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PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information.PDF generated at: Wed, 13 Apr 2011 19:39:05 UTC

The Divine RatioDesign Structure

ContentsArticles

Golden Ratio 1

Golden ratio 1

Phidias 21

Irrational number 25

Golden rectangle 33

Golden spiral 35

Golden angle 37

Golden rhombus 39

Logarithmic spiral 40

Canons of page construction 45

List of works designed with the golden ratio 50

ReferencesArticle Sources and Contributors 58

Image Sources, Licenses and Contributors 60

1

Golden Ratio

Golden ratio

The golden section is a line segment divided according to thegolden ratio: The total length a + b is to the length of thelonger segment a as the length of a is to the length of the

shorter segment b.

In mathematics and the arts, two quantities are in the goldenratio if the ratio of the sum of the quantities to the largerquantity is equal to the ratio of the larger quantity to thesmaller one. The golden ratio is an irrational mathematicalconstant, approximately 1.6180339887.[1] Other namesfrequently used for the golden ratio are the golden section(Latin: sectio aurea) and golden mean.[2] [3] [4] Other termsencountered include extreme and mean ratio,[5] medialsection, divine proportion, divine section (Latin: sectiodivina), golden proportion, golden cut,[6] golden number,and mean of Phidias.[7] [8] [9] In this article the golden ratiois denoted by the Greek lowercase letter phi ( ) , while itsreciprocal, or , is denoted by the uppercase

variant Phi ( ).

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has one positive solution in the set of algebraic irrational numbers:

[1]

At least since the Renaissance, many artists and architects have proportioned their works to approximate the goldenratioÄespecially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the goldenratioÄbelieving this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because ofits unique and interesting properties.

Golden ratio 2

Construction of a golden rectangle:1. Construct a unit square (red).

2. Draw a line from the midpoint of one side to anopposite corner.

3. Use that line as the radius to draw an arc thatdefines the long dimension of the rectangle.

A golden rectangle with longer side a and shorterside b, when placed adjacent to a square with

sides of length a, will produce a similar goldenrectangle with longer side a + b and shorter side

a. This illustrates the relationship

List of numbers Å Irrational and suspected irrational numbersÄ Å Å(3) Å Ç2 Å Ç3 Å Ç5 Å Ç Å É Å ÑS Å Ö Å e Å Ü Å Ñ

Binary 1.1001111000110111011É

Decimal 1.6180339887498948482É

Continued fraction

Algebraic form

Infinite series

Two quantities a and b are said to be in the golden ratio Ä if:

This equation unambiguously defines Ä.

The fraction on the left can be converted to

Multiplying through by Ä produces

which can be rearranged to

Golden ratio 3

The only positive solution to this quadratic equation is

Also interestingly enough,

The Product of their sum and differences is Equal to the Product of the numerals themselves.

History

Mathematician Mark Barr proposed using the firstletter in the name of Greek sculptor Phidias, phi, tosymbolize the golden ratio. Usually, the lowercase

form (Ç) is used. Sometimes, the uppercase form (á) isused for the reciprocal of the golden ratio, 1/Ç.[10]

The golden ratio has fascinated Western intellectuals of diverseinterests for at least 2,400 years. According to Mario Livio:

Some of the greatest mathematical minds of all ages, fromPythagoras and Euclid in ancient Greece, through themedieval Italian mathematician Leonardo of Pisa and theRenaissance astronomer Johannes Kepler, to present-dayscientific figures such as Oxford physicist Roger Penrose,have spent endless hours over this simple ratio and itsproperties. But the fascination with the Golden Ratio is notconfined just to mathematicians. Biologists, artists,musicians, historians, architects, psychologists, and evenmystics have pondered and debated the basis of its ubiquityand appeal. In fact, it is probably fair to say that the GoldenRatio has inspired thinkers of all disciplines like no othernumber in the history of mathematics.[11]

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance ingeometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry ofregular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or hisfollowers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid's Elements (Greek: àâäãåçéÖ) provides the first known written definition of what is now called the goldenratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greatersegment, so is the greater to the less."[5] Euclid explains a construction for cutting (sectioning) a line "in extreme andmean ratio", i.e. the golden ratio.[12] Throughout the Elements, several propositions (theorems in modernterminology) and their proofs employ the golden ratio.[13] Some of these propositions show that the golden ratio is anirrational number.

The name "extreme and mean ratio" was the principal term used from the 3rd century BC[5] until about the 18thcentury.

The modern history of the golden ratio starts with Luca Pacioli's De divina proportione of 1509, which captured theimagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of thegolden ratio.

Golden ratio 4

Michael Maestlin, first to publish a decimalapproximation of the golden ratio, in 1597.

The first known approximation of the (inverse) golden ratio by adecimal fraction, stated as "about 0.6180340," was written in 1597 byProf. Michael Maestlin of the University of Tèbingen in a letter to hisformer student Johannes Kepler.[14]

Since the twentieth century, the golden ratio has been represented bythe Greek letter Ä or Å (phi, after Phidias, a sculptor who is said tohave employed it) or less commonly by Ç (tau, the first letter of theancient Greek root âäêëÄmeaning cut).[2] [15]

Timeline

Timeline according to Priya Hemenway.[16]

í Phidias (490Å430 BC) made the Parthenon statues that seem toembody the golden ratio.

í Plato (427Å347 BC), in his Timaeus, describes five possible regularsolids (the Platonic solids: the tetrahedron, cube, octahedron,dodecahedron and icosahedron), some of which are related to thegolden ratio.[17]

í Euclid (c. 325Åc. 265 BC), in his Elements, gave the first recordeddefinition of the golden ratio, which he called, as translated intoEnglish, "extreme and mean ratio" (Greek: ìîÉäï îÖñ êóòäïôöÄäï).[5]

í Fibonacci (1170Å1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio ofsequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.

í Luca Pacioli (1445Å1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.í Johannes Kepler (1571Å1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci

numbers,[18] and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is theTheorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compareto a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Keplertriangle.

í Charles Bonnet (1720Å1793) points out that in the spiral phyllotaxis of plants going clockwise andcounter-clockwise were frequently two successive Fibonacci series.

í Martin Ohm (1792Å1872) is believed to be the first to use the term goldener Schnitt (golden section) to describethis ratio, in 1835.[19]

í Edouard Lucas (1842Å1891) gives the numerical sequence now known as the Fibonacci sequence its presentname.

í Mark Barr (20th century) suggests the Greek letter phi (Ä), the initial letter of Greek sculptor Phidias's name, as asymbol for the golden ratio.[20]

í Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodictilings, which led to new discoveries about quasicrystals.

Golden ratio 5

Applications and observations

AestheticsDe Divina Proportione, a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, wasknown mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportioneexplored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio'sapplication to yield pleasing, harmonious proportions, Livio points out that that interpretation has been traced to anerror in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[2] Pacioli also sawCatholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids byLeonardo Da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence ongenerations of artists and architects alike.

Architecture

Many of the proportions of the Parthenon are alleged to exhibit thegolden ratio.

The Parthenon's facade as well as elements of its facadeand elsewhere are said by some to be circumscribed bygolden rectangles.[21] Other scholars deny that theGreeks had any aesthetic association with golden ratio.For example, Midhat J. Gazalõ says, "It was not untilEuclid, however, that the golden ratio's mathematicalproperties were studied. In the Elements (308 BC) theGreek mathematician merely regarded that number asan interesting irrational number, in connection with themiddle and extreme ratios. Its occurrence in regularpentagons and decagons was duly observed, as well asin the dodecahedron (a regular polyhedron whosetwelve faces are regular pentagons). It is indeedexemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that numberfor what it is, without attaching to it other than its factual properties."[22] And Keith Devlin says, "Certainly, the oftrepeated 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 knowfor sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate itsvalue."[23] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in wholenumbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratiothroughout the design, according to Boussora and Mazouz.[24] It is found in the overall proportion of the plan and inthe dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlierarchaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio byapplying these constructions to the plan of the mosque to test their hypothesis.

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his designphilosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universewas closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eyeand clear in their relations with one another. And these rhythms are at the very root of human activities. Theyresound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the GoldenSection by children, old men, savages and the learned."[25]

Golden ratio 6

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He sawthis system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work ofLeon Battista Alberti, and others who used the proportions of the human body to improve the appearance andfunction 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 toan extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, thensubdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in theModulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. Thevilla's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[26]

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses hedesigned 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.[27]

In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e JahanSquare and the adjacent Lotfollah mosque.[28]

Painting

The drawing of a man's body in a pentagramsuggests relationships to the golden ratio.

The 16th-century philosopher Heinrich Agrippa drew a man over apentagram inside a circle, implying a relationship to the golden ratio.

Golden ratio 7

Illustration from Luca Pacioli's De DivinaProportione applies geometric proportions to the

human face.

Leonardo da Vinci's illustrations of polyhedra in De DivinaProportione (On the Divine Proportion) and his views that somebodily proportions exhibit the golden ratio have led some scholars tospeculate that he incorporated the golden ratio in his paintings.[29] Butthe suggestion that his Mona Lisa, for example, employs golden ratioproportions, is not supported by anything in Leonardo's ownwritings.[30]

Salvador Dalú, influenced by the works of Matila Ghyka,[31] explicitlyused the golden ratio in his masterpiece, The Sacrament of the LastSupper. The dimensions of the canvas are a golden rectangle. A hugedodecahedron, in perspective so that edges appear in golden ratio toone another, is suspended above and behind Jesus and dominates thecomposition.[2] [32]

Mondrian has been said to have used the golden section extensively inhis geometrical paintings,[33] though other experts (including criticYve-Alain Bois) have disputed this claim.[2]

A statistical study on 565 works of art of different great painters,performed in 1999, found that these artists had not used the goldenratio in the size of their canvases. The study concluded that the averageratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to1.46 (Bellini).[34] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4,and 6.[35]

Book design

Depiction of the proportions in a medievalmanuscript. According to Jan Tschichold: "Pageproportion 2:3. Margin proportions 1:1:2:3. Text

area proportioned in the Golden Section."[36]

According to Jan Tschichold,[37]

There was a time when deviations from the truly beautifulpage proportions 2:3, 1:Ç3, and the Golden Section wererare. Many books produced between 1550 and 1770 showthese proportions exactly, to within half a millimetre.

Perceptual studies

Studies by psychologists, starting with Fechner, have been devised totest the idea that the golden ratio plays a role in human perception ofbeauty. While Fechner found a preference for rectangle ratios centeredon the golden ratio, later attempts to carefully test such a hypothesishave been, at best, inconclusive.[2] [38]

Golden ratio 8

MusicJames Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardlyglissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equaltempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutivetones are a lower or higher pitch already, or soon to be, produced.

Ernù Lendvai analyzes Bõla Bartûk's works as being based on two opposing systems, that of the golden ratio and theacoustic scale,[39] though other music scholars reject that analysis.[2] In Bartok's Music for Strings, Percussion andCelesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[40] French composer Erik Satie used thegolden ratio in several of his pieces, including Sonneries de la Rose+Croix.

The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau(Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals34, 21, 13 and 8, and the main climax sits at the phi position."[40]

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the goldensection.[41] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidencesuggests that Debussy consciously sought such proportions.[42] Also, many works of Chopin, mainly Etudes (studies)and Nocturnes, are formally based on the golden ratio. This results in the biggest climax of both musical expressionand technical difficulty after about 2/3 of the piece.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claimsthat this arrangement improves bass response and has applied for a patent on this innovation.[43]

In the opinion of author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematicshave involved Fibonacci numbers and the related golden ratio."[44]

Industrial designSome sources claim that the golden ratio is commonly used in everyday design, for example in the shapes ofpostcards, playing cards, posters, wide-screen televisions, photographs, and light switch plates.[45] [46] [47] [48]

NatureAdolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in thearrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletonsof animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometryof crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operatingas a universal law.[49] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrotein 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty andcompleteness 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 fullestrealization, however, in the human form."[50]

In 2003, Volkmar Weiss and Harald Weiss analyzed psychometric data and theoretical considerations and concludedthat the golden ratio underlies the clock cycle of brain waves.[51] In 2008 this was empirically confirmed by a groupof neurobiologists.[52]

In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance ofspins in cobalt niobate crystals.[53]

Several researchers have proposed connections between the golden ratio and human genome DNA.[54] [55] [56]

Golden ratio 9

Mathematics

Golden ratio conjugateThe negative root of the quadratic equation for Ä (the "conjugate root") is

The absolute value of this quantity (Ñ 0.618) corresponds to the length ratio taken in reverse order (shorter segmentlength over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[10] It is denotedhere by the capital Phi (Å):

Alternatively, Å can be expressed as

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

or its inverse:

This means 0.61803...:1 = 1:1.61803...

Short proofs of irrationality

Contradiction from an expression in lowest terms

Recall that:

the whole is the longer part plus the shorter part;

the whole is to the longer part as the longer part is to the shorter part.

If we call the whole n and the longer part m, then the second statement above becomes

n is to m as m is to nüÖüm,

or, algebraically

To say that Ä is rational means that Ä is a fraction n/m where n and m are integers. We may take n/m to be in lowestterms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(nüÖüm) isin still lower terms. That is a contradiction that follows from the assumption that Ä is rational.

Golden ratio 10

Derivation from irrationality of Ä5

Another short proofÄperhaps more commonly knownÄof the irrationality of the golden ratio makes use of theclosure of rational numbers under addition and multiplication. If is rational, then is

also rational, which is a contradiction if it is already known that the square root of a non-square natural number isirrational.

Alternate formsThe formula Ä = 1 + 1/Ä can be expanded recursively to obtain a continued fraction for the golden ratio:[57]

and its reciprocal:

The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, É , or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, É) areratios of successive Fibonacci numbers.

The equation Ä2 = 1 + Ä likewise produces the continued square root, or infinite surd, form:

An infinite series can be derived to express phi:[58]

Also:

These correspond to the fact that the length of the diagonal of a regular pentagon is Ç times the length of its side, andsimilar relations in a pentagram.

Golden ratio 11

GeometryThe number Ç turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of aregular pentagon's diagonal is Ç times its side. The vertices of a regular icosahedron are those of three mutuallyorthogonal golden rectangles.

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of severaldefinitions of even distribution (see, for example, Thomson problem). However, a useful approximation results fromdividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by agolden section of the circle, i.e. 360†/Ç Ü 222.5†. This method was used to arrange the 1500 mirrors of thestudent-participatory satellite Starshine-3.[59]

Golden triangle, pentagon and pentagram

Golden triangle

Golden triangle

The golden triangle can be characterized as an isosceles triangle ABCwith the property that bisecting the angle C produces a new triangleCXB which is a similar triangle to the original.

If angle BCX = Ö, then XCA = Ö because of the bisection, and CAB =Ö because of the similar triangles; ABC = 2Ö from the originalisosceles symmetry, and BXC = 2Ö by similarity. The angles in atriangle add up to 180†, so 5Ö = 180, giving Ö = 36†. So the angles ofthe golden triangle are thus 36†-72†-72†. The angles of the remainingobtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36†-36†-108†.

Suppose XB has length 1, and we call BC length Ç. Because of the isosceles triangles BC=XC and XC=XA, so theseare also length Ç. Length AC = AB, therefore equals Ç+1. But triangle ABC is similar to triangle CXB, so AC/BC =BC/BX, and so AC also equals Ç2. Thus Ç2 = Ç+1, confirming that Ç is indeed the golden ratio.

Similarly (pun is inherent), the ratio of the area of the larger triangle to the smaller is equal to Ç, while the inverseratio is 1 - Ç. The proof is left to the reader, utilizing the links in the previous sentence.

A pentagram colored to distinguish its linesegments of different lengths. The four lengths

are in golden ratio to one another.

Pentagram

The golden ratio plays an important role in regular pentagons andpentagrams. Each intersection of edges sections other edges in thegolden ratio. Also, the ratio of the length of the shorter segment to thesegment bounded by the 2 intersecting edges (a side of the pentagon inthe pentagram's center) is Ç, as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and fiveobtuse isosceles triangles. In all of them, the ratio of the longer side tothe shorter side is Ç. The acute triangles are golden triangles. Theobtuse isosceles triangles are golden gnomon.

