Golden Ratio - The Divine Ratio - Design Structure jx4r

The Divine Ratio Design Structure

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

Contents Articles 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

References Article Sources and Contributors

58

Image Sources, Licenses and Contributors

60

Article Licenses License

61

1

Golden Ratio Golden ratio In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.[1] Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.[2] [3] [4] Other

encountered include extreme and mean ratio,[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[6] golden number, [7] [8] [9]

and mean of Phidias. In this article the golden ratio is denoted by the Greek lowercase letter phi ( ) , while its reciprocal, variant Phi (

or

The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b.

, is denoted by the uppercase

).

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 golden ratioÄespecially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratioÄbelieving this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

Golden ratio

2

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship

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

List of numbers Å Irrational and suspected irrational numbers Ä 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

Hexadecimal

1.9E3779B97F4A7C15F39É

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 The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.[11]

Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (Ç) is used. Sometimes, the uppercase form (á) is [10] used for the reciprocal of the golden ratio, 1/Ç.

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. 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 golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."[5] Euclid explains a construction for cutting (sectioning) a line "in extreme and

mean ratio", i.e. the golden ratio.[12] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[13] Some of these propositions show that the golden ratio is an irrational number. The name "extreme and mean ratio" was the principal term used from the 3rd century BC[5] until about the 18th century. The modern history of the golden ratio starts with Luca Pacioli's De divina proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

Golden ratio

4

The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340," was written in 1597 by Prof. Michael Maestlin of the University of Tèbingen in a letter to his former student Johannes Kepler.[14] Since the twentieth century, the golden ratio has been represented by the Greek letter Ä or Å (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by Ç (tau, the first letter of the

ancient Greek root âäêëÄmeaning cut).[2] [15]

Timeline Timeline according to Priya Hemenway.[16] í Phidias (490Å430 BC) made the Parthenon statues that seem to embody the golden ratio. í Plato (427Å347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron), some of which are related to the golden ratio.

[17]

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

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

í Fibonacci (1170Å1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential 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 the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle. í Charles Bonnet (1720Å1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-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 describe this ratio, in 1835.[19] í Edouard Lucas (1842Å1891) gives the numerical sequence now known as the Fibonacci sequence its present name. í Mark Barr (20th century) suggests the Greek letter phi (Ä), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.[20] í Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

Golden ratio

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

Architecture The Parthenon's facade as well as elements of its facade and elsewhere are said by some to be circumscribed by golden rectangles.[21] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalõ says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular Many of the proportions of the Parthenon are alleged to exhibit the pentagons and decagons was duly observed, as well as golden ratio. in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."[22] And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not ed by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its

value."[23] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[24] It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis. The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."[25]

5

Golden ratio

6

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[26] Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[27] In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.[28]

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

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

Golden ratio

7

Leonardo da Vinci's illustrations of polyhedra in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.[29] But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not ed by anything in Leonardo's own writings.[30] Salvador Dalú, influenced by the works of Matila Ghyka,[31] explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[2] [32] Mondrian has been said to have used the golden section extensively in his geometrical paintings,[33] though other experts (including critic Yve-Alain Bois) have disputed this claim.[2] A statistical study on 565 works of art of different great painters,

Illustration from Luca Pacioli's De Divina Proportione applies geometric proportions to the human face.

performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[34] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.[35]

Book design According to Jan Tschichold,[37] There was a time when deviations from the truly beautiful page proportions 2:3, 1:Ç3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimetre.

Perceptual studies Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[2] [38]

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text [36] area proportioned in the Golden Section."

Golden ratio

Music James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced. Ernù Lendvai analyzes Bõla Bartûk's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[39] though other music scholars reject that analysis.[2] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[40] French composer Erik Satie used the golden 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 intervals 34, 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 golden section.[41] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests 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 expression and technical difficulty after about 2/3 of the piece. Pearl Drums positions the air vents on its Masters models based on the golden ratio. The company claims that 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 mathematics have involved Fibonacci numbers and the related golden ratio."[44]

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

Nature Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[49] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form."[50] In 2003, Volkmar Weiss and Harald Weiss analyzed psychometric data and theoretical considerations and concluded that the golden ratio underlies the clock cycle of brain waves.[51] In 2008 this was empirically confirmed by a group of neurobiologists.[52] In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.[53] Several researchers have proposed connections between the golden ratio and human genome DNA.[54] [55] [56]

8

Golden ratio

Mathematics Golden ratio conjugate The 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 segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[10] It is denoted here 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 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 lowest and n and m to be positive. But if n/m is in lowest , then the identity labeled (*) above says m/(nüÖüm) is in still lower . That is a contradiction that follows from the assumption that Ä is rational.

