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Encyclopedia > Golden ratio
The golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. This article is about the European Renaissance of the 14th-17th centuries. ... The definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art. ... For other uses, see Architect (disambiguation). ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ... Aesthetics is commonly perceived as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter $varphi$ (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically: A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Look up Î¦, Ï† in Wiktionary, the free dictionary. ...

$frac{a+b}{a} = frac{a}{b} = varphi,.$

This equation has as its unique positive solution the algebraic irrational number In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

$varphi = frac{1 + sqrt{5}}{2}approx 1.61803,39887dots,$ [1]

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi ($varphi$).[2][3][4] Other terms encountered include extreme and mean ratio, medial section, divine proportion (Italian: proporzione divina), divine section (Latin: sectio divina), golden proportion, golden cut,[5] and mean of Phidias.[6][7][8] Look up Î¦, Ï† in Wiktionary, the free dictionary. ... Phidias Showing the Frieze of the Parthenon to his Friends by Sir Lawrence Alma-Tadema Phidias (or Pheidias) (in ancient Greek, ) (c. ...

Construction of a golden rectangle:
1. Construct a unit square.
2. Draw a line from the midpoint of one side to an opposite corner.
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ...

 List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ Binary 1.1001111000110111011... Decimal 1.6180339887498948482... Hexadecimal 1.9E3779B97F4A7C15F39... Continued fraction $1 + frac{1}{1 + frac{1}{1 + frac{1}{1 + frac{1}{ddots}}}}$ Algebraic form $frac{1 + sqrt{5}}{2}$

Two quantities (positive numbers) a and b are said to be in the golden ratio $varphi$ if This is a list of articles about numbers (not about numerals). ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ... In mathematics, ApÃ©rys constant is a curious number that occurs in a variety of situations. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... The square root of 3 is equal to the length across the flat sides of a regular hexagon with sides of length 1. ... The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. ... The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... For other uses, see Decimal (disambiguation). ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0â€“9 and Aâ€“F, or aâ€“f. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...

$frac{a+b}{a} = frac{a}{b} = varphi,.$

This equation unambiguously defines $varphi,$.

The right equation shows that $a=bvarphi$, which can be substituted in the left part, giving

$frac{bvarphi+b}{bvarphi}=frac{bvarphi}{b},.$

Cancelling b yields

$frac{varphi+1}{varphi}=varphi.$

Multiplying both sides by $varphi$ and rearranging terms leads to:

$varphi^2 - varphi - 1 = 0.$

The only positive solution to this quadratic equation is In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

$varphi = frac{1 + sqrt{5}}{2} approx 1.61803,39887dots,$

## History

Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form ($varphi$) is used. Sometimes, the uppercase form ($Phi,$) is used for the reciprocal of the golden ratio, $1/varphi$.

The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years: Image File history File links Phi_uc_lc. ... Image File history File links Phi_uc_lc. ... This page is a candidate for speedy deletion. ... Phidias Showing the Frieze of the Parthenon to his Friends by Sir Lawrence Alma-Tadema Phidias (or Pheidias) (in ancient Greek, ) (c. ... The reciprocal function: y = 1/x. ...

 “ Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. ” — Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number
Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The ratio is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of the ratio to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol. Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... For other uses, see Euclid (disambiguation). ... The term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. ... Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... Kepler redirects here. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Mario Livio (born 1945) is an astrophysicist and an author of works that popularize science and mathematics. ... Michael Maestlin (1550-1631) was a German astronomer and mathematician. ... The term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. ... For other uses, see Geometry (disambiguation). ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Look up pentagon in Wiktionary, the free dictionary. ... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Bust of Pythagoras Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. ...

Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."[9] Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio.[10] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[11] Some of these propositions show that the golden ratio is an irrational number. For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

The name "extreme and mean ratio" was the principal term used from the 3rd century BC[9] until about the 18th century. The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ...

The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio. Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ...

The first known calculation of the (conjugate) golden ratio as a decimal, of "about 0.6180340," was written in 1597 by Prof. Michael Maestlin of the University of Tübingen to his former student Johannes Kepler.[12] For other uses, see Decimal (disambiguation). ... Michael Maestlin (1550-1631) was a German astronomer and mathematician. ... Eberhard Karls University of TÃ¼bingen (German: Eberhard-Karls-UniversitÃ¤t TÃ¼bingen) is a state-supported university located on the Neckar river, in the city of TÃ¼bingen, Baden-WÃ¼rttemberg, Germany. ... Kepler redirects here. ...