Golden ratio 12

Ptolemy's theorem

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

The golden ratio can also be confirmed by applying Ptolemy's theoremto the quadrilateral formed by removing one vertex from a regularpentagon. If the quadrilateral's long edge and diagonals are b, and shortedges are a, then Ptolemy's theorem gives b2ü=üa2ü+üab which yields

Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to bethe smaller of the two ratios a/b and b/c. The scalenity is always less than Ç and can be made as close as desired toÇ.[60]

Triangle whose sides form a geometric progression

If the side lengths of a triangle form a geometric progression and are in the ratio 1 : r : r2, where r is the commonratio, then r must lie in the range ÇÖ1 < r < Ç , which is a consequence of the triangle inequality (the sum of any twosides of a triangle must be strictly bigger than the length of the third side). If r = Ç then the shorter two sides are 1and Ç but their sum is Ç2, thus r < Ç. A similar calculation shows that r> ÇÖ1. A triangle whose sides are in the ratio1 : ÇÇ : Ç is a right triangle (because 1 + Ç = Ç2) known as a Kepler triangle.[61]

Golden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron's rhombi

A golden rhombus is a rhombus whose diagonals are in the goldenratio. The rhombic triacontahedron is a convex polytope that has a veryspecial property: all of its faces are golden rhombi. In the rhombictriacontahedron the dihedral angle between any two adjacent rhombi is144†, which is twice the isosceles angle of a golden triangle and fourtimes its most acute angle.

Relationship to Fibonacci sequence

The mathematics of the golden ratio and of the Fibonacci sequence areintimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ÉThe closed-form expression (known as Binet's formula, even though it was already known by Abraham de Moivre)for the Fibonacci sequence involves the golden ratio:

Golden ratio 13

The dual polyhedron of the Catalan solid ofgolden rhombi

A Fibonacci spiral which approximates the golden spiral, usingFibonacci sequence square sizes up to 34.

The golden ratio is the limit of the ratios of successiveterms of the Fibonacci sequence (or any Fibonacci-likesequence), as originally shown by Kepler:[18]

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximatesÇ; e.g., 987/610üÑü1.6180327868852. These approximations are alternately lower and higher than Ç, and convergeon Ç as the Fibonacci numbers increase, and:

More generally:

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when .

Furthermore, the successive powers of Ç obey the Fibonacci recurrence:

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

Golden ratio 14

However, this is no special property of Ç, because polynomials in any solution x to a quadratic equation can bereduced in an analogous manner, by applying:

for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rationalcoefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial ofdegree n á 1. Phrased in terms of field theory, if Ö is a root of an irreducible nth-degree polynomial, then has

degree n over , with basis .

Symmetries

The golden ratio and inverse golden ratio have a set of symmetries that preserve and

interrelate them. They are both preserved by the fractional linear transformations Åthis fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the threemaps Å they are reciprocals, symmetric about , and (projectively) symmetric about

2.More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on

3 letters, corresponding to the stabilizer of the set of 3 standard points on the projective line, and

the symmetries correspond to the quotient map Å the subgroup consisting of the 3-cycles andthe identity fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

Other propertiesThe golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of anyirrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange'sapproximation theorem. This may be the reason angles close to the golden ratio often show up in phyllotaxis (thegrowth of plants).

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractionalpart in common with Ç:

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

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

If , then:

When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary orÄ-nary), every integer has a terminating representation, despite Ç being irrational, but every fraction has anon-terminating representation.

Golden ratio 15

The golden ratio is a fundamental unit of the algebraic number field and is a PisotÅVijayaraghavan

number.[62]

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an idealtriangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by thepoints of tangency of a circle inscribed within the ideal triangle, is 4ülnüÇ.[63]

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

with Ç5 Ñ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method, startingwith an initial estimate such as xÇ = 2 and iterating

for n = 1, 2, 3, É, until the difference between xn and xnÖ1 becomes zero, to the desired number of digits.

The Babylonian algorithm for Ç5 is equivalent to Newton's method for solving the equation x2üÖü5 = 0. In its moregeneral form, Newton's method can be applied directly to any algebraic equation, including the equation x2üÖüxüÖü1 =0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

for an appropriate initial estimate xÇ such as xÇ = 1. A slightly faster method is to rewrite the equation as xüÖü1üÖü1/x= 0, in which case the Newton iteration becomes

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. Thegolden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits ofthe golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster thanknown algorithms for the transcendental numbers Ç and e.

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonaccinumbers and divide them. The ratio of Fibonacci numbers F25001 and F25000, each over 5000 digits, yields over10,000 significant digits of the golden ratio.

The golden ratio Ä has been calculated to an accuracy of several millions of decimal digits (sequence A001622 [64] inOEIS). Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.[65]

Golden ratio 16

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. Mathematicalproportions b:h:a of and and

are of particular interest in relation to

Egyptian pyramids.

Both Egyptian pyramids and those mathematicalregular square pyramids that resemble them can beanalyzed with respect to the golden ratio and otherratios.

Mathematical pyramids and triangles

A pyramid in which the apothem (slant height along thebisector of a face) is equal to Ç times the semi-base(half the base width) is sometimes called a goldenpyramid. The isosceles triangle that is the face of sucha pyramid can be constructed from the two halves of adiagonally split golden rectangle (of size semi-base byapothem), joining the medium-length edges to make theapothem. The height of this pyramid is times the

semi-base (that is, the slope of the face is ); the

square of the height is equal to the area of a face, Çtimes the square of the semi-base.

The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own

right, demonstrating via the Pythagorean theorem the relationship or . This

"Kepler triangle"[66] is the only right triangle proportion with edge lengths in geometric progression,[61] just as the3Å4Å5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle withtangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827Édegrees (51† 49' 38").[67]

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (thesource of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;[68] theface slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).[69] The slantheight or apothem is 5/3 or 1.666É 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 irrationalnumbers,[70] and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional unitsof palms per cubit) was used in the building of pyramids.[68]

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equalto 2Ü times the height, or h:b = 4:Ü. This triangle has a face angle of 51.854† (51†51'), very close to the 51.827† ofthe Kepler triangle. This pyramid relationship corresponds to the coincidental relationship .

Egyptian pyramids very close in proportion to these mathematical pyramids are known.[69]

Golden ratio 17

Egyptian pyramidsIn the mid nineteenth century, R°ber studied various Egyptian pyramids including Khafre, Menkaure and some ofthe Giza, Sakkara and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid isthe middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematicaltheories of the shape of the pyramids have also been explored.[61]

One Egyptian pyramid is remarkably close to a "golden pyramid" Å the Great Pyramid of Giza (also known as thePyramid of Cheops or Khufu). Its slope of 51† 52' is extremely close to the "golden" pyramid inclination of 51† 50'and the Ü-based pyramid inclination of 51† 51'; other pyramids at Giza (Chephren, 52† 20', and Mycerinus, 50†47')[68] are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accidentremains open to speculation.[71] Several other Egyptian pyramids are very close to the rational 3:4:5 shape.[69]

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematicianand historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slantheight 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 wasthe only right triangle known to the Egyptians and they did not know the Pythagorean theorem nor any way to reasonabout irrationals such as Ü or Ç.[72]

Michael Rice[73] asserts that principal authorities on the history of Egyptian architecture have argued that theEgyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citingGiedon (1957).[74] Historians of science have always debated whether the Egyptians had any such knowledge or not,contending rather that its appearance in an Egyptian building is the result of chance.[75]

In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ratio is representedby the ratio of the length of the face (the slope height), inclined at an angle ¢ to the ground, to half the length of theside of the square base, equivalent to the secant of the angle ¢.[76] The above two lengths were about 186.4 and 115.2meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the originalmeasurements. Similarly, Howard Vyse, according to Matila Ghyka,[77] 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 datavariability.

Disputed observationsExamples of disputed observations of the golden ratio include the following:

í 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.[78]

í Some specific proportions in the bodies of many animals (including humans[79] [80] ) and parts of the shells ofmollusks[4] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in thereal measures of these elements in specific individuals, and the proportion in question is often significantlydifferent from the golden ratio.[79] The ratio of successive phalangeal bones of the digits and the metacarpal bonehas been said to approximate the golden ratio.[80] The nautilus shell, the construction of which proceeds in alogarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, butsometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previousone;[81] however, measurements of nautilus shells do not support this claim.[82]

í The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figuresinside flowers) are often claimed to show the golden ratio proportion in several species.[83] In practice, there aresignificant variations between individuals, seasonal variations, and age variations in these species. While thegolden ratio may be found in some proportions in some individuals at particular times in their life cycles, there isno consistent ratio in their proportions.

Golden ratio 18

í In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, orresistance to price increases, of a stock or commodity; after significant price changes up or down, new supportand resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[84]

The use of the golden ratio in investing is also related to more complicated patterns described by Fibonaccinumbers; see, e.g. Elliott wave principle. See Fibonacci retracement. However, other market analysts havepublished analyses suggesting that these percentages and patterns are not supported by the data.[85]

References and footnotes[1] 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 (xü+ü1)/x = x/1 or (multiplying by x) yields: xü+ü1 = x2, or thus a quadratic equation: x2üÖüxüÖü1ü=ü0. Then, by the quadratic formula, forpositive x = (Öbü+üÇ(b2üÖü4ac))/(2a) with aü=ü1, bü=üÖ1, cü=üÖ1, the solution for x is: (Ö(Ö1)ü+üÇ((Ö1)2üÖü4£1£(Ö1)))/(2£1) or (1ü+üÇ(5))/2.

[2] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number (http:/ / books. google. com/books?id=w9dmPwAACAAJ). New York: Broadway Books. ISBNü0-7679-0815-5. .

[3] Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: AssociatedUniversity Presses, 1996

[4] Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997[5] Euclid, Elements (http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ toc. html), Book 6, Definition 3.[6] Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 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 areintellectually fruitful and suggest the rhythms of modular design."

[7] Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920[8] William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport

Publishers, 2003[9] Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.[10] Weisstein, Eric W., " Golden Ratio Conjugate (http:/ / mathworld. wolfram. com/ GoldenRatioConjugate. html)" from MathWorld.[11] Mario Livio,The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, p.6[12] Euclid, Elements (http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ toc. html), Book 6, Proposition 30.[13] Euclid, Elements (http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ toc. html), Book 2, Proposition 11; Book 4, Propositions 10Å11;

Book 13, Propositions 1Å6, 8Å11, 16Å18.[14] "The Golden Ratio" (http:/ / www-history. mcs. st-andrews. ac. uk/ HistTopics/ Golden_ratio. html). The MacTutor History of Mathematics

archive. . Retrieved 2007-09-18.[15] Weisstein, Eric W., " Golden Ratio (http:/ / mathworld. wolfram. com/ GoldenRatio. html)" from MathWorld.[16] Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp.ü20Å21. ISBNü1-4027-3522-7.[17] Plato (360 BC) (Benjamin Jowett trans.). "Timaeus" (http:/ / classics. mit. edu/ Plato/ timaeus. html). The Internet Classics Archive. .

Retrieved May 30, 2006.[18] James Joseph Tattersall (2005). Elementary number theory in nine chapters (http:/ / books. google. com/ ?id=QGgLbf2oFUYC&

pg=PA29& dq=golden-ratio+ limit+ fibonacci+ ratio+ kepler& q=golden-ratio limit fibonacci ratio kepler) (2nd ed.). Cambridge UniversityPress. p.ü28. ISBNü9780521850148. .

[19] Underwood Dudley (1999). Die Macht der Zahl: Was die Numerologie uns weismachen will (http:/ / books. google. com/?id=r6WpMO_hREYC& pg=PA245& dq="goldener+ Schnitt"+ ohm). Springer. pp.ü245. ISBNü3-7643-5978-1. .

[20] Cook, Theodore Andrea (1979) [1914]. The Curves of Life (http:/ / books. google. com/ ?id=ea-TStM-07EC& pg=PA420& dq=phi+ mark+barr+ intitle:The+ intitle:Curves+ intitle:of+ intitle:Life). New York: Dover Publications. ISBNü0-486-23701-X. .

[21] Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic",Communication Quarterly, Vol. 46 No. 2, 1998, pp 194-213.

[22] Midhat J. Gazalõ , Gnomon, Princeton University Press, 1999. ISBN 0-691-00514-1[23] 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[24] 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 (http:/ / www. nexusjournal. com/ BouMaz. html)[25] 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[26] 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".[27] Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X[28] Jason Elliot (2006). Mirrors of the Unseen: Journeys in Iran (http:/ / books. google. com/ ?id=Gcs4IjUx3-4C& pg=PA284&

dq=intitle:"Mirrors+ of+ the+ Unseen"+ golden-ratio+ maidan). Macmillan. pp.ü277, 284. ISBNü9780312301910. .[29] Leonardo da Vinci's Polyhedra, by George W. Hart (http:/ / www. georgehart. com/ virtual-polyhedra/ leonardo. html)

Golden ratio 19

[30] Livio, Mario. "The golden ratio and aesthetics" (http:/ / plus. maths. org/ issue22/ features/ golden/ ). . Retrieved 2008-03-21.[31] Salvador Dali. (2008) (in English) (DVD). The Dali Dimension: Decoding the Mind of a Genius (http:/ / www. dalidimension. com/ eng/

index. html). Media 3.14-TVC-FGSD-IRL-AVRO. .[32] Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X[33] 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[34] Olariu, Agata, Golden Section and the Art of Painting Available online (http:/ / arxiv. org/ abs/ physics/ 9908036/ )[35] Tosto, Pablo, La composiciÅn Çurea en las artes plÇsticas Ä El nÉmero de oro, Librerúa Hachette, 1969, p. 134Å144[36] Jan Tschichold. The Form of the Book, 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. Thelower outer corner of the text area is fixed by a diagonal as well."

[37] Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.[38] The golden ratio and aesthetics (http:/ / plus. maths. org/ issue22/ features/ golden/ ), by Mario Livio[39] Lendvai, Ernù (1971). BÑla BartÅk: An Analysis of His Music. London: Kahn and Averill.[40] Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (http:/ / books. google. com/ books?id=ZgftUKoMnpkC& pg=PA83&

dq=bartok+ intitle:The+ intitle:Dynamics+ intitle:of+ intitle:Delight+ intitle:Architecture+ intitle:and+ intitle:Aesthetics& as_brr=0&ei=WkkSR5L6OI--ogLpmoyzBg& sig=Ijw4YifrLhkcdQSMVAjSL5g4zVk) (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X

[41] Roy Howat (1983). Debussy in Proportion: A Musical Analysis (http:/ / books. google. com/ ?id=4bwKykNp24wC& pg=PA169&dq=intitle:Debussy+ intitle:in+ intitle:Proportion+ golden+ la-mer). Cambridge University Press. ISBNü0-521-31145-4. .

[42] Simon Trezise (1994). Debussy: La Mer (http:/ / books. google. com/ ?id=THD1nge_UzcC& pg=PA53& dq=inauthor:Trezise+ golden+evidence). Cambridge University Press. pp.ü53. ISBNü0-521-44656-2. .

[43] "Pearl Masters Premium" (http:/ / www. pearldrum. com/ premium-birch. asp). Pearl Corporation. . Retrieved December 2, 2007.[44] Leon Harkleroad (2006). The Math Behind the Music (http:/ / books. google. com/ ?id=C3dsb7Qysh4C& pg=RA4-PA120& dq=misguided+

music+ mathematics+ "golden+ ratio"). Cambridge University Press. ISBNü0-521-81095-7. .[45] Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist 11: 44Å52. "Who would suspect, for example, that

the switch plate for single light switches are standardized in terms of a Golden Rectangle?".[46] Art Johnson (1999). Famous problems and their mathematicians (http:/ / books. google. com/ ?id=STKX4qadFTkC& pg=PA45&

dq=switch+ "golden+ ratio"#v=onepage& q=switch "golden ratio"& f=false). Libraries Unlimited. p.ü45. ISBNü9781563084461. . "TheGolden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates."

[47] Alexey Stakhov, Scott Olsen, Scott Anthony Olsen (2009). The mathematics of harmony: from Euclid to contemporary mathematics andcomputer science (http:/ / books. google. com/ ?id=K6fac9RxXREC& pg=PA21& dq="credit+ card"+ "golden+ ratio"+rectangle#v=onepage& q="credit card" "golden ratio" rectangle& f=false). World Scientific. p.ü21. ISBNü9789812775825. . "A credit card hasa form of the golden rectangle."

[48] Simon Cox (2004). Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel (http:/ / books.google. com/ ?id=TbjwhwLCEeAC& q="golden+ ratio"+ postcard& dq="golden+ ratio"+ postcard). Barnes & Noble Books.ISBNü9780760759318. . "The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards andphotographs all commonly conform to its proportions."

[49] Richard Padovan (1999). Proportion (http:/ / books. google. com/ ?id=Vk_CQULdAssC& pg=PA306& dq="contained+ the+ground-principle+ of+ all+ formative+ striving"). Taylor & Francis. pp.ü305Å306. ISBNü9780419227809. .