9

Golden ratio Derivation from irrationality of Ä5 Another short proofÄperhaps more commonly knownÄof the irrationality of the golden ratio makes use of the closure 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 is irrational.

Alternate forms The 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) are ratios 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, and similar relations in a pentagram.

10

Golden ratio

11

Geometry The number Ç turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is Ç times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360†/Ç Ü 222.5†. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[59] Golden triangle, pentagon and pentagram Golden triangle The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original. If angle BCX = Ö, then XCA = Ö because of the bisection, and CAB = Ö because of the similar triangles; ABC = 2Ö from the original isosceles symmetry, and BXC = 2Ö by similarity. The angles in a triangle add up to 180†, so 5Ö = 180, giving Ö = 36†. So the angles of the golden triangle are thus 36†-72†-72†. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36†-36†-108†.

Golden triangle

Suppose XB has length 1, and we call BC length Ç. Because of the isosceles triangles BC=XC and XC=XA, so these are 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 inverse ratio is 1 - Ç. The proof is left to the reader, utilizing the links in the previous sentence. Pentagram The golden ratio plays an important role in regular pentagons and pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is Ç, as the four-color illustration shows. The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is Ç. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomon. A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

Golden ratio

12

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

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

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

Ç.[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 common ratio, then r must lie in the range ÇÖ1 < r < Ç , which is a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If r = Ç then the shorter two sides are 1 and Ç but their sum is Ç2, thus r < Ç. A similar calculation shows that r> ÇÖ1. A triangle whose sides are in the ratio 1 : ÇÇ : Ç is a right triangle (because 1 + Ç = Ç2) known as a Kepler triangle.[61] Golden triangle, rhombus, and rhombic triacontahedron A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144†, which is twice the isosceles angle of a golden triangle and four times its most acute angle.

Relationship to Fibonacci sequence The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

One of the rhombic triacontahedron's rhombi

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, É The 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 of golden rhombi

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

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

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

More generally:

where above, the ratios of consecutive 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 be reduced in an analogous manner, by applying: for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n á 1. Phrased in 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 Å

interrelate them. They are both preserved by the fractional linear transformations

this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps Å they are reciprocals, symmetric about , and (projectively) symmetric about 2. More deeply, these maps form a subgroup of the modular group 3 letters,

corresponding to the stabilizer of the set

the symmetries correspond to the quotient map the identity

isomorphic to the symmetric group on of 3 standard points on the projective line, and

Å the subgroup

consisting of the 3-cycles and

fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

Other properties The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange's approximation theorem. This may be the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part 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 Ç is 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 the constant 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 or Ä-nary), every integer has a terminating representation, despite Ç being irrational, but every fraction has a non-terminating representation.

Golden ratio

15

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

and is a PisotÅVijayaraghavan

[62]

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

Decimal expansion The 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, starting with 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 more general 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. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers Ç and e. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F25001 and F25000, each over 5000 digits, yields over 10,000 significant digits of the golden ratio. The golden ratio Ä has been calculated to an accuracy of several millions of decimal digits (sequence A001622 [64] in OEIS). Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.[65]

Golden ratio

16

Pyramids Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.

Mathematical pyramids and triangles A pyramid in which the apothem (slant height along the bisector of a face) is equal to Ç times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), ing the medium-length edges to make the apothem. The height of this pyramid is times the semi-base (that is, the slope of the face is

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of and and

); the

are of particular interest in relation to Egyptian pyramids.

square of the height is equal to the area of a face, Ç times the square of the semi-base. The medial right triangle of this "golden" pyramid (see diagram), with sides right, demonstrating via the Pythagorean theorem the relationship

is interesting in its own or

. This

"Kepler triangle"[66] is the only right triangle proportion with edge lengths in geometric progression,[61] just as the 3Å4Å5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827É degrees (51† 49' 38").[67] A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;[68] the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).[69] The slant height 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 irrational numbers,[70] and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of 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 equal to 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† of the 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

Egyptian pyramids In the mid nineteenth century, R°ber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories 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 the Pyramid 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 accident remains 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, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have ed the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem nor any way to reason about irrationals such as Ü or Ç.[72] Michael Rice[73] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing

Giedon (1957).[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 represented by the ratio of the length of the face (the slope height), inclined at an angle ¢ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle ¢.[76] The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. 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 data variability.

