist.

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Human genome

The human genome is the complete set of nucleic acid sequences for humans, encoded as DNA within the 23 chromosome pairs in cell nuclei and in a small DNA molecule found within individual mitochondria.

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Hurwitz's theorem (number theory)

In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle.

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In geometry, an icosahedron is a polyhedron with 20 faces.

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Ideal triangle

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points.

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Inorganic compound

An inorganic compound is typically a chemical compound that lacks C-H bonds, that is, a compound that is not an organic compound, but the distinction is not defined or even of particular interest.

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International Style (architecture)

The International Style is the name of a major architectural style that developed in the 1920s and 1930s and strongly related to Modernism and Modern architecture.

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Intersecting chords theorem

The intersecting chords theorem or just chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle.

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Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

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Irreducible fraction

An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).

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Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

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Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.

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Ivan Moscovich

Ivan Moscovich is a designer and commercial developer of puzzles, games, toys, and educational aids.

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Jan Tschichold

Jan Tschichold (2 April 1902 Leipzig, Germany – 11 August 1974 Locarno, Switzerland) (born as Johannes Tzschichhold, also Iwan Tschichold, Ivan Tschichold) was a calligrapher, typographer and a book designer.

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Jesus, also referred to as Jesus of Nazareth and Jesus Christ, was a first-century Jewish preacher and religious leader.

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Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

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John Taylor (English publisher)

John Taylor (31 July 1781–1864) was a publisher, essayist, and writer.

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Keith Devlin

Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science writer.

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Kepler triangle

A Kepler triangle is a right triangle with edge lengths in geometric progression.

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La mer (Debussy)

La mer, trois esquisses symphoniques pour orchestre (French for The sea, three symphonic sketches for orchestra), or simply La mer (i.e. The Sea), L. 109, is an orchestral composition by the French composer Claude Debussy.

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Le Corbusier

Charles-Édouard Jeanneret (6 October 1887 – 27 August 1965), known as Le Corbusier, was a Swiss-French architect, designer, painter, urban planner, writer, and one of the pioneers of what is now called modern architecture.

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Leon Battista Alberti

Leon Battista Alberti (February 14, 1404 – April 25, 1472) was an Italian humanist author, artist, architect, poet, priest, linguist, philosopher and cryptographer; he epitomised the Renaissance Man.

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Leonardo da Vinci

Leonardo di ser Piero da Vinci (15 April 14522 May 1519), more commonly Leonardo da Vinci or simply Leonardo, was an Italian polymath of the Renaissance, whose areas of interest included invention, painting, sculpting, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, writing, history, and cartography.

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Liber Abaci

Liber Abaci (1202, also spelled as Liber Abbaci) is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.

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Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

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Linear fractional transformation

In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.

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List of works designed with the golden ratio

Many works of art are claimed to have been designed using the golden ratio.

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Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature.

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Luca Pacioli

Fra Luca Bartolomeo de Pacioli (sometimes Paccioli or Paciolo; 1447–1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and a seminal contributor to the field now known as accounting.

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Lucas number

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers.

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Mario Botta

Mario Botta (born April 1, 1943) is a Swiss architect.

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Mario Livio

Mario Livio (born 1945 in Bucharest) is an Israeli-American astrophysicist and an author of works that popularize science and mathematics.

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Martin Ohm

Martin Ohm (May 6, 1792, Erlangen – April 1, 1872, Berlin) was a German mathematician and a younger brother of physicist Georg Ohm.

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Mathematical coincidence

A mathematical coincidence is said to occur when two expressions show a near-equality which has no known theoretical explanation.

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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Matila Ghyka

Prince Matila Costiescu Ghyka, KCVO, M.C. (born Matila Costiescu; 13 September 1881 – 14 July 1965), was a Romanian novelist, mathematician, historian, philosopher, diplomat and Plenipotentiary Minister in the United Kingdom during the late 1930s and until 1940.

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Möbius function

The classical Möbius function is an important multiplicative function in number theory and combinatorics.

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Methods of computing square roots

In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number.

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Michael Maestlin

Michael Maestlin (also Mästlin, Möstlin, or Moestlin) (30 September 1550, Göppingen – 20 October 1631, Tübingen) was a German astronomer and mathematician, known for being the mentor of Johannes Kepler.

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Minaret (مناره, minarə, minare), from منارة, "lighthouse", also known as Goldaste (گلدسته), is a distinctive architectural structure akin to a tower and typically found adjacent to mosques.

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Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial.

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Modernism is a philosophical movement that, along with cultural trends and changes, arose from wide-scale and far-reaching transformations in Western society during the late 19th and early 20th centuries.