[50] Zeising, Adolf, Neue Lehre van den Proportionen des meschlischen KÖrpers, Leipzig, 1854, preface.[51] Weiss, Volkmar; Weiss, Harald (2003). "The golden mean as clock cycle of brain waves" (http:/ / www. v-weiss. de/ chaos. html). Chaos,

Solitons and Fractals 18 (4): 643Å652. doi:10.1016/S0960-0779(03)00026-2. .[52] Roopun, Anita K.; et al.; Carracedo, LM; Kaiser, M; Davies, CH; Traub, RD; Kopell, NJ; Whittington, MA (2008). "Temporal interactions

between cortical rhythms". Frontiers in Neuroscience 2 (2): 145Å154. doi:10.3389/neuro.01.034.2008. PMCü2622758. PMIDü19225587.[53] Golden ratio discovered in a quantum world (http:/ / www. eurekalert. org/ pub_releases/ 2010-01/ haog-grd010510. php)[54] J.C. Perez (1991), "Chaos DNA and Neuro-computers: A Golden Link" (http:/ / golden-ratio-in-dna. blogspot. com/ 2008/ 01/

1991-first-publication-related-to. html), in Speculations in Science and Technology vol. 14 no. 4, ISSNü0155-7785.[55] Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), "Nucleotide Frequencies in Human Genome and Fibonacci Numbers" (http:/ /

www. springerlink. com/ content/ p140352473151957/ ?p=d5b18a2dfee949858e2062449e9ccfad& pi=0), in Bulletin of MathematicalBiology, ISSNü0092-8240 (print), ISSNü1522-9602 (online). PDF full text (http:/ / www. springerlink. com/ content/ p140352473151957/fulltext. pdf)

[56] Perez, J.-C. (September 2010). "Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the GoldenRatio 1.618". Interdisciplinary Sciences: Computational Life Science 2 (3): 228Å240. doi:10.1007/s12539-010-0022-0. PMIDü20658335.

[57] Max. Hailperin, Barbara K. Kaiser, and Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme(http:/ / books. google. com/ ?id=yYyVRueWlZ8C& pg=PA63& dq=continued-fraction+ substitute+ golden-ratio). Brooks/Cole Pub. Co.ISBNü0-534-95211-9. .

[58] Brian Roselle, "Golden Mean Series" (http:/ / sites. google. com/ site/ goldenmeanseries/ )[59] "A Disco Ball in Space" (http:/ / science. nasa. gov/ headlines/ y2001/ ast09oct_1. htm). NASA. 2001-10-09. . Retrieved 2007-04-16.[60] American Mathematical Monthly, pp. 49-50, 1954.

Golden ratio 20

[61] Roger Herz-Fischler (2000). The Shape of the Great Pyramid (http:/ / books. google. com/ ?id=066T3YLuhA0C& pg=PA81&dq=kepler-triangle+ geometric). Wilfrid Laurier University Press. ISBNü0-88920-324-5. .

[62] Weisstein, Eric W., " Pisot Number (http:/ / mathworld. wolfram. com/ PisotNumber. html)" from MathWorld.[63] Horocycles exinscrits : une propriõtõ hyperbolique remarquable (http:/ / www. cabri. net/ abracadabri/ GeoNonE/ GeoHyper/ KBModele/

Biss3KB. html), cabri.net, retrieved 2009-07-21.[64] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa001622[65] The golden number to 17 000 000 000 digits (http:/ / www. matematicas. unal. edu. co/ airlande/ phi. html. en). Universidad Nacional de

Colombia. 2008. .[66] Radio, Astraea Web (2006). The Best of Astraea: 17 Articles on Science, History and Philosophy (http:/ / books. google. com/

?id=LDTPvbXLxgQC& pg=PA93& dq=kepler-triangle). Astrea Web Radio. ISBNü1-4259-7040-0. .[67] Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999[68] Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000[69] "The Great Pyramid, The Great Discovery, and The Great Coincidence" (http:/ / www. petrospec-technologies. com/ Herkommer/ pyramid/

pyramid. htm). . Retrieved 2007-11-25.[70] Lancelot Hogben, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, Lost Discoveries: The

Ancient Roots of Modern ScienceÅfrom the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56[71] Burton, David M. (1999). The history of mathematics: an introduction (http:/ / books. google. com/ books?id=GKtFAAAAYAAJ) (4 ed.).

WCB McGraw-Hill. p.ü56. ISBNü0-070-09468-3. .[72] Eric Temple Bell, The Development of Mathematics, New York: Dover, 1940, p.40[73] Rice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp. 24 Routledge, 2003, ISBN 0-415-26876-1[74] 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[75] Markowsky, George (January 1992). "Misconceptions about the Golden Ratio" (http:/ / www. umcs. maine. edu/ ~markov/ GoldenRatio.

pdf) (PDF). College Mathematics Journal (Mathematical Association of America) 23 (1): 1. doi:10.2307/2686193. JSTORü2686193. .[76] Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859[77] Matila Ghyka The Geometry of Art and Life, New York: Dover, 1977[78] Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166Å167, Wiley, ISBN 0-471-21823-5. "The half-folio page

(30.7 § 44.5 cm) was made up of two rectanglesÄthe whole page and its text areaÄbased on the so called 'golden section', which specifies acrucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8."

[79] Pheasant, Stephen (1998). Bodyspace. London: Taylor & Francis. ISBNü0748400672.[80] van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach GmbH.[81] Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004[82] Peterson, Ivars. "Sea shell spirals" (http:/ / www. sciencenews. org/ view/ generic/ id/ 6030/ title/ Sea_Shell_Spirals). Science News. ..[83] Derek Thomas, Architecture and the Urban Environment: A Vision for the New Age, Oxford: Elsevier, 2002[84] 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" (http:/ / ftp. ny. frb. org/ research/ epr/ 00v06n2/ 0007osle. pdf)(PDF). Federal Reserve Bank of New York Economic Policy Review 6 (2): 53Å68. .

[85] Roy Batchelor and Richard Ramyar, " Magic numbers in the Dow (http:/ / www. cass. city. ac. uk/ media/ stories/ resources/Magic_Numbers_in_the_Dow. pdf)," 25th International Symposium on Forecasting, 2005, p. 13, 31. " Not since the 'big is beautiful' dayshave giants looked better (http:/ / www. telegraph. co. uk/ money/ main. jhtml?xml=/ money/ 2006/ 09/ 26/ ccinv26. xml)", Tom Stevenson,The Daily Telegraph, Apr. 10, 2006, and "Technical failure", The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor andRamyar's research.

Further readingí Doczi, Gy°rgy (2005) [1981]. The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture.

Boston: Shambhala Publications. ISBNü1-59030-259-1.í Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Proportion. New York: Dover

Publications. ISBNü0-486-22254-3.í Joseph, George G. (2000) [1991]. The Crest of the Peacock: The Non-European Roots of Mathematics (New ed.).

Princeton, NJ: Princeton University Press. ISBNü0-691-00659-8.í Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and

Design (3rd Rev. ed.). Charleston, SC: BookSurge. ISBNü1-4196-2157-2.í Schneider, Michael S. (1994). A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of

Nature, Art, and Science. New York: HarperCollins. ISBNü0-06-016939-7.

Golden ratio 21

í Stakhov, A. P. (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and ComputerScience. Singapore: World Scientific Publishing. ISBNü978-981-277-582-5.

í Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans.. Washington, DC: TheMathematical Association of America. ISBNü0-88385-534-8.

External linksí "Golden Section" (http:/ / demonstrations. wolfram. com/ GoldenSection/ ) by Michael Schreiber, Wolfram

Demonstrations Project, 2007.í Green, Thomas M. (updated June 20, 2005). "The Pentagram & The Golden Ratio" (http:/ / web. archive. org/

web/ 20071105084747/ http:/ / www. contracosta. cc. ca. us/ math/ pentagrm. htm). Archived from the original(http:/ / www. contracosta. cc. ca. us/ math/ pentagrm. htm) on November 5, 2007. Retrieved December 1, 2007.Geometry instruction with problems to solve.

í Knott, Ron. "The Golden section ratio: Phi" (http:/ / www. mcs. surrey. ac. uk/ Personal/ R. Knott/ Fibonacci/ phi.html). Information and activities by a mathematics professor.

í Weisstein, Eric W., " Golden Ratio (http:/ / mathworld. wolfram. com/ GoldenRatio. html)" from MathWorld.í "Researcher explains mystery of golden ratio" (http:/ / www. physorg. com/ news180531747. html). PhysOrg.

December 21, 2009.

Phidias

Phidias Showing the Frieze of the Parthenon to his Friends (1868) by SirLawrence Alma-Tadema.

Phidias or Pheidias (in Ancient Greek,ÅÉÑÖÜáà; circa 480 BCüÅ 430 BC), was aGreek sculptor, painter and architect, wholived in the 5th century BC, and iscommonly regarded as one of the greatest ofall sculptors of Classical Greece:[1] Phidias'Statue of Zeus at Olympia was one of theSeven Wonders of the Ancient World.Phidias also designed the statues of thegoddess Athena on the Athenian Acropolis,namely the Athena Parthenos inside theParthenon and the Athena Promachos, acolossal bronze statue of Athena whichstood between it and the Propylaea,[2] amonumental gateway that served as theentrance to the Acropolis in Athens. Phidias was the son of a certain Charmides of Athens.[3]

The ancients believed that his masters were Hegias[4] and Hageladas.

Prior to the Peloponnesian war, Phidias was accused of embezzling gold intended for the statue of Athena inside theParthenon. Pericles' enemies found a false witness against Phidias, named Menon. Phidias died in prison, althoughPericles' companion, Aspasia, was acquitted of her own charges.

Phidias 22

Works

The Acropolis of Athens

A Roman period, 2nd century CEsculpture found near the Varvakeion

school reflects the type of the restoredAthena Parthenos presently in the

(National Archaeological Museum ofAthens).

Although no original works in existence can be confidentlyattributed to him with certainty, numerous Roman copies invarying degrees of supposed fidelity are known to exist. This isnot uncommon. Almost all classical Greek paintings andsculptures have been destroyed, and only Roman copies or notesof them exist, like the passages of Plato that ascribe Phidias' worksto him. The ancient Romans frequently copied and furtherdeveloped Greek art.

Ancient critics take a very high view of the merits of Phidias.What they especially praise is the ethos or permanent moral levelof his works as compared with those of the later so called"pathetic" school. Demetrius calls his statues sublime, and at thesame time precise.

Of his life we know little apart from his works. His firstcommission was a group of national heroes with Miltiades as acentral figure.

The famous statesman Pericles also commissioned severalsculptures for Athens from him in 447 BC, to celebrate Greekvictory against the Persians at the Battle of Marathon during theGreco-Persian Wars (490 BC). Pericles used some of the moneyfrom the maritime League of Delos,[5] to rebuild and decorateAthens to celebrate this victory.

In 1958 archaeologists found the workshop at Olympia wherePhidias assembled the gold and ivory Zeus. There were still someshards of ivory at the site, moulds and other casting equipment,and a black glaze drinking cup[6] engraved "I belong to Phidias".[7]

The Golden Ratio has been represented by the Greek letter (phi), after Phidias, who is said to have employed it. The GoldenRatio is an irrational number approximating 1.6180[8] which whenstudied has special mathematical properties. The golden spiral isalso said to hold aesthetic values.

Early works

The earliest of the works of Phidias were dedications in memory of Marathon, celebrating the Greek victory. AtDelphi he erected a great group in bronze including the figures of Greek gods Apollo and Athena, several Atticheroes, and General Miltiades the Younger. On the Acropolis of Athens Pheidias set up a colossal bronze statue ofAthena, the Athena Promachos, which was visible far out at sea. Athena was the goddess of wisdom and warriorsand the protectress of Athens. At Pellene in Achaea, and at Plataea Pheidias made two other statues of Athena, aswell as a statue of the goddess Aphrodite in ivory and gold for the people of Elis.

Phidias 23

A reconstruction of Phidias' statue of Zeus, in anengraving made by Philippe Galle in 1572, from a

drawing by Maarten van Heemskerck.

Zeus at Olympia and the Athena Parthenos

Among the ancient Greeks themselves two works of Phidias faroutshone all others, the colossal chryselephantine figures of Zeuscirca 432 BC on the site where it was erected in the temple ofZeus,[9] at Olympia, Greece, and of Athena Parthenos (literally,"Athena the Virgin") a sculpture of the Greek virgin goddessAthena named after an epithet for the goddess herself, and washoused in the Parthenon in Athens. Both sculpture belong to aboutthe middle of the 5th century BC. A number of replicas and worksinspired by it, both ancient and modern, have been made. From the5th century BC, the copies of the statue of Zeus found were smallcopies on coins of Elis, which give us but a general notion of thepose, and the character of the head. The god was seated on athrone, every part of which was used as a ground for sculptural decoration. His body was of ivory, his robe of gold.His head was of somewhat archaic type: the Otricoli mask which used to be regarded as a copy of the head of theOlympian statue is certainly more than a century later in style.

Materials and theoriesIn antiquity Phidias was celebrated for his statues in bronze, and his chryselephantine works (statues made of goldand ivory). In the Hippias Major, Plato claims that Phidias seldom, if ever, have executed works in marble thoughmany of the sculptures of his times were executed in marble. Plutarch tells us that he superintended the great worksof Pericles on the Acropolis. Inscriptions prove that the marble blocks intended for the pedimental statues of theParthenon were not brought to Athens until 434 BC, which was probably after the death of Phidias. It is thereforepossible that most sculptural decoration of the Parthenon was the work of Phidias' atelier but supposedly made bypupils of Phidias, such as Alcamenes and Agoracritus. Our actual knowledge of the works of Phidias is very small.There are many stately figures in the Roman and other museums which clearly belong to the same school as theParthenos. These are copies of the Roman age.

According to geographer Pausanias (1.28.2), the original bronze Lemnian Athena was created by Phidias circa450-440 BCE, for Athenians living on Lemnos. Adolf Furtw•ngler proposed to find, in a statue of which the head isat Bologna, and of which the body is at Dresden, a copy of the Lemnian Athena of Phidias. Some 5th century torsosof Athena found at Athens. The torso of Athena in the ¶cole des Beaux-Arts at Paris, which has unfortunately lost itshead, may perhaps best serve to help our imagination in reconstructing the original statue.

Phidias 24

Gallery

Head of Aphrodite Zeus in Olympia, representationon coin

Another copy of onePhidias or his pupils

work, head of Athena,found around Pnyx,now in the National

ArcheologicalMuseum of Athens

Reconstruction ofAthena Lemnia,

Dresden.

Head of Athena, Roman copy Wounded Amazon - MuseiCapitolini, Rome

Referencesí üThis articleüincorporates text from a publication now in the public domain:üChisholm, Hugh, ed (1911).

EncyclopÜdia Britannica (Eleventh ed.). Cambridge University Press.í Andrew Stewart, One Hundred Greek Sculptors: Their Careers and Extant Works, Part III of Stewart's Greek

Sculpture, (Yale University Press) (on-line text at Perseus [10]).

Notes[1] Phidias (http:/ / www. bartleby. com/ 65/ ph/ Phidias. html)[2] Birte Lundgreen, "A Methodological Enquiry: The Great Bronze Athena by Pheidias" The Journal of Hellenic Studies[3] Not the Charmides who participated in the tyranny at Athens.[4] Not to be confused with Hegias the neoplatonic philosopher.[5] The Delian League was an association of approximately 150 Greek city-states under the leadership of Athens, whose purpose was to continue

fighting the Persian Empire.[6] Image of the cup (http:/ / www. lgpn. ox. ac. uk/ image_archive/ vases/ v5. html)[7] The Oxford Art Dictionary, s.v. "Phidias" (http:/ / www. enotes. com/ oxford-art-encyclopedia/ phidias)[8] 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 Ö x Ö 1 = 0. Then, by the quadratic formula, forpositive x = (Öb + Ç(b2 Ö 4ac))/(2a) with a = 1, b = Ö1, c = Ö1, the solution for x is: (Ö(Ö1) + Ç((Ö1)2 Ö 4£1£(Ö1)))/(2£1) or (1 + Ç(5))/2.

Phidias 25

[9] Statue of Zeus (http:/ / www. britannica. com/ eb/ article-9078346/ Statue-of-Zeus) from encyclopßdiabritannica.com (http:/ / www.britannica. com/ ). Retrieved 22 November 2006.