Disputed observations Examples 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 of mollusks[4] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in specific individuals, and the proportion in question is often significantly different from the golden ratio.[79] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[80] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;[81] however, measurements of nautilus shells do not this claim.[82] í The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species.[83] In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.

17

Golden ratio í In investing, some practitioners of technical analysis use the golden ratio to indicate of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new and 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 Fibonacci numbers; see, e.g. Elliott wave principle. See Fibonacci retracement. However, other market analysts have published analyses suggesting that these percentages and patterns are not ed 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, for [2] [3] [4] [5] [6]

positive 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. 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. . Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: Associated University Presses, 1996 Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997 Euclid, Elements (http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ toc. html), Book 6, Definition 3. 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 are intellectually 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 University Press. 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)

18

Golden ratio [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. The lower 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 " (http:/ / www. pearldrum. com/ -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 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. . "The Golden 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 and computer 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 has a 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 and photographs 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 Mathematical Biology, 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 Golden Ratio 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.

19

Golden ratio [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 a crucial 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). " 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' days have 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 s of Batchelor and Ramyar's research.

Further reading í 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.

20

Golden ratio í Stakhov, A. P. (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. 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: The Mathematical Association of America. ISBNü0-88385-534-8.

External links í "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 or Pheidias (in Ancient Greek, ÅÉÑÖÜáà; circa 480 BCüÅ 430 BC), was a Greek sculptor, painter and architect, who lived in the 5th century BC, and is commonly regarded as one of the greatest of all sculptors of Classical Greece:[1] Phidias' Statue of Zeus at Olympia was one of the Seven Wonders of the Ancient World. Phidias also designed the statues of the goddess Athena on the Athenian Acropolis, namely the Athena Parthenos inside the Parthenon and the Athena Promachos, a colossal bronze statue of Athena which Phidias Showing the Frieze of the Parthenon to his Friends (1868) by Sir stood between it and the Propylaea,[2] a Lawrence Alma-Tadema. monumental gateway that served as the entrance 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 the Parthenon. Pericles' enemies found a false witness against Phidias, named Menon. Phidias died in prison, although Pericles' companion, Aspasia, was acquitted of her own charges.

21

Phidias

22

Works Although no original works in existence can be confidently attributed to him with certainty, numerous Roman copies in varying degrees of supposed fidelity are known to exist. This is not uncommon. Almost all classical Greek paintings and sculptures have been destroyed, and only Roman copies or notes of them exist, like the ages of Plato that ascribe Phidias' works to him. The ancient Romans frequently copied and further developed Greek art. Ancient critics take a very high view of the merits of Phidias. What they especially praise is the ethos or permanent moral level of his works as compared with those of the later so called "pathetic" school. Demetrius calls his statues sublime, and at the same time precise.

The Acropolis of Athens

Of his life we know little apart from his works. His first commission was a group of national heroes with Miltiades as a central figure. The famous statesman Pericles also commissioned several sculptures for Athens from him in 447 BC, to celebrate Greek victory against the Persians at the Battle of Marathon during the Greco-Persian Wars (490 BC). Pericles used some of the money from the maritime League of Delos,[5] to rebuild and decorate Athens to celebrate this victory. In 1958 archaeologists found the workshop at Olympia where Phidias assembled the gold and ivory Zeus. There were still some shards 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 Golden Ratio is an irrational number approximating 1.6180[8] which when studied has special mathematical properties. The golden spiral is also said to hold aesthetic values.

Early works

A Roman period, 2nd century CE sculpture found near the Varvakeion school reflects the type of the restored Athena Parthenos presently in the (National Archaeological Museum of Athens).