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Modular group

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.

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The Modulor is an anthropometric scale of proportions devised by the Swiss-born French architect Le Corbusier (1887–1965).

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Mona Lisa

The Mona Lisa (Monna Lisa or La Gioconda, La Joconde) is a half-length portrait painting by the Italian Renaissance artist Leonardo da Vinci that has been described as "the best known, the most visited, the most written about, the most sung about, the most parodied work of art in the world".

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Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

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Naqsh-e Jahan Square

Naqsh-e Jahan Square (میدان نقش جهان Maidān-e Naqsh-e Jahān; trans: "Image of the World Square"), also known as Meidan Emam, is a square situated at the center of Isfahan city, Iran.

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Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

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The nautilus (from the Latin form of the original ναυτίλος, 'sailor') is a pelagic marine mollusc of the cephalopod family Nautilidae, the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.

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Nested radical

In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression.

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Newton's method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

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Numeral system

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

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In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

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Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it.

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Organic chemistry

Organic chemistry is a chemistry subdiscipline involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.

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Origlio is a municipality in the district of Lugano in the canton of Ticino in Switzerland.

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In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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The Parthenon (Παρθενών; Παρθενώνας, Parthenónas) is a former temple, on the Athenian Acropolis, Greece, dedicated to the goddess Athena, whom the people of Athens considered their patron.

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A patent is a set of exclusive rights granted by a sovereign state or intergovernmental organization to an inventor or assignee for a limited period of time in exchange for detailed public disclosure of an invention.

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Patterns in nature

Patterns in nature are visible regularities of form found in the natural world.

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Pearl Drums

, simply known as Pearl, is a multinational corporation based in Japan with a wide range of products, predominantly percussion instruments.

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Penrose tiling

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles.

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In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon or 5-gon.

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A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a five-pointed star drawn with five straight strokes.

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Phi (uppercase Φ, lowercase φ or ϕ; ϕεῖ pheî; φι fi) is the 21st letter of the Greek alphabet.

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Phidias or Pheidias (Φειδίας, Pheidias; 480 – 430 BC) was a Greek sculptor, painter, and architect.

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In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon "leaf" and táxis "arrangement").

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Phys.org is a science, research and technology news aggregator where much of the content is republished directly from press releases and news agencies-in a practice known as churnalism.

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The number is a mathematical constant.

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Piet Mondrian

Pieter Cornelis "Piet" Mondriaan, after 1906 Mondrian (later; 7 March 1872 – 1 February 1944), was a Dutch painter and theoretician who is regarded as one of the greatest artists of the 20th century.

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Pisot–Vijayaraghavan number

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 all of whose Galois conjugates are less than 1 in absolute value.

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Plastic number

In mathematics, the plastic number (also known as the plastic constant, the minimal Pisot number, the platin number, Siegel's number or, in French, le nombre radiant) is a mathematical constant which is the unique real solution of the cubic equation It has the exact value Its decimal expansion begins with.

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Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

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Proportion (architecture)

Proportion is a central principle of architectural theory and an important connection between mathematics and art.

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Proportionality (mathematics)

In mathematics, two variables are proportional if there is always a constant ratio between them.

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Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle).

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Pyramidology (or pyramidism) refers to various religious or pseudoscientific speculations regarding pyramids, most often the Giza pyramid complex and the Great Pyramid of Giza in Egypt.

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Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

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Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

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Quadratic equation

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to.

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Quadratic formula

In elementary algebra, the quadratic formula is the solution of the quadratic equation.

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A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.

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Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence.

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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.

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Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

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In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

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Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

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Reflets dans l'eau

Claude Debussy's Reflets dans l'eau ("Reflections in the Water") is the first of three piano pieces from his first volume of Images, which are frequently performed separately.

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Regular polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.

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Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of Egyptian mathematics.

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Rhombic triacontahedron

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces.

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In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length.

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Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

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Roger Penrose

Sir Roger Penrose (born 8 August 1931) is an English mathematical physicist, mathematician and philosopher of science.

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Roy Batchelor

Roy A. Batchelor (born 23 March 1947) is HSBC Professor of Banking and Finance in the Cass Business School of City, University of London.

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Roy Howat

Roy Howat (born 1951, Ayrshire) is a Scottish pianist and musicologist who specializes in French music.

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Sacred geometry

Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions.

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Salvador Dalí

Salvador Domingo Felipe Jacinto Dalí i Domènech, 1st Marquess of Dalí de Púbol (11 May 190423 January 1989), known professionally as Salvador Dalí, was a prominent Spanish surrealist born in Figueres, Catalonia, Spain.