[10] http:/ / perseus. mpiwg-berlin. mpg. de/ cgi-bin/ ptext?doc=Perseus%3Atext%3A1999. 04. 0008%3Ahead%3D%2329

External linksí Phidias as a First Name in USA (http:/ / www. pokemyname. com/ firstname_35958_phidias. htm)

Irrational numberIn mathematics, an irrational number is any real number which cannot be expressed as a fraction a/b, where a andb are integers, with b non-zero, and is therefore not a rational number. Informally, this means that an irrationalnumber cannot be represented as a simple fraction. Irrational numbers are precisely those real numbers that cannotbe represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers areuncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1] Perhaps thebest-known irrational numbers are Ü, e and Ç2.[2] [3] [4] When the ratio of lengths of two line segments is irrational,the line segments are also described as being incommensurable, meaning they share no measure in common. Ameasure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole numberof copies of J laid end-to-end occupy the same length as I.

The number is irrational.

History

It has been suggested that the concept of irrationality was implicitlyaccepted by Indian mathematicians since the 7th century BC, whenManava (c. 750Å690 BC) believed that the square roots of numberssuch as 2 and 61 could not be exactly determined,[5] but such claimsare not well substantiated and unlikely to be true.[6]

Ancient Greece

The first proof of the existence of irrational numbers is usuallyattributed to a Pythagorean (possibly Hippasus of Metapontum),[7] whoprobably discovered them while identifying sides of the pentagram.[8]

The then-current Pythagorean method would have claimed that theremust be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other.However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure,and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if thehypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must beboth odd and even, which is impossible. His reasoning is as follows:

í The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest unitspossible.

í By the Pythagorean theorem: a2 = 2b2.í Since a2 is even, a must be even.í Since a:b is in its lowest terms, b must be odd.í Since a is even, let a = 2y.í Then a2 = 4y2 = 2b2

í b2 = 2y2 so b2 must be even, therefore b is even.í However we asserted b must be odd. Here is the contradiction.[9]

Irrational number 26

Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus,however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and wassubsequently thrown overboard by his fellow Pythagoreans àÉfor having produced an element in the universe whichdenied theÉdoctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.â[10]

Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasushimself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumptionthat number and geometry were inseparableÅa foundation of their theory.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of thediscrete to the continuous. Brought into light by Zeno of Elea, he questioned the conception that quantities arediscrete, and composed of a finite number of units of a given size. Past Greek conceptions dictated that theynecessarily must be, for àwhole numbers represent discrete objects, and a commensurable ratio represents a relationbetween two collections of discrete objects.â[11] However Zeno found that in fact à[quantities] in general are notdiscrete collections of units; this is why ratios of incommensurable [quantities] appearÉ.[Q]uantities are, in otherwords, continuous.â[11] What this means is that, contrary to the popular conception of the time, there cannot be anindivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily beinfinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of thehalf in half, and so on. This process can continue infinitely, for there is always another half to be split. The moretimes the segment is halved, the closer the unit of measure will come to zero, but it will never reach exactly zero.This is exactly what Zeno sought to prove. He sought to prove this by formulating four paradoxes, whichdemonstrated the contradictions inherent in the mathematical thought of the time. While Zenoäs paradoxes accuratelydemonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of thealternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity ofanother, and therefore further investigation had to occur.

The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into accountcommensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitudeand number. A magnitude àwas not a number but stood for entities such as line segments, angles, areas, volumes, andtime which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped fromone value to another, as from 4 to 5.â[12] Numbers are composed of some smallest, indivisible unit, whereasmagnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was thenable to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude,and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, heavoided the trap of having to express an irrational number as a number. àEudoxusä theory enabled the Greekmathematicians to make tremendous progress in geometry by supplying the necessary logical foundation forincommensurable ratios.â[13]

As a result of the distinction between number and magnitude, geometry became the only method that could take intoaccount incommensurable ratios. Because previous numerical foundations were still incompatible with the conceptof incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focusedalmost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometricalterms. This may account for why we still conceive of x2 or x3 as x squared and x cubed instead of x second powerand x third power. Also crucial to Zenoäs work with incommensurable magnitudes was the fundamental focus ondeductive reasoning which resulted from the foundational shattering of earlier Greek mathematics. The realizationthat some basic conception within the existing theory was at odds with reality necessitated a complete and thoroughinvestigation of the axioms and assumptions that comprised that theory. Out of this necessity Eudoxus developed hismethod of exhaustion, and kind of reductio ad absurdum which àestablished the deductive organization on the basisof explicit axiomsÉâ as well as àÉreinforced the earlier decision to rely on deductive reasoning for proof.â[14] Thismethod of exhaustion is said to be the first step in the creation of calculus.

Irrational number 27

Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probablybecause the algebra he used couldn't be applied to the square root of 17.[15] It wasn't until Eudoxus developed atheory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundationof irrational numbers was created.[16] A magnitude "was not a number but stood for entities such as line segments,angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed tonumbers, which jumped from one value to another, as from 4 to 5."[17] Numbers are composed of some smallest,indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned tomagnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining aratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values(numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. àEudoxusätheory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessarylogical foundation for incommensurable ratios.â[18] Euclid's Elements Book 10 is dedicated to classification ofirrational magnitudes.

Middle AgesIn the Middle ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treatedas "algebraic objects".[19] Muslim mathematicians also merged the concepts of "number" and "magnitude" into amore general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, andextended the concept of number to ratios of continuous magnitude.[20] In his commentary on Book 10 of theElements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubicirrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. Hedealt with them freely but explains them in geometric terms as follows:[21]

"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value ispronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounceand represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are notsquares, the sides of numbers which are not cubes etc."

In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rationalmagnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach tothe concept of irrationality, as he attributes the following to irrational magnitudes:[21]

"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting amagnitude of this kind from an irrational one, or of a rational magnitude from it."

The Egyptian mathematician Ab® K©mil Shuj© ibn Aslam (c. 850Å930) was the first to accept irrational numbers assolutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots andfourth roots.[22] In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather thangeometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmeticalfunctions.[23] Ab® Ja'far al-Kh©zin (900Å971) provides a definition of rational and irrational magnitudes, stating thatif a definite quantity is:[24]

"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to arational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of thegiven magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rationalmagnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as onenumber to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), orparts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."

Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. Al-Hass©r, a Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where

Irrational number 28

the numerator and denominator are separated by a horizontal bar. This same fractional notation appears soon after inthe work of Fibonacci in the 13th century. During the 14th to 16th centuries, Madhava of Sangamagrama and theKerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as piand certain irrational values of trigonometric functions. Jyesthadeva provided proofs for these infinite series in theYuktibháàá.[25]

Modern periodThe 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, andespecially of Leonhard Euler. The completion of the theory of complex numbers in the nineteenth century entailedthe differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence oftranscendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored sinceEuclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), EduardHeine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Mõray had taken in 1869 the samepoint of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has beencompletely set forth by Salvatore Pincherle in 1880,[26] and Dedekind's has received additional prominence throughthe author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine basetheir theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of realnumbers, separating all rational numbers into two groups having certain characteristic properties. The subject hasreceived later contributions at the hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Mõray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the handsof Euler, and at the opening of the nineteenth century were brought into prominence through the writings of JosephLouis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of thesubject.

Johann Heinrich Lambert proved (1761) that Ü cannot be rational, and that en is irrational if n is rational (unlessnü=ü0).[27] While Lambert's proof is often said to be incomplete, modern assessments support it as satisfactory, and infact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the BesselÅCliffordfunction, provided a proof to show that Ü2 is irrational, whence it follows immediately that Ü is irrational also. Theexistence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873)proved their existence by a different method, that showed that every interval in the reals contains transcendentalnumbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting fromHermite's conclusions, showed the same for Ü. Lindemann's proof was much simplified by Weierstrass (1885), stillfurther by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan.

Example proofs

Square rootsThe square root of 2 was the first number to be proved irrational and that article contains a number of proofs. Thegolden ratio is the next most famous quadratic irrational and there is a simple proof of its irrationality in its article.The square roots of all numbers which are not perfect squares are irrational and a proof may be found in quadraticirrationals.

The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, calledan argument by reductio ad absurdum. The following argument appeals twice to the fact that the square of an oddinteger is always odd.

If Ç2 is rational it has the form m/n for integers m, n not both even. Then m2 = 2n2, hence m is even, say m = 2p.Thus 4p2 = 2n2 so 2p2 = n2, hence n is also even, a contradiction.

Irrational number 29

General rootsThe proof above for the square root of two can be generalized using the fundamental theorem of arithmetic whichwas proved by Gauss in 1798. This asserts that every integer has a unique factorization into primes. Using it we canshow that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms theremust be a prime in the denominator which does not divide into the numerator whatever power each is raised to.Therefore if an integer is not an exact kth power of another integer then its kth root is irrational.

LogarithmsPerhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio adabsurdum that log2ü3 is irrational. Notice that log2ü3 Ñü1.58ü>ü0.

Assume log2ü3 is rational. For some positive integers m and n, we have

It follows that

However, the number 2 raised to any positive integer power must be even (because it will be divisible byü2) and thenumberü3 raised to any positive integer power must be odd (since none of its prime factors will beü2). Clearly, aninteger can not be both odd and even at the same time: we have a contradiction. The only assumption we made wasthat log2ü3 is rational (and so expressible as a quotient of integers m/n with nüãü0). The contradiction means that thisassumption must be false, i.e. log2ü3 is irrational, and can never be expressed as a quotient of integers m/n with nüãü0.

Cases such as log10ü2 can be treated similarly.

Transcendental and algebraic irrationalsAlmost all irrational numbers are transcendental and all transcendental numbers are irrational: the article ontranscendental numbers lists several examples. eür and Üür are irrational if rüãü0 is rational; eÜ is irrational.

Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials withinteger coefficients: start with a polynomial equation

where the coefficients ai are integers. Suppose you know that there exists some real number x with p(x)ü=ü0 (forinstance if n is odd and an is non-zero, then because of the intermediate value theorem). The only possible rationalroots of this polynomial equation are of the form r/s where r is a divisor of a0 and s is a divisor of an; there are onlyfinitely many such candidates which you can all check by hand. If neither of them is a root of p, then x must beirrational. For example, this technique can be used to show that xü=ü(21/2ü+ü1)1/3 is irrational: we have (x3üÖü1)2 = 2and hence x6üÖü2x3üÖü1ü=ü0, and this latter polynomial does not have any rational roots (the only candidates to checkare ™1).

Because the algebraic numbers form a field, many irrational numbers can be constructed by combiningtranscendental and algebraic numbers. For example 3Üü+ü2, Üü+üÇ2 and eÇ3 are irrational (and even transcendental).

Irrational number 30

Decimal expansionsThe decimal expansion of an irrational number never repeats or terminates, unlike a rational number.

To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the divisionof n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 neveroccurs, then the algorithm can run at most m Ö 1 steps without using any remainder more than once. After that, aremainder must recur, and then the decimal expansion repeats.

Conversely, suppose we are faced with a recurring decimal, we can prove that it is a fraction of two integers. Forexample:

Here the length of the repitend is 3. We multiply by 103:

Note that since we multiplied by 10 to the power of the length of the repeating part, we shifted the digits to the left ofthe decimal point by exactly that many positions. Therefore, the tail end of 1000A matches the tail end of A exactly.Here, both 1000A and A have repeating 162 at the end.

Therefore, when we subtract A from both sides, the tail end of 1000A cancels out of the tail end of A:

Then

(135 is the greatest common divisor of 7155 and 9990). Alternatively, since 0.5 = 1/2, one can clear fractions bymultiplying the numerator and denominator by 2:

(27 is the greatest common divisor of 1431 and 1998).

53/74 is a quotient of integers and therefore a rational number.

Irrational powersDov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab isrational.[28]

Indeed, if Ç2Ç2 is rational, then take a = b = Ç2. Otherwise, take a to be the irrational number Ç2Ç2 and b = Ç2. Thenab = (Ç2Ç2)Ç2 = Ç2Ç2£Ç2 = Ç22 = 2 which is rational.

Although the above argument does not decide between the two cases, the GelfondÅSchneider theorem implies thatÇ2Ç2 is transcendental, hence irrational. This theorem states that all non-rational algebraic powers of algebraicnumbers other than 0 or 1 are transcendental.

Irrational number 31

Open questionsIt is not known whether Ü + e or Ü Ö e is irrational or not. In fact, there is no pair of non-zero integers m and n forwhich it is known whether mÜ + ne is irrational or not. Moreover, it is not known whether the set {Ü, e} isalgebraically independent over Q.

It is not known whether Üe, Ü/e, 2e, Üe, ÜÇ2, ln Ü, Catalan's constant, or the EulerÅMascheroni gamma constant Ä areirrational.[29] [30] [31]

The set of all irrationalsSince the reals form an uncountable set, of which the rationals are a countable subset, the complementary set ofirrationals is uncountable.

Under the usual (Euclidean) distance function d(x,üy) = |xüÖüy|, the real numbers are a metric space and hence also atopological space. Restricting the Euclidean distance function gives the irrationals the structure of a metric space.Since the subspace of irrationals is not closed, the induced metric is not complete. However, being a G-deltasetÄi.e., a countable intersection of open subsetsÄin a complete metric space, the space of irrationals istopologically complete: that is, there is a metric on the irrationals inducing the same topology as the restriction of theEuclidean metric, but with respect to which the irrationals are complete. One can see this without knowing theaforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines ahomeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seento be completely metrizable.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals have a basis of clopensets so the space is zero-dimensional.

References[1] Cantor, Georg (1955, 1915). Philip Jourdain. ed. Contributions to the Founding of the Theory of Transfinite Numbers (http:/ / www. archive.

org/ details/ contributionstot003626mbp). New York: Dover. ISBNü978-0486600451.[2] The 15 Most Famous Transcendental Numbers (http:/ / sprott. physics. wisc. edu/ Pickover/ trans. html). by Clifford A. Pickover. URL

retrieved 24 October 2007.[3] http:/ / www. mathsisfun. com/ irrational-numbers. html; URL retrieved 24 October 2007.[4] Weisstein, Eric W., " Irrational Number (http:/ / mathworld. wolfram. com/ IrrationalNumber. html)" from MathWorld. URL retrieved 26

October 2007.[5] T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411Å2, in Selin, Helaine; D'Ambrosio, Ubiratan (2000).

Mathematics Across Cultures: The History of Non-western Mathematics. Springer. ISBNü1402002602.[6] Boyer (1991). "China and India". p.ü208. "It has been claimed also that the first recognition of incommensurables is to be found in India

during the Sulbasutra period, but such claims are not well substantiated. The case for early Hindu awareness of incommensurable magnitudesis rendered most unlikely by the lack of evidence that Indian mathematicians of that period had come to grips with fundamental concepts."

[7] Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.[8] James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal..[9] Kline, M. (1990). Mathematical Thought from Ancient to Modern Times, Vol. 1. New York: Oxford University Press. (Original work

published 1972). p.33.[10] Kline 1990, p. 32.[11] Kline 1990, p.34.[12] Kline 1990, p.48.[13] Kline 1990, p.49.[14] Kline 1990, p.50.[15] Robert L. McCabe (1976). "Theodorus' Irrationality Proofs". Mathematics Magazine..[16] Charles H. Edwards (1982). The historical development of the calculus. Springer.[17] Kline 1990, p.48.[18] Kline 1990, p.49.[19] O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?" (http:/ / www-history. mcs. st-andrews. ac. uk/

HistTopics/ Arabic_mathematics. html), MacTutor History of Mathematics archive, University of St Andrews, ..

Irrational number 32

[20] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy ofSciences 500: 253Å277 [254]. doi:10.1111/j.1749-6632.1987.tb37206.x.

[21] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy ofSciences 500: 253Å277 [259]. doi:10.1111/j.1749-6632.1987.tb37206.x

[22] Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000). Mathematics Across Cultures: The Historyof Non-western Mathematics. Springer. ISBNü1402002602.

[23] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy ofSciences 500: 253Å277 [260]. doi:10.1111/j.1749-6632.1987.tb37206.x.

[24] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy ofSciences 500: 253Å277 [261]. doi:10.1111/j.1749-6632.1987.tb37206.x.

[25] Katz, V. J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Mathematical Association of America) 68 (3): 163Å74.[26] Salvatore Pincherle (1880). "Saggio di una introduzione alla teorica delle funzioni analitiche secondo i principi del prof. Weierstrass".