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

Phidias

Zeus at Olympia and the Athena Parthenos Among the ancient Greeks themselves two works of Phidias far outshone all others, the colossal chryselephantine figures of Zeus circa 432 BC on the site where it was erected in the temple of Zeus,[9] at Olympia, Greece, and of Athena Parthenos (literally, "Athena the Virgin") a sculpture of the Greek virgin goddess Athena named after an epithet for the goddess herself, and was housed in the Parthenon in Athens. Both sculpture belong to about the middle of the 5th century BC. A number of replicas and works inspired by it, both ancient and modern, have been made. From the A reconstruction of Phidias' statue of Zeus, in an 5th century BC, the copies of the statue of Zeus found were small engraving made by Philippe Galle in 1572, from a copies on coins of Elis, which give us but a general notion of the drawing by Maarten van Heemskerck. pose, and the character of the head. The god was seated on a throne, 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 the Olympian statue is certainly more than a century later in style.

Materials and theories In antiquity Phidias was celebrated for his statues in bronze, and his chryselephantine works (statues made of gold and ivory). In the Hippias Major, Plato claims that Phidias seldom, if ever, have executed works in marble though many of the sculptures of his times were executed in marble. Plutarch tells us that he superintended the great works of Pericles on the Acropolis. Inscriptions prove that the marble blocks intended for the pedimental statues of the Parthenon were not brought to Athens until 434 BC, which was probably after the death of Phidias. It is therefore possible that most sculptural decoration of the Parthenon was the work of Phidias' atelier but supposedly made by pupils 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 the Parthenos. These are copies of the Roman age. According to geographer Pausanias (1.28.2), the original bronze Lemnian Athena was created by Phidias circa 450-440 BCE, for Athenians living on Lemnos. Adolf Furtw•ngler proposed to find, in a statue of which the head is at Bologna, and of which the body is at Dresden, a copy of the Lemnian Athena of Phidias. Some 5th century torsos of Athena found at Athens. The torso of Athena in the ¶cole des Beaux-Arts at Paris, which has unfortunately lost its head, may perhaps best serve to help our imagination in reconstructing the original statue.

23

Phidias

24

Gallery

Head of Aphrodite

Zeus in Olympia, representation on coin

Head of Athena, Roman copy

Wounded Amazon - Musei Capitolini, Rome

Another copy of one Phidias or his pupils work, head of Athena, found around Pnyx, now in the National Archeological Museum of Athens

Reconstruction of Athena Lemnia, Dresden.

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

Notes [1] [2] [3] [4] [5]

Phidias (http:/ / www. bartleby. com/ 65/ ph/ Phidias. html) Birte Lundgreen, "A Methodological Enquiry: The Great Bronze Athena by Pheidias" The Journal of Hellenic Studies Not the Charmides who participated in the tyranny at Athens. Not to be confused with Hegias the neoplatonic philosopher. 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, for positive 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 links í Phidias as a First Name in USA (http://www.pokemyname.com/firstname_35958_phidias.htm)

Irrational number In mathematics, an irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero, and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are precisely those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1] Perhaps the

best-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. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

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

Ancient Greece The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[7] who

probably discovered them while identifying sides of the pentagram.[8] The number is irrational. The then-current Pythagorean method would have claimed that there must 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 the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both 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 units possible. í By the Pythagorean theorem: a2 = 2b2. í Since a2 is even, a must be even. í Since a:b is in its lowest , 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 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 was subsequently thrown overboard by his fellow Pythagoreans àÉfor having produced an element in the universe which denied 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 Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that 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 the discrete to the continuous. Brought into light by Zeno of Elea, he questioned the conception that quantities are discrete, and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for àwhole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects.â[11] However Zeno found that in fact à[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appearÉ.[Q]uantities are, in other words, continuous.â[11] What this means is that, contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times 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, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zenoäs paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, 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 commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude àwas 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 to numbers, which jumped from one value to another, as from 4 to 5.â[12] Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to for both commensurable and incommensurable ratios by defining a ratio in 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. àEudoxusä theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.â[13] As a result of the distinction between number and magnitude, geometry became the only method that could take into incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometrical . This may for why we still conceive of x2 or x3 as x squared and x cubed instead of x second power and x third power. Also crucial to Zenoäs work with incommensurable magnitudes was the fundamental focus on deductive reasoning which resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that comprised that theory. Out of this necessity Eudoxus developed his method of exhaustion, and kind of reductio ad absurdum which àestablished the deductive organization on the basis of explicit axiomsÉâ as well as àÉreinforced the earlier decision to rely on deductive reasoning for proof.â[14] This method of exhaustion is said to be the first step in the creation of calculus.