Since the twentieth century, the golden ratio has been represented by the Greek letter $varphi$ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut). The Greek alphabet (Greek: ) is an alphabet consisting of 24 letters that has been used to write the Greek language since the late 8th or early 8th century BC. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel... This article needs to be cleaned up to conform to a higher standard of quality. ... Phidias Showing the Frieze of the Parthenon to his Friends by Sir Lawrence Alma-Tadema Phidias (or Pheidias) (in ancient Greek, ) (c. ... Look up Î¤, Ï„ in Wiktionary, the free dictionary. ... Beginning of Homers Odyssey The Ancient Greek language is the historical stage of the Greek language[1] as it existed during the Archaic (9thâ€“6th centuries BC) and Classical (5thâ€“4th centuries BC) periods in Ancient Greece. ...

### Timeline

Timeline according to Priya Hemenway[13].

• Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
• Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron), some of which are related to the golden ratio.
• Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ακρος και μεσος λογος).[9]
• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the Fibonacci sequence is closely related to the golden ratio.
• Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
• Johannes Kepler (1571–1630) describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel."
• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
• Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.[14]
• Edouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
• Mark Barr (20th century) uses the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.
• Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

Phidias Showing the Frieze of the Parthenon to his Friends by Sir Lawrence Alma-Tadema Phidias (or Pheidias) (in ancient Greek, ) (c. ... For other uses, see Parthenon (disambiguation). ... For other uses, see Plato (disambiguation). ... Timaeus (Honour) (or TimÃ¦us) is a name that appears in several ancient (Greek) sources: Timaeus (dialogue), a Socratic dialogue by Plato Timaeus of Locri, the 5th-century Pythagorean philosopher, appearing in Platos s Timaeus. ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... For the number sequence, see Fibonacci number. ... For other senses of this word, see sequence (disambiguation). ... Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is... Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ... Kepler redirects here. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Bonnet Charles Bonnet (March 13, 1720 â€“ May 20, 1793), Swiss naturalist and philosophical writer, was born at Geneva, of a French family driven into Switzerland by the religious persecution in the 16th century. ... In botany, phyllotaxis is the arrangement of the leaves on the shoot of a plant. ... The Clockwise direction A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... Martin Ohm (1792â€“1872) was a German mathematician and a younger brother of physicist Georg Ohm. ... FranÃ§ois Ã‰douard Anatole Lucas (April 4, 1842 in Amiens - October 3, 1891) was a French mathematician. ... This page is a candidate for speedy deletion. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... This article or section is in need of attention from an expert on the subject. ... Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. ...

## Aesthetics

Further information: List of works designed with golden ratio

Beginning in the Renaissance, a body of literature on the aesthetics of the golden ratio has developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works. This article is about the European Renaissance of the 14th-17th centuries. ... Aesthetics is commonly perceived as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste. ...

The first and most influential of these was De Divina Proportione by Luca Pacioli, a three-volume work published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that that interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[2] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by Leonardo Da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects. Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ... 1509 was a common year starting on Friday (see link for calendar) of the Gregorian calendar. ... The Order of Friars Minor and other Franciscan movements are disciples of Saint Francis of Assisi. ... A friar is a member of a religious mendicant order of men. ... â€œDa Vinciâ€ redirects here. ...

### Architecture

Some studies of the Acropolis, including the Parthenon, conclude that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by golden rectangles.[15] To the extent that classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the golden ratio and consciously employed it in their designs. Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio. On the other hand, such retrospective analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and that these choices affect the proportions observed. Acropolis (Gr. ... For other uses, see Parthenon (disambiguation). ...

Some scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 B.C.) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."[16] And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value."[17] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions. In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Keith Devlin is an English mathematician and writer. ... Marcus Vitruvius Pollio (born ca. ...

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[18] It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis. Minaret of Mosque of Uqba Mosque of Uqba (Arabic: â€Ž) is a mosque located in Kairouan, Tunisia. ... This article or section does not cite any references or sources. ...

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."[19] For other uses, see Architect (disambiguation). ... Charles-Edouard Jeanneret, who chose to be known as Le Corbusier (October 6, 1887 â€“ August 27, 1965), was a Swiss-born architect and writer, who is famous for his contributions to what now is called Modern Architecture. ... For Christian theological modernism, see Liberal Christianity and Modernism (Roman Catholicism). ... The Weissenhof Estate in Stuttgart, Germany (1927) The Weissenhof Estate in Stuttgart, Germany (1930) The International style was a major architectural style of the 1920s and 1930s. ...

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[20] // The Modulor is a scale of proportions devised by the French architect Le Corbusier (1887â€“1965). ... The concept of scale is applicable if a system is represented proportionally by another system. ... Architectural practice has often used proportional systems to generate or constrain the forms considered suitable for inclusion in a building. ... Marcus Vitruvius Pollio (born ca. ... Leonardo da Vincis Vitruvian Man (1492). ... Leone Battista Alberti (February 1404 - 25th April 1472), Italian painter, poet, linguist, philosopher, cryptographer, musician, architect, and general Renaissance polymath . ... This article is about building architecture. ... Illustration from The Speaking Portrait (Pearsons Magazine, Vol XI, January to June 1901) demonstrating the principles of Bertillons anthropometry. ... // The Modulor is a scale of proportions devised by the French architect Le Corbusier (1887â€“1965). ... Garches is a city in suburban Paris in France Sites of interest The northern part of the suburban city wsa marked by the combat of January 19, 1871 when the Parisian besieged and tried to force the German blockade to join the French troops of Versailles. ...