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In the context of spaceflight, a satellite is an artificial object which has been intentionally placed into orbit.

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Scale (ratio)

The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original.

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Section d'Or

The Section d'Or ("Golden Section"), also known as Groupe de Puteaux (or Puteaux Group), was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism.

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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

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Silver ratio

In mathematics, two quantities are in the silver ratio (also silver mean or silver constant) if the ratio of the sum of the smaller and twice the larger of those quantities, to the larger quantity, is the same as the ratio of the larger one to the smaller one (see below).

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Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.

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The skeleton is the body part that forms the supporting structure of an organism.

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Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.

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Square pyramid

In geometry, a square pyramid is a pyramid having a square base.

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Square root of 5

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5.

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STARSHINE (satellite)

The STARSHINE (Student Tracked Atmospheric Research Satellite Heuristic International Networking Experiment) series of three artificial satellites were student participatory missions sponsored by the United States Naval Research Laboratory (NRL).

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Structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized.

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In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

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Surface area

The surface area of a solid object is a measure of the total area that the surface of the object occupies.

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A symbol is a mark, sign or word that indicates, signifies, or is understood as representing an idea, object, or relationship.

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Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

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Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.

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Tau (uppercase Τ, lowercase τ; ταυ) is the 19th letter of the Greek alphabet.

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Technical analysis

In finance, technical analysis is an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume.

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In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

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The Daily Telegraph

The Daily Telegraph, commonly referred to simply as The Telegraph, is a national British daily broadsheet newspaper published in London by Telegraph Media Group and distributed across the United Kingdom and internationally.

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The Economist

The Economist is an English-language weekly magazine-format newspaper owned by the Economist Group and edited at offices in London.

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The New York Times

The New York Times (sometimes abbreviated as The NYT or The Times) is an American newspaper based in New York City with worldwide influence and readership.

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The Sacrament of the Last Supper

The Sacrament of the Last Supper is a painting by Salvador Dalí.

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In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

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Thomson problem

The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law.

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Timaeus (dialogue)

Timaeus (Timaios) is one of Plato's dialogues, mostly in the form of a long monologue given by the title character Timaeus of Locri, written c. 360 BC.

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Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

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A triangle is a polygon with three edges and three vertices.

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Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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University of Tübingen

The University of Tübingen, officially the Eberhard Karls University of Tübingen (Eberhard Karls Universität Tübingen; Universitas Eberhardina Carolina), is a German public research university located in the city of Tübingen, Baden-Württemberg.

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Visual appearance

The visual appearance of objects is given by the way in which they reflect and transmit light.

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Vitruvian Man

The Vitruvian Man (Le proporzioni del corpo umano secondo Vitruvio, which is translated to "The proportions of the human body according to Vitruvius"), or simply L'Uomo Vitruviano, is a drawing by Leonardo da Vinci around 1490.

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Marcus Vitruvius Pollio (c. 80–70 BC – after c. 15 BC), commonly known as Vitruvius, was a Roman author, architect, civil engineer and military engineer during the 1st century BC, known for his multi-volume work entitled De architectura.

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Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

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Yve-Alain Bois

Yve-Alain Bois (born April 16, 1952) is a professor of Art History at the School of Historical Studies at the Institute for Advanced Study in Princeton, New Jersey.

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1509 in literature

This article presents lists of the literary events and publications in 1509.

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3 (three) is a number, numeral, and glyph.

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833 cents scale

The 833 cents scale is a musical tuning and scale proposed by Heinz Bohlen based on combination tones, an interval of 833.09 cents, and, coincidentally, the Fibonacci sequence.

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.618, 0.618, 1+1/phi, 1,618, 1.6, 1.618, 1.61803..., 1.618034, A+b is to a as a is to b, Divine Proportion, Divine proportion, Divine ratio, Divine section, Extreme ratio, Golden Ratio, Golden Rule (Mathematics), Golden Section, Golden and extreme ratio, Golden cut, Golden mean number, Golden proportion, Golden ratio conjugate, Golden ratio, the, Golden section, Golden thirds, Golden-ratio, Goldenratio, Mean of Phidias, Phi (Golden Number), Phi (Golden Ratio), Phi (golden ratio), Phi (mathematical constant), Phi (number), Phi ratio, Sectio aurea, Sectio divina, Sqrt(1+phi), The Golden Mean/Rectangle, The Golden Ratio, The golden ratio.


[1] https://en.wikipedia.org/wiki/Golden_ratio

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