26

Irrational number Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used couldn't be applied to the square root of 17.[15] It wasn't until Eudoxus developed a theory of proportion that took into irrational as well as rational ratios that a strong mathematical foundation of 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 to numbers, 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 to magnitudes, Eudoxus was then able to for both commensurable and incommensurable ratios by defining a ratio in 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. àEudoxusä theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.â[18] Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes.

Middle Ages In the Middle ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as "algebraic objects".[19] Muslim mathematicians also merged the concepts of "number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and

extended the concept of number to ratios of continuous magnitude.[20] In his commentary on Book 10 of the Elements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric as follows:[21] "It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, 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 rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the 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 a magnitude 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 as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.[22] In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.[23] Ab® Ja'far al-Kh©zin (900Å971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:[24] "contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (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

27

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

Modern period The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. The completion of the theory of complex numbers in the nineteenth century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Mõray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880,[26] and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received 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 hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. Johann Heinrich Lambert proved (1761) that Ü cannot be rational, and that en is irrational if n is rational (unless nü=ü0).[27] While Lambert's proof is often said to be incomplete, modern assessments it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the BesselÅClifford function, provided a proof to show that Ü2 is irrational, whence it follows immediately that Ü is irrational also. The existence 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 transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for Ü. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan.

Example proofs Square roots The square root of 2 was the first number to be proved irrational and that article contains a number of proofs. The golden 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 quadratic irrationals. The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, called an argument by reductio ad absurdum. The following argument appeals twice to the fact that the square of an odd integer 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.

28

Irrational number

29

General roots The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic which was proved by Gauss in 1798. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest there must 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.

Logarithms Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum 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 the numberü3 raised to any positive integer power must be odd (since none of its prime factors will beü2). Clearly, an integer can not be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2ü3 is rational (and so expressible as a quotient of integers m/n with nüãü0). The contradiction means that this

assumption 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 irrationals Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental 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 with integer 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 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem). The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor of a and s is a divisor of a ; there are only 0

n

finitely many such candidates which you can all check by hand. If neither of them is a root of p, then x must be irrational. For example, this technique can be used to show that xü=ü(21/2ü+ü1)1/3 is irrational: we have (x3üÖü1)2 = 2 and hence x6üÖü2x3üÖü1ü=ü0, and this latter polynomial does not have any rational roots (the only candidates to check are ™1). Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. For example 3Üü+ü2, Üü+üÇ2 and eÇ3 are irrational (and even transcendental).

Irrational number

Decimal expansions The 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 division of n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m Ö 1 steps without using any remainder more than once. After that, a remainder 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. For example: 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 of the 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 by multiplying 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 powers Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab is rational.[28] Indeed, if Ç2Ç2 is rational, then take a = b = Ç2. Otherwise, take a to be the irrational number Ç2Ç2 and b = Ç2. Then

ab = (Ç2Ç2)Ç2 = Ç2Ç2£Ç2 = Ç22 = 2 which is rational.

Although the above argument does not decide between the two cases, the GelfondÅSchneider theorem implies that Ç2Ç2 is transcendental, hence irrational. This theorem states that all non-rational algebraic powers of algebraic numbers other than 0 or 1 are transcendental.

30

Irrational number

Open questions It 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 for which it is known whether mÜ + ne is irrational or not. Moreover, it is not known whether the set {Ü, e} is algebraically 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 are irrational.[29] [30] [31]

The set of all irrationals Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable. Under the usual (Euclidean) distance function d(x,üy) = |xüÖüy|, the real numbers are a metric space and hence also a topological 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-delta setÄi.e., a countable intersection of open subsetsÄin a complete metric space, the space of irrationals is topologically complete: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable. Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals have a basis of clopen sets 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 magnitudes is 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, ..