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[21] Mario Botta (born April 1, 1943) is a famous modern architect born in Mendrisio, Ticino canton, Switzerland. ... Origlio is a municipality in the district of Lugano in the canton of Ticino in Switzerland. ...

### Art

Leonardo Da Vinci's illustration from De Divina Proportione applies the golden ratio to the human face.

Image File history File links Download high resolution version (1000x1482, 210 KB) File links The following pages link to this file: Golden ratio Luca Pacioli Golden Mean ... Image File history File links Download high resolution version (1000x1482, 210 KB) File links The following pages link to this file: Golden ratio Luca Pacioli Golden Mean ...

#### Painting

Leonardo da Vinci's illustrations in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own paintings. Some suggest that his Mona Lisa, for example, employs the golden ratio in its geometric equivalents.[22] Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate. The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive[citation needed]. â€œDa Vinciâ€ redirects here. ... For other uses, see Mona Lisa (disambiguation). ...

Salvador Dalí explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[23][2] Salvador Domingo Felipe Jacinto DalÃ­ i DomÃ¨nech, 1st Marquis of PÃºbol (May 11, 1904 â€“ January 23, 1989), was a Spanish surrealist painter of Catalan descent born in Figueres, Catalonia (Spain). ... Completed in 1955 after nine months of work, Salvador Daliâ€™s painting The Sacrament of the Last Supper has remained one of his most popular compositions. ...

Mondrian used the golden section extensively in his geometrical paintings.[24] Piet Mondrian, 1924 Pieter Cornelis (Piet) Mondriaan, after 1912 Mondrian, (pronounced: Dutch IPA: , later Pete Mon-dree-on, IPA: ) (b. ...

Interestingly, a statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[25]

### Book design

See Canons of page construction.
Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."[26]

According to Jan Tschichold,[27] "There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimetre." Recto page from a rare Blackletter Bible (1497) In the field of book design, proportions of pages, type areas (print spaces), and margins of medieval books have been analyzed by scholars, and several canons of page construction have been described by them to represent the ways in which these books... Image File history File links Medieval_manuscript_framework. ... Image File history File links Medieval_manuscript_framework. ... Titlepage for Typographische Gestaltung written and designed by Jan Tschichold using City Medium and Bodoni. ...

### Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[2][28] Gustav Theodor Fechner (April 19, 1801 - November 28, 1887), was a German experimental psychologist. ... For beauty as a characteristic of a persons appearance, see Physical attractiveness. ...

### Music

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced. James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ... For Ann (rising) is a piece created by James Tenney in 1969. ... Glissando (plural: glissandi) is a musical term that refers to either a continuous sliding from one pitch to another (a true glissando), or an incidental scale played while moving from one melodic note to another (an effective glissando). ... Figure 1: Shepard tones forming a Shepard scale, illustrated in a sequencer A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. ... Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... A minor sixth is the smaller of two commonly occuring musical intervals that span six diatonic scale degrees. ... The musical interval of a major sixth is the relationship between the first note (the root or tonic) and the sixth note in a Major scale. ...

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[29] though other music scholars reject that analysis.[2] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[30] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry. Erno Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartoks music. ... Bartok redirects here. ... In music the acoustic scale is a seven note scale, starting on C: C D E F# G A Bb which is formed from a major triad (C E G) with an added minor seventh and raised fourth (Bb and F#, drawn from the overtone series) and major second and... Music for Strings, Percussion and Celesta is a piece of classical music by BÃ©la BartÃ³k. ... Selfportrait of Erik Satie. ...

The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."[30] Claude Debussy Claude Achille Debussy (August 22, 1862 – March 25, 1918), composer of impressionistic classical music. ...

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.[31] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[32] La Mer is an orchestral composition by the French composer Claude Debussy. ...

This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled "1.618" in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean. This Binary Universe is the fifth album by composer and electronica artist Brian Transeau (BT) released on August 29, 2006. ... Brian Wayne Transeau (born October 4, 1971 in Rockville, Maryland) is a trance musician, better known by his stage name, BT. He has been called the Father of Trance for his pioneering in the trance genre [1],[2] and Prince of Dance Music for his multi-instrumentalist skills [3], and...

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[33] The Pearl Musical Instrument Company ) is a world leader in the manufacturing of percussion equipment, including drum kits, hand drums, drum hardware, bass drum pedals, mallet percussion, and other auxiliary percussion instruments. ... For other uses, see Patent (disambiguation). ...