Giornale di Matematiche.[27] J. H. Lambert (1761). "Mõmoire sur quelques propriõtõs remarquables des quantitõs transcendentes circulaires et logarithmiques". Histoire

de l'AcadÑmie Royale des Sciences et des Belles-Lettres der Berlin: 265Å276.[28] George, Alexander; Velleman, Daniel J. (2002). Philosophies of mathematics. Blackwell. pp.ü3Å4. ISBNü0-631-19544-0.[29] Weisstein, Eric W., " Pi (http:/ / mathworld. wolfram. com/ Pi. html)" from MathWorld.[30] Weisstein, Eric W., " Irrational Number (http:/ / mathworld. wolfram. com/ IrrationalNumber. html)" from MathWorld.[31] Some unsolved problems in number theory (http:/ / www. math. ou. edu/ ~jalbert/ courses/ openprob2. pdf)

Further readingí Adrien-Marie Legendre, âlÑments de GÑometrie, Note IV, (1802), Parisí Rolf Wallisser, "On Lambert's proof of the irrationality of Ü", in Algebraic Number Theory and Diophantine

Analysis, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyer

External Linksí Zeno's Paradoxes and Incommensurability. (n.d.). Retrieved April 1, 2008, from http:/ / www. dm. uniba. it/

~psiche/ bas2/ node5. html

External linksí Weisstein, Eric W., " Irrational Number (http:/ / mathworld. wolfram. com/ IrrationalNumber. html)" from

MathWorld.í Square root of 2 is irrational (http:/ / www. cut-the-knot. org/ proofs/ sq_root. shtml)

Golden rectangle 33

Golden rectangle

A golden rectangle with longer side a and shorter side b, whenplaced adjacent to a square with sides of length a, will

produce a similar golden rectangle with longer side a + b andshorter side a. This illustrates the relationship

A golden rectangle is one whose side lengths are in the golden ratio, or approximately 1:1.618.

A distinctive feature of this shape is that when a square section is removed, the remainder is another goldenrectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which casecorresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmicspiral with this property.

According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's DivinaProportione in 1509,[1] when "with Pacioli's book, the Golden Ratio started to become available to artists intheoretical treatises that were not overly mathematical, that they could actually use,"[2] many artists and architectshave been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. Theproportions of the golden rectangle have been observed in works predating Pacioli's publication.[3]

Golden rectangle 34

Construction

A method to construct a golden rectangle. Thesquare is outlined in red. The resulting

dimensions are in the golden ratio.

A golden rectangle can be constructed with only straightedge andcompass by this technique:

1. Construct a simple square2. Draw a line from the midpoint of one side of the square to an

opposite corner3. Use that line as the radius to draw an arc that defines the height of

the rectangle4. Complete the golden rectangle

Applications

í Le Corbusier's 1927 Villa Stein in Garches features a rectangularground plan, elevation, and inner structure that are closelyapproximate to golden rectangles.[4]

í The flag of Togo was designed to approximate a golden rectangleclosely.[5]

References[1] Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.[2] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books.

ISBNü0-7679-0815-5.[3] Van Mersbergen, Audrey M., Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic, Communication

Quarterly, Vol. 46, 1998 ("a 'Golden Rectangle' has a ratio of the length of its sides equal to 1:1.61803+. The Parthenon is of thesedimensions.")

[4] 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".

[5] "Flag of Togo" (http:/ / www. fotw. us/ flags/ tg. html). FOTW.us. Flags Of The World. . Retrieved 2007-06-09.

External linksí Golden Ratio at MathWorld (http:/ / mathworld. wolfram. com/ GoldenRatio. html)í The Golden Mean and the Physics of Aesthetics (http:/ / uk. arxiv. org/ abs/ physics/ 0411195)í Golden rectangle demonstration (http:/ / www. mathopenref. com/ rectanglegolden. html) With interactive

animation

Golden spiral 35

Golden spiral

Approximate and true golden spirals: the greenspiral is made from quarter-circles tangent to theinterior of each square, while the red spiral is a

golden spiral, a special type of logarithmic spiral.Overlapping portions appear yellow. The lengthof the side of a larger square to the next smaller

square is in the golden ratio.

In geometry, a golden spiral is a logarithmic spiral whose growthfactor b is related to Ç, the golden ratio.[1] Specifically, a golden spiralgets wider (or further from its origin) by a factor of Ç for every quarterturn it makes.

Formula

The polar equation for a golden spiral is the same as for otherlogarithmic spirals, but with a special value of b:[2]

or

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when ¢ is aright angle (a quarter turn in either direction):

Therefore, b is given by

The numerical value of b depends on whether the right angle is measured as 90 degrees or as radians; and since the

angle can be in either direction, it is easiest to write the formula for the absolute value of (that is, b can also be thenegative of this value):

A Fibonacci spiral approximates the goldenspiral; unlike the "whirling rectangle diagram"based on the golden ratio, above, this one uses

squares of integer Fibonacci-number sizes, shownfor square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

for ¢ in degrees;

Golden spiral 36

An alternate formula for a logarithmic and golden spiral is:[3]

where the constant c is given by:

which for the golden spiral gives c values of:

if ¢ is measured in degrees, and

if ¢ is measured in radians.

Approximations of the golden spiral

There are several similar spirals that approximate, but do not exactly equal, a goldenspiral.[4] These are often confused with the golden spiral.

For example, a golden spiral can be approximated by a "whirling rectangle diagram,"in which the opposite corners of squares formed by spiraling golden rectangles areconnected by quarter-circles. The result is very similar to a true golden spiral (Seeimage on top right).

Another approximation is a Fibonacci spiral, which is not a true logarithmic spiral.Every quarter turn a Fibonacci spiral gets wider not by Ç, but by a changing factorrelated to the ratios of consecutive terms in the Fibonacci sequence. The ratios ofconsecutive terms in the Fibonacci series approach Ç, so that the two spirals are very similar in appearance. (Seeimage on top right).

Spirals in natureApproximate logarithmic spirals can occur in nature (for example, the arms of spiral galaxies). It is sometimes statedthat nautilus shells get wider in the pattern of a golden spiral, and hence are related to both Ç and the Fibonacciseries. In truth, nautilus shells (and many mollusc shells) exhibit logarithmic spiral growth, but at an angle distinctlydifferent from that of the golden spiral.[5] This pattern allows the organism to grow without changing shape. Spiralsare common features in nature; golden spirals are one special case of these.

References[1] Chang, Yu-sung, " Golden Spiral (http:/ / demonstrations. wolfram. com/ GoldenSpiral/ )", The Wolfram Demonstrations Project.[2] Priya Hemenway (2005). Divine Proportion: ä Phi in Art, Nature, and Science. Sterling Publishing Co. pp.ü127Å129. ISBNü1402735227.[3] Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science (http:/ / books. google. com/

books?id=rqzaQo6CaA0C& pg=PA200& ots=8airJXF_BB& dq="golden+ spiral"+ log& as_brr=3& sig=3jQ4u9WBBv-taoGZR8jtu_5Nv9o).Walter de Gruyter. pp.ü45, 199Å200. ISBNü3110129906. .

[4] Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis (http:/ / books. google. com/books?id=JhnERQLm4lUC& dq=rectangles+ approximate+ golden-spiral). High Art Press. pp.ü14Å16. ISBNü0967172764. .

[5] Oberon Zell-Ravenheart (2004). Grimoire for the Apprentice Wizard (http:/ / books. google. com/ books?id=cMuQADen69UC& dq=).Career Press. pp.ü274. ISBNü1564147118. .

Golden angle 37

Golden angle

The golden angle is the angle subtended by thesmaller (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 createdby sectioning the circumference of a circle according to the goldensection; that is, into two arcs such that the ratio of the length of thelarger arc to the length of the smaller arc is the same as the ratio of thefull circumference to the length of the larger arc.

Algebraically, let c be the circumference of a circle, divided into alonger arc of length a and a smaller arc of length b such that

and

The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.51†, orabout 2.399963 radians.

The name comes from the golden angle's connection to the golden ratio Ä; the exact value of the golden angle is

or

where the equivalences follow from well-known algebraic properties of the golden ratio.

DerivationThe golden ratio is equal to Äü=üa/b given the conditions above.

Let ã be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided bythe angular measurement of the circle.

But since

it follows that

This is equivalent to saying that Äü2 golden angles can fit in a circle.

The fraction of a circle occupied by the golden angle is therefore:

Golden angle 38

The golden angle g can therefore be numerically approximated in degrees as:

Golden angle in nature

The angle between successive florets in some flowers is the golden angle.

The golden angle plays a significant role in thetheory of phyllotaxis. Perhaps most notably, thegolden angle is the angle separating the florets ona sunflower.

References

í Vogel, H (1979). "A better way to constructthe sunflower head". MathematicalBiosciences 44 (44): 179Å189.doi:10.1016/0025-5564(79)90080-4.

í Prusinkiewicz, Przemyslaw; Lindenmayer,Aristid (1990). [[The Algorithmic Beauty ofPlants [1]]]. Springer-Verlag. pp.å101Ä107.ISBNå978-0387972978.

References[1] http:/ / algorithmicbotany. org/ papers/ #webdocs

Golden rhombus 39

Golden rhombus

The golden rhombus.

A golden rhombus is a rhombus whose diagonals are in the ratio , with as the golden ratio. The plural

of rhombus is rhombi, and a polyhedron whose faces are golden rhombi is a golden rhombohedron. One suchpolyhedron is the rhombic triacontahedron.The internal angles of the rhombus are approximately 63†26å and 116†34å. The dihedral angle between adjacentrhombi of the rhombic triacontahedron is 144†, which can be constructed by placing the short sides of two goldentriangles back-to-back.

External linksí Weisstein, Eric W., "Golden Rhombus [1]" from MathWorld.í Weisstein, Eric W., "Golden Rhombohedron [2]" from MathWorld.

References[1] http:/ / mathworld. wolfram. com/ GoldenRhombus. html[2] http:/ / mathworld. wolfram. com/ GoldenRhombohedron. html

Logarithmic spiral 40

Logarithmic spiral

Logarithmic spiral (pitch 10†)

Cutaway of a nautilus shell showing the chambersarranged in an approximately logarithmic spiral

Romanesco broccoli, which grows in a logarithmicspiral

A logarithmic spiral, equiangular spiral or growthspiral is a special kind of spiral curve which oftenappears in nature. The logarithmic spiral was firstdescribed by Descartes and later extensivelyinvestigated by Jacob Bernoulli, who called it Spiramirabilis, "the marvelous spiral".

Definition

In polar coordinates the curve can be written

as[1]

Logarithmic spiral 41

A section of the Mandelbrot set following alogarithmic spiral

A low pressure area over Iceland shows anapproximately logarithmic spiral pattern

The arms of spiral galaxies often have the shape of alogarithmic spiral, here the Whirlpool Galaxy

Logarithmic spiral 42

or

with being the base of natural logarithms, and and being arbitrary positive real constants.

In parametric form, the curve is

with real numbers and .

The spiral has the property that the angle Ä between the tangent and radial line at the point is constant. This

property can be expressed in differential geometric terms as

The derivative of is proportional to the parameter . In other words, it controls how "tightly" and in which

direction the spiral spirals. In the extreme case that ( ) the spiral becomes a circle of radius .

Conversely, in the limit that approaches infinity (Ä ç 0) the spiral tends toward a straight half-line. Thecomplement of Ä is called the pitch.

Spira mirabilis and Jacob BernoulliSpira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve hadalready been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given tothis curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of thespiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly asa result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such asnautilus shells and sunflower heads. Jakob Bernoulli wanted such a spiral engraved on his headstone along with thephrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiralwas placed there instead.[2] [3]

Logarithmic spiral 43

PropertiesThe logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between theturnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances areconstant.

Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling themgives the same result as rotating them). Scaling by a factor gives the same as the original, without rotation.They are also congruent to their own involutes, evolutes, and the pedal curves based on their centers.Starting at a point and moving inward along the spiral, one can circle the origin an unbounded number of timeswithout reaching it; yet, the total distance covered on this path is finite; that is, the limit as goes toward isfinite. This property was first realized by Evangelista Torricelli even before calculus had been invented.[4] The totaldistance covered is , where is the straight-line distance from to the origin.

The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, toall logarithmic spirals in the complex plane with centre at 0. (Up to adding integer multiples of to the lines, themapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle betweenthe line and the imaginary axis.

The function , where the constant is a complex number with non-zero imaginary part, maps the real lineto a logarithmic spiral in the complex plane.One can construct a golden spiral, a logarithmic spiral that grows outward by a factor of the golden ratio for every 90degrees of rotation (pitch about 17.03239 degrees), or approximate it using Fibonacci numbers.

Logarithmic spirals in natureIn several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows someexamples and reasons:

í The approach of a hawk to its prey. Their sharpest view is at an angle to their direction of flight; this angle is thesame as the spiral's pitch.[5]

í The approach of an insect to a light source. They are used to having the light source at a constant angle to theirflight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will resultin a practically straight line.[6]

í The arms of spiral galaxies.[7] Our own galaxy, the Milky Way, has several spiral arms, each of which is roughlya logarithmic spiral with pitch of about 12 degrees.[8]

í The nerves of the cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epitheliallayer of the cornea in a logarithmic spiral pattern).[9]

í The arms of tropical cyclones, such as hurricanes.[10]

í Many biological structures including the shells of mollusks.[11] In these cases, the reason may be constructionfrom expanding similar shapes, as shown for polygonal figures in the accompanying graphic.

í Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay,California is an example of such a type of beach.[12]

Logarithmic spiral 44

References[1] Priya Hemenway (2005). Divine Proportion: ä Phi in Art, Nature, and Science. Sterling Publishing Co. ISBNü1402735227.[2] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books.

ISBNü0-7679-0815-5.[3] Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes." p. 206[4] Carl Benjamin Boyer (1949). The history of the calculus and its conceptual development (http:/ / books. google. com/

books?id=KLQSHUW8FnUC& pg=PA133). Courier Dover Publications. p.ü133. ISBNü9780486605098. .[5] Chin, Gilbert J. (8 December 2000), "Organismal Biology: Flying Along a Logarithmic Spiral" (http:/ / www. sciencemag. org/ cgi/ content/

short/ 290/ 5498/ 1857c), Science 290 (5498): 1857, doi:10.1126/science.290.5498.1857c,[6] John Himmelman (2002). Discovering Moths: Nighttime Jewels in Your Own Backyard (http:/ / books. google. com/

books?id=iGn6ohfKhbAC& pg=PA63). Down East Enterprise Inc. p.ü63. ISBNü9780892725281. .[7] G. Bertin and C. C. Lin (1996). Spiral structure in galaxies: a density wave theory (http:/ / books. google. com/ books?id=06yfwrdpTk4C&

pg=PA78). MIT Press. p.ü78. ISBNü9780262023962. .[8] David J. Darling (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes (http:/ / books. google. com/

books?id=nnpChqstvg0C& pg=PA188). John Wiley and Sons. p.ü188. ISBNü9780471270478. .[9] C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure

and regeneration," Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.[10] Andrew Gray (1901). Treatise on physics, Volume 1 (http:/ / books. google. com/ books?id=ArELAAAAYAAJ& pg=PA357). Churchill.

p.ü356Å357. .[11] Michael Cortie (1992). "The form, function, and synthesis of the molluscan shell" (http:/ / books. google. com/ books?id=Ga8aoiIUx1gC&

pg=PA370). In Istv´n Hargittai and Clifford A. Pickover. Spiral symmetry. World Scientific. p.ü370. ISBNü9789810206154. .[12] Allan Thomas Williams and Anton Micallef (2009). Beach management: principles and practice (http:/ / books. google. com/

books?id=z_vKEMeJXKYC& pg=PA14). Earthscan. p.ü14. ISBNü9781844074358. .

í Weisstein, Eric W., " Logarithmic Spiral (http:/ / mathworld. wolfram. com/ LogarithmicSpiral. html)" fromMathWorld.

í Jim Wilson, Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves (http:/ / jwilson. coe. uga. edu/EMT668/ EMAT6680. F99/ Erbas/ KURSATgeometrypro/ related curves/ related curves. html), University ofGeorgia (1999)

í Alexander Bogomolny, Spira Mirabilis - Wonderful Spiral (http:/ / www. cut-the-knot. org/ Curriculum/Geometry/ Mirabilis. shtml), at cut-the-knot

External linksí Spira mirabilis (http:/ / jwilson. coe. uga. edu/ EMT668/ EMAT6680. F99/ Erbas/ KURSATgeometrypro/ golden

spiral/ logspiral-history. html) history and mathí Astronomy Picture of the Day (http:/ / antwrp. gsfc. nasa. gov/ apod/ ap030925. html), Hurricane Isabel vs. the

Whirlpool Galaxyí Astronomy Picture of the Day (http:/ / antwrp. gsfc. nasa. gov/ apod/ ap080517. html), Typhoon Rammasun vs.

the Pinwheel Galaxyí SpiralZoom.com (http:/ / SpiralZoom. com), an educational website about the science of pattern formation, spirals

in nature, and spirals in the mythic imagination.í Online exploration using JSXGraph (JavaScript) (http:/ / jsxgraph. uni-bayreuth. de/ wiki/ index. php/

Logarithmic_spiral)

Canons of page construction 45

Canons of page construction

Recto page from a rare Blackletter Bible (1497)

The canons of page construction are a set of principles in the field ofbook design used to describe the ways that page proportions, marginsand type areas (print spaces) of books are constructed.