31

Irrational number [20] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences 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 of Sciences 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 History of Non-western Mathematics. Springer. ISBNü1402002602. [23] Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences 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 of Sciences 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 reading í Adrien-Marie Legendre, âlÑments de GÑometrie, Note IV, (1802), Paris í 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 Links í Zeno's Paradoxes and Incommensurability. (n.d.). Retrieved April 1, 2008, from http://www.dm.uniba.it/ ~psiche/bas2/node5.html

External links í 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)

32

Golden rectangle

33

Golden rectangle

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter 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 golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property. According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509,[1] when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use,"[2] many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.[3]

Golden rectangle

34

Construction A golden rectangle can be constructed with only straightedge and com by this technique: 1. Construct a simple square 2. Draw a line from the midpoint of one side of the square to an opposite corner 3. Use that line as the radius to draw an arc that defines the height of the rectangle 4. Complete the golden rectangle

Applications í Le Corbusier's 1927 Villa Stein in Garches features a rectangular ground plan, elevation, and inner structure that are closely approximate to golden rectangles.[4] í The flag of Togo was designed to approximate a golden rectangle closely.[5]

A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio.

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 these dimensions.") [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 links í 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 In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to Ç, the golden ratio.[1] Specifically, a golden spiral gets wider (or further from its origin) by a factor of Ç for every quarter turn it makes.

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

Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.

or

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when ¢ is a right 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 negative of this value):

A Fibonacci spiral approximates the golden spiral; unlike the "whirling rectangle diagram" based on the golden ratio, above, this one uses squares of integer Fibonacci-number sizes, shown for square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

for ¢ in degrees;

(that is, b can also be the

Golden spiral

36

for ¢ in radians. 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 golden spiral.[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 are connected by quarter-circles. The result is very similar to a true golden spiral (See image 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 factor related to the ratios of consecutive in the Fibonacci sequence. The ratios of consecutive in the Fibonacci series approach Ç, so that the two spirals are very similar in appearance. (See image on top right).

Spirals in nature Approximate logarithmic spirals can occur in nature (for example, the arms of spiral galaxies). It is sometimes stated that nautilus shells get wider in the pattern of a golden spiral, and hence are related to both Ç and the Fibonacci series. In truth, nautilus shells (and many mollusc shells) exhibit logarithmic spiral growth, but at an angle distinctly different from that of the golden spiral.[5] This pattern allows the organism to grow without changing shape. Spirals are 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 In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc. Algebraically, let c be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that

and

The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio

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

Derivation The 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 by the 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: or in radians as:

Golden angle in nature The golden angle plays a significant role in the theory of phyllotaxis. Perhaps most notably, the golden angle is the angle separating the florets on a sunflower.

References í Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences 44 (44): 179Å189. doi:10.1016/0025-5564(79)90080-4. í Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). [[The Algorithmic Beauty of Plants [1]]]. Springer-Verlag. pp.å101Ä107. ISBNå978-0387972978.

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

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 such polyhedron is the rhombic triacontahedron. The internal angles of the rhombus are approximately 63†26å and 116†34å. The dihedral angle between adjacent rhombi of the rhombic triacontahedron is 144†, which can be constructed by placing the short sides of two golden triangles back-to-back.

External links í 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 A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Definition In polar coordinates

the curve can be written

[1]

as

Logarithmic spiral (pitch 10†)

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

Romanesco broccoli, which grows in a logarithmic spiral

Logarithmic spiral

A section of the Mandelbrot set following a logarithmic spiral

A low pressure area over Iceland shows an approximately logarithmic spiral pattern

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

41

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 as

The derivative of

is proportional to the parameter

direction the spiral spirals. In the extreme case that Conversely, in the limit that

. In other words, it controls how "tightly" and in which (

) the spiral becomes a circle of radius

.

approaches infinity (Ä ç 0) the spiral tends toward a straight half-line. The

complement of Ä is called the pitch.

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

Logarithmic spiral

43

Properties The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant. Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives 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 times

without reaching it; yet, the total distance covered on this path is finite; that is, the limit as

goes toward

[4]

finite. This property was first realized by Evangelista Torricelli even before calculus had been invented. distance covered is , where is the straight-line distance from to the origin.

is The total

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

, where the constant

is a complex number with non-zero imaginary part, maps the real line

to 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 90 degrees of rotation (pitch about 17.03239 degrees), or approximate it using Fibonacci numbers.