According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio."[34]

## Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[35] Zeising wrote in 1854:

[The Golden Ratio is a universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.[36]

See also History of aesthetics (pre-20th-century) This description of the history of aesthetics before the twentieth century is based on an article from the 1911 edition of the EncyclopÃ¦dia Britannica. ...

## Mathematics

### Golden ratio conjugate

The negative root of the quadratic equation for φ (the "conjugate root") is $1 - varphi approx -0.618$. The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[37] It is denoted here by the capital Phi (Φ):

$Phi = {1 over varphi} approx 0.61803,39887,.$

Alternatively, Φ can be expressed as

$Phi = varphi -1,.$

This illustrates the unique property of the golden ratio among positive numbers, that

${1 over varphi} = varphi - 1,$

or its inverse:

${1 over Phi} = Phi + 1,.$

### Short proofs of irrationality

Recall that we denoted the "larger part" by a and the "smaller part" by b. If the golden ratio is a positive rational number, then it must be expressible as a fraction in lowest terms in the form a / b where a and b are coprime positive integers. The algebraic definition of the golden ratio then indicates that if a / b = φ, then In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and âˆ’1, or equivalently, if their greatest common divisor is 1. ...

$frac{a}{b} = frac{a+b}{a},$.

Multiplying both sides by ab leads to:

$a^2 = ab+b^2,.$

Subtracting ab from both sides and factoring out a gives:

$a(a-b) = b^2,.$

Finally, dividing both sides by b(ab) yields:

$frac{a}{b} = frac{b}{a-b},.$

This last equation indicates that a / b could be further reduced to b / (ab), where ab is still positive, which is an equivalent fraction with smaller numerator and denominator. But since a / b was already given in lowest terms, this is a contradiction. Thus this number cannot be so written, and is therefore irrational.

Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If $textstylefrac{1 + sqrt{5}}{2}$ is rational, then $textstyle2left(frac{1 + sqrt{5}}{2} - frac{1}{2}right) = sqrt{5}$ is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational. In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

### Alternate forms

The formula $varphi = 1 + 1/varphi$ can be expanded recursively to obtain a continued fraction for the golden ratio:[38] In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

$varphi = [1; 1, 1, 1, dots] = 1 + cfrac{1}{1 + cfrac{1}{1 + cfrac{1}{1 + ddots}}}$

and its reciprocal:

$varphi^{-1} = [0; 1, 1, 1, dots] = 0 + cfrac{1}{1 + cfrac{1}{1 + cfrac{1}{1 + ddots}}},.$

The convergents of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, ..., or 1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers. A convergent is one of a sequence of rational values obtained by evaluating successive truncations of a continued fraction. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ...

The equation $varphi^2 = 1 + varphi$ likewise produces the continued square root form: In mathematics, a square root (âˆš) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

$varphi = sqrt{1 + sqrt{1 + sqrt{1 + sqrt{1 + cdots}}}},.$

Also:

$varphi = 1+2sin(pi/10) = 1 + 2sin 18^circ$
$varphi = {1 over 2}csc(pi/10) = {1 over 2}csc 18^circ$
$varphi = 2cos(pi/5)=2cos 36^circ.,$

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram. A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ...

If x agrees with $varphi$ to n decimal places, then $frac{x^2+2x}{x^2+1}$ agrees with it to 2n decimal places.

An equation derived in 1994 connects the golden ratio to the Number of the Beast (666):[2] For other uses, see Number of the Beast (disambiguation). ...

$-frac{varphi}{2}=sin666^circ=cos(6cdot 6 cdot 6^circ).$

Which can be combined into the expression:

$-varphi=sin666^circ+cos(6cdot 6 cdot 6^circ).$

This relationship depends upon the choice of the degree as the measure of angle, and will not hold when using other units of angular measure. This article describes the unit of angle. ...

### Geometry

The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. For other uses, see Geometry (disambiguation). ... Sphere symmetry group o. ... Look up pentagon in Wiktionary, the free dictionary. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... For the numeral, see 3 (number). ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ...

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ 222.5°. This approach was used to arrange mirrors on the Starshine 3 satellite[39]. Flowcharts are often used to graphically represent algorithms. ... The Thomson problem is that of determining the minimum (ground state) energy configuration of N classical electrons on the surface of the 2-sphere . The electrons repel each other with a force given by Coulombs law. ...

#### Golden triangle, pentagon and pentagram

##### Golden triangle
Golden triangle

The golden triangle can be characterised as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For the bisection theorem, see ham sandwich theorem. ... Several equivalence relations in mathematics are called similarity. ...

If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°. A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ... A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ...

Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles BC=XC and XC=XA, so these are also length φ. Length AC = AB, therefore equals φ+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ2. Thus φ2 = φ+1, confirming that φ is indeed the golden ratio.

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

Image File history File links Pentagram-phi. ... Image File history File links Pentagram-phi. ...

##### Pentagram
For more details on this topic, see Pentagram.

The golden ratio plays an important role in regular pentagons and pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows. A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ...