The notion of canons, or laws of form, of book page construction waspopularized by Jan Tschichold in the mid to late twentieth century,based on the work of J. A. van de Graaf, Ra¨l M. Rosarivo, HansKayser, and others.[1] Tschichold wrote àThough largely forgottentoday, methods and rules upon which it is impossible to improve havebeen developed for centuries. To produce perfect books these ruleshave to be brought to life and applied.â[2] Kayser's 1946 Einharmonikaler Teilungskanon[3] had earlier used the term canon in thiscontext.

Typographers and book designers apply these principles to this day,with variations related to the availability of standardized paper sizes,and the diverse types of commercially printed books.[4]

Van de Graaf canon

Van de Graaf devised this construction to showhow Gutenberg and others may have divided their

page to achieve margins of one-ninth andtwo-ninths and a type area in the same

proportions as the page.

The Van de Graaf canon is a historical reconstruction of a method thatmay have been used in book design to divide a page in pleasingproportions.[5] This canon is also known as the "secret canon" used inmany medieval manuscripts and incunabula.

The geometrical solution of the construction of Van de Graaf's canon,which works for any page width:height ratio, enables the book designerto position the text body in a specific area of the page. Using thecanon, the proportions are maintained while creating pleasing andfunctional margins of size 1/9 and 2/9 of the page size.[6] The resultinginside margin is one-half of the outside margin, and of proportions2:3:4:6 (inner:top:outer:bottom) when the page proportion is 2:3 (moregenerally 1:R:2:2R for page proportion 1:R[7] ). This method wasdiscovered by Van de Graaf, and used by Tschichold and othercontemporary designers; they speculate that it may be older.[8]

The page proportions vary, but most commonly used is the 2:3 proportion. Tschichold writes "For purposes of bettercomparison I have based his figure on a page proportion of 2:3, which Van de Graaf does not use."[9] In this canonthe text area and page size are of same proportions, and the height of the text area equals the page width. This canonwas popularized by Jan Tschichold in his book The Form of the Book.[1]

Robert Bringhurst, in his The Elements of Typographic Style, asserts that the proportions that are useful for theshapes of pages are equally useful in shaping and positioning the textblock. This was often the case in medievalbooks, although later on in the Renaissance, typographers preferred to apply a more polyphonic page in which theproportions of page and textblock would differ.[10]

Canons of page construction 46

Golden canon

Tschichold's "golden canon of page construction"here illustrated by a synthesis of Tschichold'sfigure thereof, with the diagonals and circle,combined with Rosarivo's construction bydivision of the page into ninths. These two

constructions rely on the 2:3 page ratio to give atype area height equal to page width as

demonstrated by the circle, and result in marginproportions 2:3:4:6. For other page ratios,

Rosarivo's method of ninths is equivalent to vande Graaf's canon, as Tschichold observed.

Medieval manuscript framework according toTschichold, in which a text area proportioned

near the golden ratio is constructed. "Pageproportion is 2:3, text area proportioned in the

Golden Section."[9]

Tschichold's "golden canon of page construction"[1] is based on simpleinteger ratios, equivalent to Rosarivo's "typographical divineproportion."[11]

Interpretation of Rosarivo

Ra¨l Rosarivo analyzed Renaissance books with the help of a draftingcompass and a ruler, and concluded in his Divina proporciÅntipogrÇfica ("Typographical Divine Proportion", first published in1947) that Gutenberg, Peter Sch°ffer, Nicolaus Jenson and others hadapplied the golden canon of page construction in their works.[12]

According to Rosarivo, his work and assertion that Gutenberg used the"golden number" 2:3, or "secret number" as he called it, to establish theharmonic relationships between the diverse parts of a work,[13] wasanalyzed by experts at the Gutenberg Museum and re-published in theGutenberg-Jahrbuch, its official magazine.[14] Ros Vicente points outthat Rosarivo "demonstrates that Gutenberg had a module differentfrom the well-known one of Luca Paccioli" (the golden ratio).[14]

Tschichold also interprets Rosarivo's golden number as 2:3, saying:

In figure 5 the height of the type area equals the width of thepage: using a page proportion of 2:3, a condition for this canon,we get one-ninth of the paper width for the inner margin,two-ninths for the outer or fore-edge margin, one-ninth of thepaper height for the top, and two-ninths for the bottom margin.Type area and paper size are of equal proportions. ... What Iuncovered as the canon of the manuscript writers, Raul Rosarivoproved to have been Gutenberg's canon as well. He finds the sizeand position of the type area by dividing the page diagonal intoninths.[9]

The figures he refers to are reproduced in combination here.

John Man's interpretation of Gutenberg

Historian John Man suggests that Gutenberg's Bible page was based on the golden ratio (commonly approximated asthe decimal 0.618 or the ratio 5:8), and that the printed area also had that shape.[15] He quotes the dimensions ofGutenberg's half-folio Bible page as 30.7 x 44.5ücm, a ratio of 1:1.45, close to Rosarivo's golden 2:3 (1.5) but not tothe golden ratio 1.618.

Canons of page construction 47

Tschichold and the golden sectionBuilding on Rosarivo's work, contemporary experts in book design such as Tschichold and Richard Hendel assert aswell that the page proportion of the golden section (21:34) has been used in book design, in manuscripts, andincunabula, mostly in those produced between 1550 and 1770. Hendel writes that since Gutenberg's time, books havebeen most often printed in an upright position, that conform loosely, if not precisely, to the golden ratio.[16]

Tschichold's drawing of an octavo-format pageproportioned in the golden ratio or golden section"34:21". The text area and margin proportions are

determined by the starting page proportions.

These page proportions based on the golden section or golden ratio, areusually described through its convergents such as 2:3, 5:8, and 21:34.

Tschichold says that common ratios for page proportion used in bookdesign include as 2:3, 1:Ç3, and the golden section. The image withcircular arcs depicts the proportions in a medieval manuscript, thataccording to Tschichold feature a "Page proportion 2:3. Marginproportions 1:1:2:3. Text area in accord with the Golden Section. Thelower outer corner of the text area is fixed by a diagonal as well."[17]

By accord with the golden section, he does not mean exactly equal to,which would conflict with the stated proportions.

Tschichold refers to a construction equivalent to van de Graaf's orRosarivo's with a 2:3 page ratio as "the Golden Canon of book pageconstruction as it was used during late Gothic times by the finest ofscribes." For the canon with the arc construction, which yields a text area ratio closer to the golden ratio, he says "Iabstracted from manuscripts that are older yet. While beautiful, it would hardly be useful today."[18]

Of the different page proportions that such a canon can be applied to, he says "Book pages come in manyproportions, i.e., relationships between width and height. Everybody knows, at least from hearsay, the proportion ofthe Golden Section, exactly 1:1.618. A ratio of 5:8 is no more than an approximation of the Golden Section. It wouldbe difficult to maintain the same opinion about a ratio of 2:3."[19]

And he expresses a preference for certain ratios over others: "The geometrically definable irrational page proportionslike 1:1.618 ( Golden Section), 1:Ç2, 1:Ç3, 1:Ç5, 1:1.538, and the simple rational proportions of 1:2, 2:3, 5:8 and 5:9I call clear, intentional and definite. All others are unclear and accidental ratios. The difference between a clear andan unclear ratio, though frequently slight, is noticeable. ... Many books show none of the clear proportions, butaccidental ones."[20]

John Man's quoted Gutenberg page sizes are in a proportion not very close to the golden ratio,[15] but Rosarivo's orvan de Graaf's construction is applied by Tschichold to make a pleasing text area on pages of arbitrary proportions,even such accidental ones.

Current applicationsRichard Hendel, associate director of the University of North Carolina Press, describes book design as a craft with itsown traditions and a relatively small body of accepted rules.[21] The dust cover of his book, On Book Design,features the Van de Graaf canon.

Christopher Burke, in his book on German typographer Paul Renner, creator of the Futura typeface, described hisviews about page proportions:

Renner still championed the traditional proportions of margins, with the largest at the bottom of a page, 'because we hold the book by the lower margin when we take it in the hand and read it'. This indicates that he envisioned a small book, perhaps a novel, as his imagined model. Yet he struck a pragmatic note by adding that the traditional rule for margin proportions cannot be followed as a doctrine: for example, wide margins for pocket books would be counter-productive. Similarly, he refuted the notion that the type area must have the

Canons of page construction 48

same proportions as the page: he preferred to trust visual judgment in assessing the placement of the type areaon the page, instead of following a pre-determined doctrine.[22]

Bringhurst describes a book page as a tangible proportion, which together with the textblock produce an antiphonalgeometry, which has the capability to bind the reader to the book, or conversely put the reader's nerve on edge ordrive the reader away.[23]

Footnotes[1] Tschichold, Jan, The Form of the Book. p.46, Hartley & Marks (1991), ISBN 0-88179-116-4.[2] As cited in Hendel, Richard. On Book Design, p.7[3] Hans Kayser, Ein harmonikaler Teilungskanon: Analyse einer geometrischen Figur im Bauhçttenbuch Villard de Honnecourt (A canon for

harmonious page division: analysis of a geometric figure in Bauhaus book of Villard de Honnecourt). Zurich: Occident-Verlag, 1946. cited byweb page loaded 2006-09-11 Writings on Villard de Honnecourt, 1900-1949 (http:/ / www. villardman. net/ bibliography/ bibliog. 1900-1949.html) "An article-length (p. 32) attempt to demonstrate the use of Pythagorian musical proportion as the basis for the geometry in three ofVillard's figures: fol. 18r, two figures at the bottom; and fol. 19r, rightmost figure in the second row from the top. While the geometric designitself is unquestionably that generated from the Pythagorian monochord, Kayser does not convince the reader that Villard understood itsmusical basis. Kayser apparently worked from photographs of the original folios, and the significance of Kayser's claim may be summarizedin his own admission (p.30) that Villard's geometry does not match that of the Pythagorean design when correctly drawn."

[4] Egger, Willi. "Help! The Typesetting Area" (http:/ / www. ntg. nl/ maps/ pdf/ 30_13. pdf) (PDF). De Nederlandstalige TeX Gebruikersgroep.. Retrieved 2008-03-16.

[5] Van de Graaf, J. A. , Nieuwe berekening voor de vormgeving. (1946) (as cited by Tschichold and others; original not examined)[6] Tschichold, Jan, The Form of the Book. pp.28,37,48,51,58,61,138,167,174, Hartley & Marks (1991), ISBN 0-88179-116-4.[7] Max, Stanley M. (2010) "The 'Golden Canon' of book-page construction: proving the proportions geometrically," Journal of Mathematics and

the Arts, 4:3, 137-141. (http:/ / dx. doi. org/ 10. 1080/ 17513470903458205)[8] Hurlburt, Allen, Grid: A Modular System for the Design and Production of Newspapers, Magazines, and Books, p.71, John Wiley and Sons

(1982) ISBN 0-471-28923-X[9] Tschichold , The Form of the Book p.45[10] Bringhurst, The Elements of Typographic Style, p.163[11] Rosarivo, Ra¨l M., Divina proporciÅn tipogrÇfica, La Plata, Argentina (1953). Previous editions: 1948 and 1947. Brief discussion about his

work, is available online in Spanish (http:/ / fabiancarreras. com. ar/ rmr/ ladivina. htm)[12] Carreras, Fabi´n, "Rosarivo 1903 - 2003" (http:/ / fabiancarreras. com. ar/ rmr/ ). . Retrieved 2008-03-16.[13] Rosarivo, Ra¨l M., Divina proporciÅn tipogrÇfica, La Plata, Argentina, "[...] el n¨mero de oro o n¨mero clave en que Gutenberg se basû

para establecer las relaciones armûnicas que guardan las diversas partes de una obra"[14] Ros, Vicente, Infodiversidad. Ral Mario Rosarivo o el amor al libro, Sociedad de Investigaciones Bibilotecolûgicas, Argentina Vol. 7

(2004) Available online (Spanish) (http:/ / redalyc. uaemex. mx/ redalyc/ pdf/ 277/ 27700106. pdf) (PDF)[15] Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp.166Å67, Wiley, ISBN 0-471-21823-5. "The half-folio page

(30.7 x 44.5 cm) was made up of two rectangles Ä the whole page and its text area Ä based on the so called 'golden section', which specifiesa crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8 (footnote: The ratiois 0.618.... ad inf commonly rounded to 0.625)"

[16] Hendel, Richard, On Book Design, p.34, Yale University Press (1998), ISBN 0-300-07570-7[17] Tschichold , The Form of the Book, p.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. Thelower outer corner of the text area is fixed by a diagonal as well." (in the Dutch version, "letterveld volgens de Gulden Snede" Ä text area inaccord with the Golden Section)

[18] Tschichold , The Form of the Book p.44[19] Tschichold , The Form of the Book, p.37[20] Tschichold , The Form of the Book pp.37Å38[21] Hendel, Richard, On Book Design pp.1Å5[22] Christopher, Burke, Paul Renner: The Art of Typography, Princeton Architectural Press, 1999, ISBN 1-56898-158-9[23] Bringhurst, The elements of typographic style (1999), p.145

Canons of page construction 49

Referencesí Bringhurst, Robert (1999). The elements of typographic style. Point Roberts, WA: Hartley & Marks. p.ü145.

ISBNü0-88179-132-6.í Burke, Christopher. Paul Renner : The Art of Typography. New York: Princeton Architectural Press.

ISBNü1-56898-158-9.í Egger, Willi, Help! The Typesetting Area (http:/ / www. ntg. nl/ maps/ pdf/ 30_13. pdf) (PDF) (shows the Van de

Graaf canon and a variant that divides the page into twelfths)í Hendel, Richard (1998). On book design. New Haven, Conn: Yale University Press. ISBNü0-300-07570-7.í Infodiversidad. Ral Mario Rosarivo o el amor al libro, Sociedad de Investigaciones Bibilotecolûgicas, Argentina

Vol. 7 (2004)í Hurlburt, Allen. Grid: A Modular System for the Design and Production of Newspapers, Magazines, and Books.

New York: Wiley. ISBNü0-471-28923-X.í Rosarivo, Ra¨l M., Divina proporciÅn tipogrÇfica, La Plata, Argentina (1953). Previous editions: 1948 and 1947í Tschichold, Jan (1991). The form of the book: essays on the morality of good design. Point Roberts, WA: Hartley

& Marks. ISBNü0-88179-116-4.

Further readingí Elam, Kimberly (2001). Geometry of design: studies in proportion and composition. New York: Princeton

Architectural Press. ISBNü1-56898-249-6.í Luca Pacioli, De Divina Proportione (1509) (the originator of the excitement over the golden ratio)í Lehmann-Haupt, Hellmut, Five Centuries of Book Design: A Survey of Styles in the Columbia Library, Columbia

University, (1931)

External linksí "A Tribute to Richard Eckersley: British-born Book Designer" (http:/ / www. bobolinkbooks. com/ DesignHist/

Eckersley. html).í "Consistent Correlation Between Book Page and Type Area" (http:/ / learning. north. londonmet. ac. uk/ epoc/

tschichd. htm). chapter from The Form of the Bookí "Rosarivo - Divina proporciûn tipogr´fica" (http:/ / rosarivo. com. ar/ ladivina. htm) (in Spanish).

List of works designed with the golden ratio 50

List of works designed with the golden ratioWorks designed with the golden ratio are works of human design that are proportioned according to the goldenratio, an irrational number that is approximately 1.618; it is often denoted by the Greek letter Ç (phi).