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomon. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ... A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ...

The golden ratio in a regular pentagon can be computed using Ptolemy's theorem.

Image File history File links Ptolemy_Pentagon. ... Image File history File links Ptolemy_Pentagon. ... In mathematics, Ptolemys theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a quadrilateral inscribed in circle. ...

##### Ptolemy's theorem

The golden ratio can also be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one vertex from a regular pentagon. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b2 = a2 + ab which yields In mathematics, Ptolemys theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a quadrilateral inscribed in circle. ...

${b over a}={{(1+sqrt{5})}over 2},.$

#### Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.[40] A triangle. ...

### Relationship to Fibonacci sequence

For more details on this topic, see Fibonacci number.
Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.
A Fibonacci spiral that approximates the golden spiral, using Fibonacci sequence square sizes up to 34.

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is: A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is... Fake spiral made of circle quarters, logarithmic spiral. ... Fake spiral made of circle quarters, logarithmic spiral. ... Approximate and true Golden Spirals. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is...

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

The explicit expression for the Fibonacci sequence involves the golden ratio:

$Fleft(nright) = {{varphi^n-(1-varphi)^n} over {sqrt 5}} = {{varphi^n-(-varphi)^{-n}} over {sqrt 5}},.$

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence): Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

$lim_{ntoinfty}frac{F(n+1)}{F(n)}=varphi.$

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

$sum_{n=1}^{infty}|F(n)varphi-F(n+1)| = varphi,.$

Furthermore, the successive powers of φ obey the Fibonacci recurrence: In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

$varphi^{n+1} = varphi^n + varphi^{n-1},.$

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

$3varphi^3 - 5varphi^2 + 4 = 3(varphi^2 + varphi) - 5varphi^2 + 4 = 3[(varphi + 1) + varphi] - 5(varphi + 1) + 4 = varphi + 2 approx 3.618,.$

### Other properties

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of the Lagrange's approximation theorem. This may be the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... In botany, phyllotaxis is the arrangement of the leaves on the shoot of a plant. ...

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:

$varphi^2 = varphi + 1 = 2.618dots,$
${1 over varphi} = varphi - 1 = 0.618dots,.$

The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally, any power of φ is equal to the sum of the two immediately preceding powers:

$varphi^n = varphi^{n-1} + varphi^{n-2} = varphi cdot operatorname{F}_n + operatorname{F}_{n-1},.$

As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ:

If $lfloor n/2 - 1 rfloor = m$, then:

$! varphi^n = varphi^{n-1} + varphi^{n-3} + cdots + varphi^{n-1-2m} + varphi^{n-2-2m}.$

When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary or φ-nary), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation. This article is about different methods of expressing numbers with symbols. ... Golden ratio base refers to the use of the golden ratio, the irrational number â‰ˆ1. ...

The golden ratio is the fundamental unit of the algebraic number field $mathbb{Q}(sqrt{5})$ and is a Pisot-Vijayaraghavan number. In mathematics, an algebraic number field (or simply number field) is a finite-dimensional (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. The study of... In mathematics, a Pisot-Vijayaraghavan number is an algebraic integer Î± which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value. ...

### Decimal expansion

The golden ratio's decimal expansion can be calculated directly from the expression

$varphi = {1+sqrt{5} over 2},$

with √5 ≈ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as x1 = 2 and iterating This article presents and explains several methods which can be used to calculate square roots. ... It has been suggested that this article or section be merged with Guess value. ...

$x_{n+1} = frac{(x_n + 5/x_n)}{2}$

for n = 1, 2, 3, ..., until the difference between xn and xn−1 becomes zero, to the desired number of digits.

The Babylonian algorithm for √5 is equivalent to Newton's method for solving the equation x2 − 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation, including the equation x2 − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself, In numerical analysis, Newtons method (also known as the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

$x_{n+1} = frac{x_n^2 + 1}{2x_n - 1},$

for an appropriate initial estimate x1 such as x1 = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes

$x_{n+1} = frac{x_n^2 + 2x_n}{x_n^2 + 1}.$

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers π and e. In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ... On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers or rational numbers (including floating-point numbers) with an arbitrary number of digits of precision, typically limited only by the available memory of the host system. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F25001 and F25000, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

Millions of digits of φ are available (sequence A001622 in OEIS). The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

## Pyramids

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of $1:sqrt{varphi}:varphi$ and $3:4:5$ and $1:4/pi:1.61899$ are of particular interest in relation to Egyptian pyramids.

Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios. Image File history File links Mathematical_Pyramid. ... Image File history File links Mathematical_Pyramid. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ...

### Mathematical pyramids and triangles

A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is $sqrt{varphi}$ times the semi-base (that is, the slope of the face is $sqrt{varphi}$); the square of the height is equal to the area of a face, φ times the square of the semi-base.

The medial right triangle of this "golden" pyramid (see diagram), with sides $1:sqrt{varphi}:varphi$ is interesting in its own right, demonstrating via the Pythagorean theorem the relationship $sqrt{varphi} = sqrt{varphi^2 - 1}$ or $varphi = sqrt{1 + varphi}$. This "Kepler triangle"[41] is the only right triangle proportion with edge lengths in geometric progression,[42] just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent $sqrt{varphi}$ corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").[43] For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio. ... Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;[44] the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).[45] The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,[46] and the rational inverse slope (run/rise, mutliplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.[44] The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ... This article or section is in need of attention from an expert on the subject. ...

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the golden triangle. This pyramid relationship corresponds to the coincidental relationship $sqrt{varphi} approx 4/pi$. A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ...

Egyptian pyramids very close in proportion to these mathematical pyramids are known.[45]

### Egyptian pyramids

The shapes of Egyptian pyramids include one that is remarkably close to a "golden pyramid". This is the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47')[44] are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains a topic of controversy. Several other Egyptian pyramids are very close to the rational 3:4:5 shape.[45] The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now Cairo, Egypt in Africa, and is the only remaining member of the Seven Wonders of the World. ...

Michael Rice[47] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the Golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957).[48] He also asserts that some recent historians of science have denied that the Egyptians had any such knowledge, contending rather that its appearance in an Egyptian building is the result of chance. For at least ten thousand years, the Nile valley has been the site of one of the most influential civilizations in the world. ...

In 1859, the Pyramidologist John Taylor (1781-1864) asserted that in the Great Pyramid of Giza built around 2600 BC, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ.[49] The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Pyramidology is a term used to refer to alternative scientific theories regaring pyramids. ... John Taylor was a publisher, essayist, lawyer, soldier, politician and writer born in East Retford, Nottinghamshire in 1781 and died in 1864. ... The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now Cairo, Egypt in Africa, and is the only remaining member of the Seven Wonders of the World. ... (Redirected from 2600 BC) (27th century BC - 26th century BC - 25th century BC - other centuries) (4th millennium BC - 3rd millennium BC - 2nd millennium BC) Events 2900 - 2334 BC – Mesopotamian wars of the Early Dynastic period. ... Note: A theta probe is a device for measuring soil moisture. ... Secant is a term in mathematics. ...

Howard Vyse, according to Matila Ghyka,[50] reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability. Colonel Richard William Howard Vyse (1784 â€“ 1872) was a British soldier, anthropologist and Egyptologist. ...

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, asserts that Egyptian mathematics as understood in modern times, would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians, and they did not know the Pythagorean theorem nor any way to reason about irrationals such as π or φ.[51] For other persons named Eric Bell, see Eric Bell (disambiguation). ...

## Disputed sightings of the golden ratio

Examples of disputed observations of the golden ratio include the following:

• Historian John Man states that the pages of the Gutenberg Bible were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.[52]
The sculpture Ratio by Andrew Rogers in Jerusalem is proportioned according to Fibonacci numbers; some call it Golden Ratio.
• Australian sculptor Andrew Rogers's 50-ton stone and gold sculpture entitled Ratio, installed outdoors in Jerusalem.[55] Despite the sculpture's sometimes being referred to as "Golden Ratio,"[56] it is not proportioned according to the golden ratio, and the sculptor does not call it that: the height of each stack of stones, beginning from either end and moving toward the center, is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8. His sculpture Ascend in Sri Lanka, also in his Rhythms of Life series, is similarly constructed, with heights 1, 1, 2, 3, 5, 8, 13, but no descending side.[55]
• It is sometimes claimed that the number of bees in a beehive divided by the number of drones yields the golden ratio.[57] In reality, the proportion of drones in a beehive varies greatly by beehive, by bee race, by season, and by beehive health status; the ratio is normally much greater than the golden ratio (usually close to 20:1 in healthy colonies).[citation needed] This misunderstanding may arise because in theory bees have approximately this ratio of male to female ancestors (See The Bee Ancestry Code) - the caveat being that ancestry can trace back to the same drone by more than one route, so the actual numbers of bees do not need to match the formula.
• Some specific proportions in the bodies of many animals (including humans[58][59]) and parts of the shells of mollusks[4] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in a specific individual and the proportion in question is often significantly different from the golden ratio.[58] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[59] The Nautilus shell, whose construction proceeds in a logarithmic spiral, is often cited, usually under the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one.[57]
• The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species.[60] In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.[citation needed]
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[61] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers; see, e.g. Elliott wave principle. However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[62]