Early historyIt is claimed that Stonehenge (3100 BC Å 2200 BC) has golden ratio proportions between its concentric circles.[1] [2]

Kimberly Elam proposes this relation as early evidence of human cognitive preference for the golden ratio.[3]

However, others point out that this interpretation of Stonehenge "may be doubtful" and that the geometricconstruction that generates it can only be surmised.[2]

Various authors discern golden ratio proportions in Egyptian, Summerian and Greek vases, Chinese pottery, Olmecsculptures, and Cretan and Mycenaean products from the late Bronze Age, which predates by about 1,000 years theGreek mathematicians who were first known to have studied the golden ratio.[2] [4] However, the historical sourcesare obscure, and the analyses are difficult to compare because they employ differing methods.[2]

The Great Pyramid of Giza (constructed c. 2570 BC by Hemiunu) exhibits the golden ratio according to variouspyramidologists, including Charles Funck-Hellet.[4] [5] John F. Pile, interior design professor and historian, hasclaimed that Egyptian designers sought the golden proportions without mathematical techniques and that it iscommon to see the 1.618:1 ratio, along with many other simpler geometrical concepts, in their architectural details,art, and everyday objects found in tombs. In his opinion, "That the Egyptians knew of it and used it seems certain."[6]

Even before the beginning of these theories, some other historians and mathematicians have always proposedalternative theories for the pyramid designs that are not related to any use of the golden ratio, and are instead basedon purely rational slopes that only approximate the golden ratio.[7] The Egyptians of those times apparently did notknow the Pythagorean theorem; the only right triangle whose proportions they knew was the 3:4:5 triangle.[8]

Olmos states that the Sculpture of King Gudea (c. 2350 BC) clearly has golden proportions between all of itssecondary elements repeated many times at its base.[4]

Greece

The Acropolis of Athens (468Å430 BC), including the Parthenon,according to some studies, has many proportions that approximate thegolden ratio.[9] Other scholars question whether the golden ratio wasknown to or used by Greek artists and architects as a principle ofaesthetic proportion.[10] Building the Acropolis is calculated to havebeen started around 600 BC, but the works said to exhibit the goldenratio proportions were created from 468 BC to 430 BC.

The Parthenon (447Å432 BC), was a temple built on the Acropolis inthe 5th century BC for the Greek goddess Athena. It is the mostimportant surviving building of Classical Greece. The Parthenon'sfacade as well as elements of its facade and elsewhere can be circumscribed by a progression of goldenrectangles.[11] Some more recent studies dispute the view that the golden ratio was employed in the design.[10] [12]

[13]

The Greek sculptor Phidias (c. 480Åc. 430 BC) used the divine proportion in some of his sculptures, according to Hemenway.[14] He created Athena Parthenos in Athens and Statue of Zeus (one of the Seven Wonders of the Ancient World) in the Temple of Zeus at Olympia. He is believed to have been in charge of other Parthenon sculptures, although they may have been executed by his alumni or peers. Many art historians conclude that Phidias made

List of works designed with the golden ratio 51

meticulous use of the golden ratio in proportioning his sculptures. For this reason, in the early 20th century,American mathematician Mark Barr proposed using the Greek letter phi (Ç), the first letter of Phidias's name, todenote the golden ratio.[15]

According to Lothar Haselberger,[4] the temple of Apollo in Didyma (c. 334 BC), designed by Daphnis of Mileto andPaionios of Efesus, have golden proportions.

Prehispanic Mesoamerican architectureOlmos claims the presence of the golden ratio in a series of olmec heads, the Aztec calendar stone, and a series ofAztec permission house plans.

In the fifties, Manuel Amabilis applied some of the analysis methods of Frederik Macody Lund and Jay Hambidge toseveral plans and sections of prehispanic buildings, such as El Toloc and La Iglesia of Las Monjas, a notablecomplex of Terminal Classic buildings constructed in the Puuc architectural style at Chichen Itza. According to hisstudies, their proportions derived from a series of successively inscribed pentagons, circles and pentagrams, just asthe Gothic churches Lund studied do. Amabilis published his studies along with several self-explanatory images ofvarious other precolumbine buildings with golden proportions in La Arquitectura Precolombina de Mexico,[16]

which was awarded the gold medal and the title of Academico by the "Real Academia de Bellas Artes de SanFernando" (Spain) in the "Fiesta de la Raza" contest of 1929.

According to John Pile, The Castle of Chichen Itza, built by the Maya civilization sometime between the 11th and13th centuries AD to serve as a temple to the god Kukulcan, has golden proportions in its interior layout with wallsplaced so that the outer spaces relate to the center chamber as 0.618:1.[17]

Islamic architectureA geometrical analysis of the Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D.) reveals a consistentapplication of the golden ratio throughout the design, according to Boussora and Mazouz, who say it is found in theoverall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret.[18]

Panorama of the minaret and the courtyard (on the right)

List of works designed with the golden ratio 52

Buddhist architectureThe Stuppa of Borobudur in Java, Indonesia (built eighth to ninth century AD), the largest known Buddhist stupa,has the dimension of the square base related to the diameter of the largest circular terrace as 1.618:1, according toPile.[19]

Gothic era

Illustration of the Notre-Dame of Laon cathedral.According to Macody Lund, the superimposed

regulator lines show that the cathedral has goldenproportions.

In his 1919 book Ad Quadratum, Frederik Macody Lund, a historianwho studied the geometry of several gothic structures, claims that theCathedral of Chartres (begun in the 12th century), the Notre-Dame ofLaon (1157Å1205), and the Notre Dame de Paris (1160) are designedaccording to the golden ratio.[4] Other scholars argue that until Pacioli's1509 publication (see next section), the golden ratio was unknown toartists and architects.[10]

A 2003 conference on medieval architecture resulted in the book AdQuadratum: The Application of Geometry to Medieval Architecture.According to a summary by one reviewer:

Most of the contributors consider that the setting out wasdone ad quadratum, using the sides of a square and itsdiagonal. This gave an incommensurate ratio of [squareroot of (2)] by striking a circular arc (which could easilybe done with a rope rotating around a peg). Most alsoargued that setting out was done geometrically rather thanarithmetically (with a measuring rod). Some consideredthat setting out also involved the use of equilateral orPythagorean triangles, pentagons, and octagons. Twoauthors believe the Golden Section (or at least its approximation) was used, but its use in medieval timesis not supported by most architectural historians.[20]

List of works designed with the golden ratio 53

Renaissance

Leonardo Da Vinci's illustration of a human headfrom De Divina Proportione[21]

De divina proportione, written by Luca Pacioli in Milan in 1496Å1498,published in Venice in 1509,[21] features 60 drawings by Leonardo daVinci, some of which illustrate the appearance of the golden ratio ingeometric figures. Starting with part of the work of Leonardo DaVinci, this architectural treatise was a major influence on generationsof artists and architects.

Vitruvian Man, created by Leonardo da Vinci around the year 1492,[22]

is based on the theories of the man after which the drawing takes itsname, Vitruvius, who in De Architectura: The Planning of Temples (c.I BC) pointed that the planning of temples depends on symmetry,which must be based on the perfect proportions of the human body.Some authors feel there is no actual evidence that Da Vinci used thegolden ratio in Vitruvian Man;[23] however, Olmos[4] (1991) observesotherwise through geometrical analysis. He also proposes Leonardo daVinci's self portrait, Michelangelo's David (1501Å1504), AlbrechtDèrer's Melencolia and the classic violin design by the Masters ofCremona, as having similar regulator lines related to the golden ratio.

Da Vinci's Mona Lisa (c. 1503Å1506) "has been the subject of so manyvolumes of contradicting scholarly and popular speculations that itvirtually impossible to reach any unambiguous conclusions" with respect to the golden ratio, according to Livio.[10]

The Tempietto chapel at the Monastery of Saint Peter in Montorio, Rome, built by Bramante, has relations to thegolden ratio in its elevation and interior lines.[24]

The Baroque and the Spanish empireJose Villagran Garcia has claimed[25] that the golden ratio is an important element in the design of the Mexico CityMetropolitan Cathedral (circa 1667Å1813). Olmos claims the same for the design of the cities of Coatepec (1579),Chicoaloapa (1579) and Huejutla (1580), as well as the Mõrida Cathedral, the Acolman Temple, Cristo Crucificadoby Diego Vel´zquez (1639) and La Madona de Media Luna of Bartolomõ Esteban Murillo.[4]

Neoclassicism and romanticismLeonid Sabaneyev hypothesizes that the separate time intervals of the musical pieces connected by the "culminationevent", as a rule, are in the ratio of the golden section.[26] However the author attributes this incidence to the instinctof the mucisians: "All such events are timed by author's instinct to such points of the whole length that they dividetemporary durations into separate parts being in the ratio of the golden section."

In Surrey's Internet site, Ron Knott[27] exposes how the golden ratio is unintentionally present in several pieces ofclassical music:

í An article of American Scientist[28] (Did Mozart use the Golden mean?, March/April 1996), reports that John Putzfound that there was considerable deviation from ratio section division in many of Mozart's sonatas and claimedthat any proximity to this number can be explained by constraints of the sonata form itself.

í Derek Haylock[29] claims that the opening motif of Ludwig van Beethoven's Symphony No. 5 in C minor, Op. 67 (c. 1804Å08), occurs exactly at the golden mean point 0.618 in bar 372 of 601 and again at bar 228 which is the other golden section point (0.618034 from the end of the piece) but he has to use 601 bars to get these figures. This he does by ignoring the final 20 bars that occur after the final appearance of the motif and also ignoring bar

List of works designed with the golden ratio 54

387.

ImpressionismMatila Ghyka[30] and others[31] contend that Georges Seurat used golden ratio proportions in paintings like LaParade, Le Pont de Courbevoie and Bathers at Asniéres. However, there is no direct evidence to support theseclaims.[23]

NeogothicAccording to the official tourism page of Buenos Aires, Argentina, the ground floor of the Palacio Barolo (1923),designed by Italian architect Mario Palanti, is built according to the golden section.[32]

CubismFrench mathematician, Henri Poincarõ, taught the properties of the golden ratio to Juan Gris, who developed Cubismfeaturing them.[33]

SurrealismThe Sacrament of the Last Supper (1955): The canvas of this surrealist masterpiece by Salvador Dalú is a goldenrectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus anddominates the composition.[10] [34]

De StijlSome works in the Dutch artistic movement called De Stijl, or neoplasticism, exhibit golden ratio proportions. PietMondrian used the golden section extensively in his neoplasticist, geometrical paintings, created circa 1918Å38.[31]

[35] Mondrian sought proportion in his paintings by observation, knowledge and intuition, rather than geometrical ormathematical methods.[36]

Juan Gris also used golden ratio proportions.[31]

Modern architecture

Mies Van der RoheThe Farnsworth House has been described as "the proportions, within the glass walls, approach 1:2"[37] and "with awidth to length ratio of 1:1.75 (nearly the golden section)"[38] and has been studied with his other works in relationto the golden ratio.[39]

Le CorbusierThe Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his designphilosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universewas closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eyeand clear in their relations with one another. And these rhythms are at the very root of human activities. Theyresound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the GoldenSection by children, old men, savages and the learned."[40]

Modulor: Le Corbusier explicitly used the golden ratio in his 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

List of works designed with the golden ratio 55

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 toan extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, thensubdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in theModulor system.[41]

In The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and MechanicsLe Corbusier reveals he used his system in the Marseilles Unite D'Habitation (in the general plan and section, thefront elevation, plan and section of the apartment, in the woodwork, the wall, the roof and some prefabricatedfurniture), a small office in 35 rue de SÆvres, a factory in Saint-Die and the United Nations Headquarters building inNew York City.[42] Many authors claim that the shape of the facade of the second is the result of three goldenrectangles;[43] however, each of the three rectangles that can actually be appreciated have different heights.

Post-modern architectureAnother Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses hedesigned 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.[44]

Contemporary musicJames Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardlyglissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equaltempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutivetones are a lower or higher pitch already, or soon to be, produced.

Ernù Lendvai analyzes Bõla Bartûk's works as being based on two opposing systems, that of the golden ratio and theacoustic scale,[45] though other music scholars reject that analysis.[10] In Bartûk's Music for Strings, Percussion andCelesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[46] The French composer Erik Satie usedthe golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music anotherworldly symmetry.

The golden ratio is also apparent in the organisation of the sections in the music of Claude Debussy's Image:Reflections in the Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and themain climax sits at the phi position."[46]

The musicologist Roy Howat has observed that the formal boundaries of La mer correspond exactly to the goldensection.[47] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidencesuggests that Debussy consciously sought such proportions.[48]

This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled "1.618" inhomage to the golden ratio. The track features musical versions of the ratio and the accompanying video [49] (linkbroken) displays various animated versions of the golden mean.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claimsthat this arrangement improves bass response and has applied for a patent on this innovation.[50]

According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics haveinvolved Fibonacci numbers and the related golden ratio."[51]

List of works designed with the golden ratio 56

References[1] Prash Trivede. The 27 Celestial Portals: The Real Secret Behind the 12 Star-Signs. Lotus Press. Page 397[2] Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science (http:/ / books. google. com/

books?id=rqzaQo6CaA0C& pg=PA118& dq=stonehenge+ golden-section& lr=& as_brr=3& ei=MY_cR9H5HKiwtAP_26WJDg&sig=zTEHmqsZD1uUuqvofxzlSYrIKNY). Walter de Gruyter. pp.ü118. ISBNü3110129906. .

[3] Kimberly Elam. Geometry of Design: Studies in Proportion and Composition By Kimberly Elam. Princeton Architectural Press. p. 6.[4] CHANFØN OLMOS, Carlos. Curso sobre ProporciÅn. Procedimientos reguladors en construcciÅn. Convenio de intercambio

UNAMÅUADY. Mõxico - Mõrica, 1991[5] Lidwell, William; Holden, Kritina; and Butler, Jill. Universal Principles of Design. Rockport Publishers. October 1, 2003. Page 96[6] Pile, John F. A history of interior design (http:/ / books. google. com/ books?id=YVQJvcI1XeoC& pg=PA26& dq=intitle:"interior+ design"+

inauthor:pile+ "everyday+ objects"& lr=& as_brr=0& ei=I1LfR63LLY6IswOYn4nvAQ& sig=o55FhLnB7odaYpCVuWP7wA7RfE8).Laurence King Publishing. 2005. Page 29.

[7] Eli Maor, Trigonometric Delights (http:/ / books. google. com/ books?id=xU0IPNGRXqEC& pg=PA7& dq=intitle:trigonometric+inauthor:maor+ seked+ rhind& lr=& as_brr=0& ei=GYo5SJHPCJnstAPNk5S8Cw& sig=dCMf0wTbtg6KwHNbxMS-2Ra_yDc), PrincetonUniv. Press, 2000

[8] Eric Temple Bell, The Development of Mathematics, New York: Dover, 1940, p.40[9] Van Mersbergen, Audrey M. (1998). "Rhetorical Prototypes in Architecture: Measuring the Acropolis" (http:/ / www. questia. com/ PM.

qst?a=o& se=gglsc& d=5001403053). Communication Quarterly (Eastern Communication Association) 46 (2): 194Å195. .[10] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books.

ISBNü0-7679-0815-5.[11] Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", Philosophical Polemic Communication

Quarterly, Vol. 46, 1998.[12] Markowsky, George (January 1992). "Misconceptions About the Golden Ratio". The College Mathematics Journal 23 (1): 2Å19.

doi:10.2307/2686193.[13] Markowsky, George http:/ / laptops. maine. edu/ GoldenRatio. pdf[14] Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science (http:/ / books. google. com/ books?id=akjSa4AdK4AC&

pg=PA96& dq=inauthor:Hemenway+ phidias+ divine& lr=& as_brr=3& ei=EpvcR_TVIabutAO5hMjwAQ&sig=DXfN9iQeiqf8HqeohZ1Oz_5qRRk#PPA96,M1). New York: Sterling. pp.ü96. ISBNü1-4027-3522-7. .

[15] Cook, Theodore Andrea (1979). The Curves of Life, p. 420. Courier Dover Publications, ISBN 0-486-23701-X.[16] Manue Amabilis (http:/ / books. google. com. mx/ books?um=1& hl=en& q=manuel+ amabilis). (1956) La Arquitectura Precolombina en

Mexico. Editorial Orion. P. 200, 202. (http:/ / www. antiqbook. com/ boox/ dailey/ 5479. shtml)[17] PILE, John F. A history of interior design (http:/ / books. google. com/ books?id=YVQJvcI1XeoC& pg=PA23& dq=chichen-itza+ +

intitle:A+ intitle:history+ intitle:of+ intitle:interior+ intitle:design& lr=& as_brr=0& ei=-CvfR5vDF5u8swPmxOXzAQ&sig=4FQS1K4hAjcgqmGBNET_BleGQIE#PPA23,M1). Laurence King Publishing. 2005. Page 23.

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[19] PILE, John F. A history of interior design . Laurence King Publishing. 2005. Page 88.[20] "The geometry of Romanesque and Gothic cathedrals. (Ad Quadratum: The Application of Geometry to Medieval Architecture) (Book

Review)". Architectural Science Review 46 (3): 337Å338. September 1, 2003.[21] Pacioli, Luca. De Divina Proportione. Venice, 1509.[22] TUBERVILLE, Joseph. A Glimmer of Light from the Eye of a Giant: Tabular Evidence of a Monument in Harmony with the Universe.

2001. Page 1[23] Keith Devlin (June 2004). "Good stories, pity they're not true" (http:/ / www. maa. org/ devlin/ devlin_06_04. html). MAA Online.