Aesthetics is commonly perceived as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste. ... The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section... In mathematics, the golden function is the upper branch of the hyperbola In functional form, Once gold(x) has been defined, the lower branch of the hyperbola can also be defined as y = âˆ’gold(âˆ’x). ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ... A golden triangle is a triangle in which two of the sides have equal length and in which the ratio of this length to that of the third, smaller side is the golden ratio. ... Diagram of a golden section search The Golden section search is a technique for finding the extremum (minimum or maximum) of a mathematical function, by successively narrowing brackets by upper bounds and lower bounds. ... Look up Î¦, Ï† in Wiktionary, the free dictionary. ... A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is... // The Modulor is a scale of proportions devised by the French architect Le Corbusier (1887â€“1965). ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... The roses of Heliogabalus by Alma-Tadema (1888), oil on canvas. ... The plastic number (also known as the plastic constant or silver number) is the unique real solution of the equation and has the value which is approximately 1. ... A Penrose tiling is pattern of tiles, discovered by Roger Penrose, which could completely cover an infinite surface, but only in a pattern which is non-repeating (aperiodic). ... Dynamic symmetry is a proportioning system originated from the classical Greek period. ... Golden ratio base refers to the use of the golden ratio, the irrational number â‰ˆ1. ... Leonardo da Vincis Vitruvian Man (1492). ... The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. ... The silver ratio is a mathematical constant. ...

## References and footnotes

1. ^ a b The golden ratio can be derived by the quadratic formula, by starting with the first number as 1, then solving for 2nd number x, where the ratios [x+1]/x = x/1 or (multiplying by x) yields: x+1 = x2, or thus a quadratic equation: x2-x-1=0. Then, by the quadratic formula, for positive x = [-b + sqrt(b2-4ac)]/2a with a=1, b=-1, c=-1, the solution for x is: [-(-1) + sqrt([-1]2 -4*1*-1)]/2*1 or [1 + sqrt(5) ]/2.
2. ^ a b c d e f Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. ISBN 0-7679-0815-5.
3. ^ Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: Associated University Presses, 1996
4. ^ a b Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
5. ^ Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) pp.37 . "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."
6. ^ Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920
7. ^ William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003
8. ^ Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
9. ^ a b c Euclid, Elements, Book 6, Definition 3.
10. ^ Euclid, Elements, Book 6, Proposition 30.
11. ^ Euclid, Elements, Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.
12. ^ The Golden Ratio. The MacTutor History of Mathematics archive. Retrieved on 2007-09-18.
13. ^ Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling, pp. 20–21. ISBN 1-4027-3522-7.
14. ^ Underwood Dudley (1999). Die Macht der Zahl: Was die Numerologie uns weismachen will. Springer, p.245. ISBN 3764359781.
15. ^ Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", Philosophical Polemic Communication Quarterly, Vol. 46, 1998.
16. ^ Midhat J. Gazalé , Gnomon, Princeton University Press, 1999. ISBN 0-691-00514-1
17. ^ Keith J. Devlin The Math Instinct: Why You're A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs) New York: Thunder's Mouth Press, 2005, ISBN 1-56025-672-9
18. ^ Boussora, Kenza and Mazouz, Said, The Use of the Golden Section in the Great Mosque of Kairouan, Nexus Network Journal, vol. 6 no. 1 (Spring 2004), Available online
19. ^ Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6
20. ^ Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".
21. ^ Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X
22. ^ Livio, Mario. The golden ratio and aesthetics. Retrieved on 2008-03-21.
23. ^ Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X
24. ^ Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp.247-8, Harcourt, Brace & World, ISBN 0-87817-259-9
25. ^ Olariu, Agata, Golden Section and the Art of Painting Available online
26. ^ Ibid. Tschichold, pp.43 Fig 4. "Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well."
27. ^ Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.
28. ^ The golden ratio and aesthetics, by Mario Livio
29. ^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
30. ^ a b Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X
31. ^ Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 0521311454.
32. ^ Simon Trezise (1994). Debussy: La Mer. Cambridge University Press, p.53. ISBN 0521446562.
33. ^ Pearl Masters Premium. Pearl Corporation. Retrieved on Dec. 2, 2007.
34. ^ Leon Harkleroad (2006). The Math Behind the Music. Cambridge University Press. ISBN 0521810957.
35. ^ Ibid. Padovan, R. Proportion: Science, Philosophy, Architecture , pp. 305-06
36. ^ Zeising, Adolf, Neue Lehre van den Proportionen des meschlischen Körpers, Leipzig, 1854, preface.
37. ^ Eric W. Weisstein, Golden Ratio Conjugate at MathWorld.
38. ^ Max. Hailperin, Barbara K. Kaiser, and Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. ISBN 0534952119.
39. ^ A Disco Ball in Space. NASA (2001-10-09). Retrieved on 2007-04-16.
40. ^ American Mathematical Monthly, pp. 49-50, 1954.
41. ^ (2006) The Best of Astraea: 17 Articles on Science, History and Philosophy. Astrea Web Radio. ISBN 1425970400.
42. ^ Roger Herz-Fischler (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0889203245.
43. ^ Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999
44. ^ a b c Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000
45. ^ a b c The Great Pyramid, The Great Discovery, and The Great Coincidence. Retrieved on 2007-11-25.
46. ^ Lancelot Hogben, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56
47. ^ Rice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp. 24 Routledge, 2003, ISBN 0-415-26876-1
48. ^ S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp.24 Routledge, 2003
49. ^ Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859
50. ^ Matila Ghyka The Geometry of Art and Life, New York: Dover, 1977
51. ^ Eric Temple Bell, The Development of Mathematics, New York: Dover, 1940, p.40
52. ^ Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166-67, Wiley, ISBN 0-471-21823-5. "The half-folio page (30.7 x 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8."
53. ^ J.C. Perez (1991), "Chaos DNA and Neuro-computers: A Golden Link", in Speculations in Science and Technology vol. 14 no. 4, ISSN 0155-7785
54. ^ Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), "Nucleotide Frequencies in Human Genome and Fibonacci Numbers", in Bulletin of Mathematical Biology, ISSN 0092-8240 (print), ISSN 1522-9602 (online).
55. ^ a b Sculptures by Andrew Rogers.
56. ^ "Golden Ratio" in Jerusalem.
57. ^ a b Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004
58. ^ a b Stephen Pheasant, Bodyspace, London: Taylor & Francis, 1998
59. ^ a b Walter van Laack, A Better History Of Our World: Volume 1 The Universe, Aachen: van Laach GmbH, 2001.
60. ^ Derek Thomas, Architecture and the Urban Environment: A Vision for the New Age, Oxford: Elsevier, 2002
61. ^ For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates". Federal Reserve Bank of New York Economic Policy Review 6 (2): 53–68.
62. ^ Roy Batchelor and Richard Ramyar, "Magic numbers in the Dow," 25th International Symposium on Forecasting, 2005, p. 13, 31. "Not since the 'big is beautiful' days have giants looked better", Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and "Technical failure", The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.