Mathematical Association of America. .[24] PILE, John F. A history of interior design . Laurence King Publishing. 2005. Page 130.[25] VILLAGRAN GARCIA, Jose. Los Trazos Reguladores de la Proporcion Arquitectonica. Memoria de el Colegio Nacional, Volume VI, No.

4, Editorial de El Colegio Nacional, Mexico, 1969[26] SABANEEV, Leonid and JOFFE, Judah A. Modern Russian Composers. 1927.[27] KNOTT, Ron, [Ron Knott's web pages on Mathematic], Fibonacci Numbers and The Golden Section in Art, Architecture and Music (http:/ /

www. mcs. surrey. ac. uk/ Personal/ R. Knott/ Fibonacci/ fibInArt. html), Surrey University[28] MAY, Mike, Did Mozart use the Golden mean?, American Scientist, March/April 1996[29] HEYLOCK, Derek. Mathematics Teaching, Volume 84, p. 56-57. 1978[30] GHYKA, Matila. The Geometry of Art and Life. 1946. Page 162[31] STASZKOW, Ronald and BRADSHAW, Robert. The Mathematical Palette. Thomson Brooks/Cole. P. 372[32] Official tourism page of the city of Buenos Aires (http:/ / www. bue. gov. ar/ recorridos/ index. php?menu_id=52& info=auto_contenido&

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List of works designed with the golden ratio 57

[35] Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp. 247-48, Harcourt, Brace & World, ISBN0-87817-259-9

[36] PADOVAN, Richard. Proportion: Science, Philosophy, Architecture. Taylor & Francis. Page 26.[37] Neil Jackson (1996). The Modern Steel House (http:/ / books. google. com/ books?id=3pyDEb9RRPwC& pg=PA71& dq="Farnsworth+

House"+ golden& ei=3KTdR4OgLIrysgP6yMXqAQ& sig=SRQygYWI50dtLCaOXRf7KOwOyQw#PPA71,M1). Taylor & Francis.ISBNü0419217207. .

[38] Leland M. Roth (2001). American Architecture: A History (http:/ / books. google. com/ books?id=pH7rd6EFImgC& pg=PA433&dq="Farnsworth+ House"+ golden& ei=3KTdR4OgLIrysgP6yMXqAQ& sig=IkeDUUCZ6ruL_SMAEkQNEhdqTaE). Westview Press.ISBNü0813336619. .

[39] SANO, Junichi. Study on the Golden Ratio in the works of Mies van der Rolle : On the Golden Ratio in the plans of House with three Courtsand IIT Chapel. Journal of Arch tecture, Planning and Environmental Engineering Ä Academic Journal ,1993 Å 453,153-158 / ,

[40] Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor andFrancis, ISBN 0-419-22780-6

[41] 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".

[42] Le Corbusier, The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics,Birkh•user, 2000, p. 130

[43] Daniel Pedoe (1983). Geometry and the Visual Arts (http:/ / books. google. com/ books?id=g24GMnuf36MC& pg=PA121&dq=golden-ratio+ united-nations& lr=& as_brr=0& ei=O3fdR7XsOIrysgP6yMXqAQ& sig=_NDCQ2VQ1fYJLepf5I4vsofCxcw). CourierDover Publications. pp.ü121. ISBNü048624458X. .

[44] Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X[45] Lendvai, Ernù (1971). BÑla BartÅk: An Analysis of His Music. London: Kahn and Averill.[46] Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (http:/ / books. google. com/ books?id=ZgftUKoMnpkC& pg=PA83&

dq=bartok+ intitle:The+ intitle:Dynamics+ intitle:of+ intitle:Delight+ intitle:Architecture+ intitle:and+ intitle:Aesthetics& as_brr=0&ei=WkkSR5L6OI--ogLpmoyzBg& sig=Ijw4YifrLhkcdQSMVAjSL5g4zVk) (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X

[47] Roy Howat (1983). Debussy in Proportion: A Musical Analysis (http:/ / books. google. com/ books?id=4bwKykNp24wC& pg=PA169&dq=intitle:Debussy+ intitle:in+ intitle:Proportion+ golden+ la-mer& lr=& as_brr=0& ei=KFKlR5b5O4bOiQGQt82pCg&sig=oBWbHkWkhTG11w_BNdx89SWjQTY#PPA169,M1). Cambridge University Press. ISBNü0521311454. .

[48] Simon Trezise (1994). Debussy: La Mer (http:/ / books. google. com/ books?id=THD1nge_UzcC& pg=PA53& dq=inauthor:Trezise+golden+ evidence& lr=& as_brr=0& ei=DlSlR-7HJJXEigGR2sS5Cg& sig=5auw0tRu24Jq0aFKOjLyZ2u7BGo). Cambridge University Press.pp.ü53. ISBNü0521446562. .

[49] http:/ / stage6. divx. com/ BT/ show_video/ 1051714[50] "Pearl Masters Premium" (http:/ / www. pearldrum. com/ premium-birch. asp). Pearl Corporation. . Retrieved December 2, 2007.[51] Leon Harkleroad (2006). The Math Behind the Music (http:/ / books. google. com/ books?id=C3dsb7Qysh4C& pg=RA4-PA120&

dq=misguided+ music+ mathematics+ "golden+ ratio"& lr=& as_brr=0& ei=9GJwR7m-HIbktAO_-ayeBw&sig=h_YrTJ6LYBsfmhjexiQvmmPZFFM). Cambridge University Press. ISBNü0521810957. .

External linksí Nexux Network Journal Å Architecture and Mathematics Online. (http:/ / www. emis. de/ journals/ NNJ/ Frings.

html) Kim Williams Books

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Phidias üSource: http://en.wikipedia.org/w/index.php?oldid=419065034 üContributors: A Nobody, ABF, Adam Bishop, Adziura, Akhilleus, Al Silonov, Alansohn, Alex.muller, Alphachimp,Amit A., AnakngAraw, Apilok, Ariobarzan, Athinaios, Aymatth2, BD2412, Belovedfreak, Ben-Zin, Binabik80, Bits4tits, BorgQueen, Brebe, CRiyl, Can't sleep, clown will eat me, Carnun,Catalographer, Ceoil, Chaleyer61, Cimon Avaro, Crakkpot, DIEGO RICARDO PEREIRA, Daderot, Daniel 1992, Danny, David Schaich, Deucalionite, Douglasfrankfort, Dpwkbw,Dudeman5685, Dysepsion, Ed Poor, Ejosse1, Enviroboy, Excirial, Extransit, Fabricationary, Fahadsadah, FaustX, FeanorStar7, Flauto Dolce, Foolycooly111, Fordmadoxfraud, FourthAve,Galoubet, Geoffg, Gil Gamesh, GoingBatty, Gradiva, HTO, Haiduc, Halibutt, Hmains, Ilikejouge, ImMAW, Insane99, J.R. Hercules, J.delanoy, J.smith, JNW, Jacquerie27, James Arboghast,Jauerback, Jeff G., Joseph Solis in Australia, Josh3580, Jpbowen, Juanpdp, June w, Kelly Martin, LeaveSleaves, Lockley, Mackensen, Malo, Margacst, Mathwhiz 29, Mdebets, Meegs,Midnightblueowl, Modeha, Mtiffany, NawlinWiki, Nazar, Neddyseagoon, Neurolysis, Neutrality, NewEnglandYankee, Nicke Lilltroll, Noit, Nuno Tavares, Omegastar, Oreo Priest, Outriggr,PGSONIC, Penwhale, Peter cohen, Peter1219, Pheidias, Philip Trueman, Poshzombie, Private Pilot, Prodego, Prof saxx, PseudoSudo, RODERICKMOLASAR, Ricagambeda, Rmky87,Roastytoast, RuM, Ryan Postlethwaite, SWAdair, Sardanaphalus, Shakko, Simeon H, SimonP, Skeppy, Solipsist, Sparkit, Stambouliote, Stan Shebs, SteinbDJ, Steven J. Anderson,Subash.chandran007, SumerII, The Nut, The Thing That Should Not Be, The wub, Thejerm, Thingg, Tomdobb, Tomisti, Twospoonfuls, Utcursch, Vegetator, WadeSimMiser, Warrington,West.andrew.g, Wetman, Whale plane, WikHead, Wumple, XCrunner1422, XJamRastafire, Zazpi, ±≤≥¥µ∂∑∏π∫, 259 anonymous edits

Irrational number üSource: http://en.wikipedia.org/w/index.php?oldid=422514545 üContributors: 2D, A Stop at Willoughby, AbcXyz, Aitias, Akanemoto, Alansohn, [email protected],Amazing Steve, AndrewKepert, Androl, Antandrus, Arthur Rubin, AstroWiki, Athenean, Atif.t2, AustinKnight, AxelBoldt, Az1568, B4hand, BL, Barneca, Bdesham, Beach drifter, BenStandeven, Beremiz, Betterusername, BiT, Bidabadi, Bilal.alsallakh, Bjankuloski06en, Blue Tie, Bowlhover, Brfettig, Btg2290, C-4, CRGreathouse, Caltas, Captainj, Card, CardinalDan,Catherineyronwode, Charles Matthews, CharlesGillingham, ChrisfromHouston, Christian List, Chuck SMITH, Colm Keogh, Cometstyles, Connormah, Conversion script, CosineKitty,Cptmurdok, Crisco 1492, Crobzub, DMacks, DYLAN LENNON, Danny5000, Daran, Dauto, Demmy, DerHexer, Dharmabum420, Dialectric, Diggers2004, Discospinster, Dmcq, Dmr2,Dondegroovily, Dr Dec, Drunken Pirate, Dwineman, Dycedarg, Dylan Lake, Dysepsion, Dysprosia, ESkog, EdC, Edward, El C, Enviroboy, Epbr123, FF2010, Falcorian, Flyingspuds, FocalPoint,Franci.cariati, Fredrik, Fresheneesz, Furrykef, Fuzzypeg, Gene Ward Smith, Giftlite, GirasoleDE, Glane23, Glenn L, GregorB, Grubber, Grue, Gscshoyru, Gurch, Hairy Dude, Haonhien, Hdt83,Henrygb, Howabout1, Ibbn, Igny, J.delanoy, JForget, JHarris, Jagged 85, Jan Hidders, JeffBobFrank, Jogers, John Reid, Johncatsoulis, JonathanFreed, Josh Parris, Joshalex88, Jusdafax, KSmrq,Kayau, Kelly Martin, Khalidmathematics, Khukri, Koeplinger, Krasniy, Ktalon, Kuru, Kutulu, L Kensington, LOL, Lambiam, LiDaobing, Loadmaster, Loonymonkey, LordFoom, Luke-Jr,MacMed, Macy, Madmath789, Martin451, Masgatotkaca, Maverick starstrider, MaybeJesusMaybeNot, Mdd, Melchoir, Michael Hardy, Mild Bill Hiccup, Minesweeper, Mjb, Momo san, Mouseis back, Mr Stephen, Mreult, Mxn, Natalie Erin, NawlinWiki, Newone, Nihiltres, Njaikrishna, Nohat, Nono64, Numbo3, Nuno Tavares, Obradovic Goran, Oleg Alexandrov, Opelio,Oxymoron83, Panoramix, Paul August, Pbroks13, Pgb23, PierreAbbat, Pizza Puzzle, Plclark, Pne, Popovvk, Potatoswatter, Psb777, Quaeler, Quoth, RJGray, Raja Hussain, RandomP,Randomblue, Recentchanges, ResearchRave, Retrovirus, RexNL, Robert2957, Romanm, Ronhjones, Rsocol, S711, Saforrest, Salix alba, Sam Hocevar, Schmock, Schutz, Seaphoto, Selfworm,Setitup, Shahkent, Shawnc, Shoy, Shreeradha, Shreevatsa, Simetrical, SimonTrew, Sir Arthur Williams, Sk•pper°d, Smiloid, Some1new4ya, Sonicsuns, SpeedyGonsales, Spiffy sperry, Starx,Steel Tortoise, Stirred-not-shaken, SuperMidget, T. Moitie, TakuyaMurata, TallNapoleon, The Anome, The Thing That Should Not Be, Thecheesykid, Thejerm, Theonlydavewilliams,Thymaridas, Tide rolls, TimDoster, Tiptoety, Tkuvho, Tobby72, Tobias Bergemann, Toby Bartels, Tommy2010, Traxs7, Trumpet marietta 45750, Ulrich Mèller, Universalss, UserGoogol,Vaughan Pratt, VladimirReshetnikov, Vonbontee, Wayne Slam, Wheelingcrows, White Shadows, Whywhenwhohow, Wolfrock, Woohookitty, Worldrimroamer, Xantharius, Xaos, Xororaz,Youssefsan, Ysangkok, Yurik, Zero sharp, Zfr, Zoso96, Zundark, ªvar Arnfj°rº Bjarmason, éèéê, 482 anonymous edits

Golden rectangle üSource: http://en.wikipedia.org/w/index.php?oldid=423141600 üContributors: 15reidhay, 21655, APNelsonL, Aknorals, Alansohn, Algebraist, Andrew Powell, Anonymous Dissident, Anskas, Artis90, Azweirdazyou, Badanedwa, Beland, Binksternet, Bjankuloski06en, Bkonrad, Blobglob, CanadianLinuxUser, Canton Viaduct, Celestianpower, Charles Matthews,

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Golden spiral üSource: http://en.wikipedia.org/w/index.php?oldid=423823211 üContributors: Bento00, Charles Matthews, Dan Hetherington, DarcNessX, Dicklyon, Doniago, Egg, Eric Burnett,Flewis, Frietjes, Goldbren, Hairy Dude, Hephaestos, HopeMaker, Icairns, Icey, Iulius, Josh Grosse, Kauffner, Lemonflash, Lucyintheskywithdada, Mckaysalisbury, Melchoir, Nae'blis, OlEnglish,Pleasantville, Preston47, Ronhjones, Svick, THF, Tamfang, The Thing That Should Not Be, The stuart, Tide rolls, Tumble, Ukexpat, Versus22, Wiki alf, Wknight94, WoollyMind, 80 anonymousedits

Golden angle üSource: http://en.wikipedia.org/w/index.php?oldid=400818230 üContributors: Alexwcovington, Ambarsande, Captain Disdain, CecilBlade, Charles Matthews, D-Notice, DavidSchaich, Dicklyon, Duncan.france, Gene Nygaard, Icairns, Jamrb, Jonathan.s.kt, Linas, Michael Hardy, Michael Tiemann, Nekura, Noe, Palica, Postdlf, Salix alba, Starie17, Superm401, TheAnome, Vsmith, Zenohockey, Zzyzx11, 20 anonymous edits

Golden rhombus üSource: http://en.wikipedia.org/w/index.php?oldid=387184697 üContributors: AnonMoos, Bearcat, Computer97, Fullstop, Michael Tiemann, Xezbeth, Zom-B, 2 anonymousedits

Logarithmic spiral üSource: http://en.wikipedia.org/w/index.php?oldid=416262758 üContributors: Acdx, AnonMoos, AugPi, AxelBoldt, B4hand, Banus, CRGreathouse, ChessA4, Chris 73,Crobichaud, Cyp, Darwinek, David Eppstein, Dicklyon, Dino, Dude1818, Egil, Ejrh, Fabiform, FernandoFHC, Fisherted1, Fredrik, Garjun, Georgewilliamherbert, Giftlite, Grafen, GregorB,HERB, Jacquerie27, JessBr, Joriki, Josh Grosse, Kf4bdy, Kwamikagami, Linas, Maedin, Mainstreetmark, MarkSweep, MathMartin, Mckaysalisbury, Melchoir, Mgiganteus1, Michael Hardy,Morn, Noe, Nsande01, Objectivesea, Oekaki, Omegatron, PAR, Patrick, Phlake, PierreAbbat, Pion, RDBury, RandomP, Revancher, Rjanag, Romanm, Sbrools, Schewek, Sligocki, Snarius, SothoTal Ker, Speight, Svick, Tosha, Unyoyega, William Evans, WilliamKF, Xiutwel, Xplat, Yghwtrrl, Yosha, 58 anonymous edits

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List of works designed with the golden ratio üSource: http://en.wikipedia.org/w/index.php?oldid=421062217 üContributors: 20-dude, 21655, 3rdAlcove, Adoniscik, ArnoLagrange,Bibliomaniac15, Biruitorul, Colonies Chris, Cuddlyable3, Dicklyon, EmanWilm, Favonian, Finell, JackofOz, Johnuniq, Jossi, KathrynLybarger, Koavf, Korg, Libcub, LilHelpa, Mandarax,Onmywaybackhome, Open2universe, Orland, Ospalh, Reinyday, Rjwilmsi, Sidhekin, TheRingess, Ulric1313, Wisdom89, 53 anonymous edits

Image Sources, Licenses and Contributors 60