In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 261st day of the year (262nd in leap years) in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance to the Gregorian calendar. ... is the 80th day of the year (81st in leap years) in the Gregorian calendar. ... Titlepage for Typographische Gestaltung written and designed by Jan Tschichold using City Medium and Bodoni. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Year 2001 (MMI) was a common year starting on Monday (link displays the 2001 Gregorian calendar). ... is the 282nd day of the year (283rd in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 106th day of the year (107th in leap years) in the Gregorian calendar. ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 329th day of the year (330th in leap years) in the Gregorian calendar. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... Roy A. Batchelor is HSBC Professor of Banking and Finance in the Sir John Cass Business School of the City University, London. ... This article concerns the British newspaper. ... The Economist is an English-language weekly news and international affairs publication owned by The Economist Newspaper Ltd and edited in London. ...

• Perez, Jean-claude (1997). L'ADN décrypté. Embourg (Belgium): Marco Pietteur. ISBN 2-87211-017-8.
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• Euclid [c. 300 BC] (David E. Joyce, ed. 1997). Elements. Retrieved on 2006-08-30.  Citations in the text are to this online edition.
• Ghyka, Matila [1946] (1977). The Geometry of Art and Life, reprint of 1946 ed., slightly corrected, New York: Dover Publications. ISBN 0-486-23542-4.
• Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Proportion. New York: Dover Publications. ISBN 0-486-22254-3.
• Joseph, George G. [1991] (2000). The Crest of the Peacock: The Non-European Roots of Mathematics, New ed., Princeton, NJ: Princeton University Press. ISBN 0-691-00659-8.
• Plato (360 BC) (Benjamin Jowett trans.). Timaeus. The Internet Classics Archive. Retrieved on May 30, 2006.
• Schneider, Michael S. (1994). A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. ISBN 0-060-16939-7.
• Walser, Hans [Der Goldene Schnitt 1993] (2001). The Golden Section, Peter Hilton trans., Washington, DC: The Mathematical Association of America. ISBN 0-88385-534-8.
• Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Random House, Inc.. ISBN 0-7679-0815-5.
• Sahlqvist, Leif (2005). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design.. Sweden: Booksurge Llc.. ISBN-13: 978-1419621574.

Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 242nd day of the year (243rd in leap years) in the Gregorian calendar. ... For other uses, see Plato (disambiguation). ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ...

Results from FactBites:

 Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence (497 words) A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio. The ratio of the lengths of the two parts of this segment is the Golden Ratio. The Golden Ratio is the ratio of BC to AB.
 Golden ratio - Wikipedia, the free encyclopedia (3564 words) The philosophy of Summum, a sect of about 200,000 adherents that has incorporated the golden ratio into the design of their Summum Pyramid winery in Utah, maintains that because it is the human mind that interprets the characteristics and qualities of the golden ratio, it should be considered in its relation to the human psyche. Nevertheless, some of these ratios are observed to be quite close to the golden ratio in the shape of the organs or parts which closely follow some basic geometrical shape (such as the Nautilus shell, whose construction proceeds in a logarithmic spiral). From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem.
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