Logarithmic spirals in nature In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples 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 the same 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 their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result [6] in a practically straight line. í The arms of spiral galaxies.[7] Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a 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 epithelial layer 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 construction from 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

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)" from MathWorld. 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 of Georgia (1999) í Alexander Bogomolny, Spira Mirabilis - Wonderful Spiral (http://www.cut-the-knot.org/Curriculum/ Geometry/Mirabilis.shtml), at cut-the-knot

External links í Spira mirabilis (http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden spiral/logspiral-history.html) history and math í Astronomy Picture of the Day (http://antwrp.gsfc.nasa.gov/apod/ap030925.html), Hurricane Isabel vs. the Whirlpool Galaxy í Astronomy Picture of the Day (http://antwrp.gsfc.nasa.gov/apod/ap080517.html), Typhoon Rammasun vs. the Pinwheel Galaxy í 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)

44

Canons of page construction

45

Canons of page construction The canons of page construction are a set of principles in the field of book design used to describe the ways that page proportions, margins and type areas (print spaces) of books are constructed. The notion of canons, or laws of form, of book page construction was popularized by Jan Tschichold in the mid to late twentieth century, based on the work of J. A. van de Graaf, Ra¨l M. Rosarivo, Hans Kayser, and others.[1] Tschichold wrote àThough largely forgotten today, methods and rules upon which it is impossible to improve have been developed for centuries. To produce perfect books these rules have to be brought to life and applied.â[2] Kayser's 1946 Ein

harmonikaler Teilungskanon[3] had earlier used the term canon in this context. Typographers and book designers apply these principles to this day, with variations related to the availability of standardized paper sizes, [4]

and the diverse types of commercially printed books.

Recto page from a rare Blackletter Bible (1497)

Van de Graaf canon The Van de Graaf canon is a historical reconstruction of a method that may have been used in book design to divide a page in pleasing proportions.[5] This canon is also known as the "secret canon" used in many 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 designer to position the text body in a specific area of the page. Using the canon, the proportions are maintained while creating pleasing and functional margins of size 1/9 and 2/9 of the page size.[6] The resulting inside margin is one-half of the outside margin, and of proportions 2:3:4:6 (inner:top:outer:bottom) when the page proportion is 2:3 (more generally 1:R:2:2R for page proportion 1:R[7] ). This method was discovered by Van de Graaf, and used by Tschichold and other contemporary designers; they speculate that it may be older.[8]

Van de Graaf devised this construction to show how Gutenberg and others may have divided their page to achieve margins of one-ninth and two-ninths and a type area in the same proportions as the page.

The page proportions vary, but most commonly used is the 2:3 proportion. Tschichold writes "For purposes of better comparison I have based his figure on a page proportion of 2:3, which Van de Graaf does not use."[9] In this canon the text area and page size are of same proportions, and the height of the text area equals the page width. This canon was 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 the shapes of pages are equally useful in shaping and positioning the textblock. This was often the case in medieval books, although later on in the Renaissance, typographers preferred to apply a more polyphonic page in which the proportions of page and textblock would differ.[10]

Canons of page construction

46

Golden canon Tschichold's "golden canon of page construction"[1] is based on simple integer ratios, equivalent to Rosarivo's "typographical divine proportion."[11]

Interpretation of Rosarivo Ra¨l Rosarivo analyzed Renaissance books with the help of a drafting com and a ruler, and concluded in his Divina proporciÅn tipogrÇfica ("Typographical Divine Proportion", first published in 1947) that Gutenberg, Peter Sch°ffer, Nicolaus Jenson and others had applied 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 the harmonic relationships between the diverse parts of a work,[13] was analyzed by experts at the Gutenberg Museum and re-published in the Gutenberg-Jahrbuch, its official magazine.[14] Ros Vicente points out that Rosarivo "demonstrates that Gutenberg had a module different from the well-known one of Luca Paccioli" (the golden ratio).[14]

Tschichold's "golden canon of page construction" here illustrated by a synthesis of Tschichold's figure thereof, with the diagonals and circle, combined with Rosarivo's construction by division of the page into ninths. These two constructions rely on the 2:3 page ratio to give a type area height equal to page width as demonstrated by the circle, and result in margin proportions 2:3:4:6. For other page ratios, Rosarivo's method of ninths is equivalent to van de Graaf's canon, as Tschichold observed.

Tschichold also interprets Rosarivo's golden number as 2:3, saying: In figure 5 the height of the type area equals the width of the page: 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 the paper height for the top, and two-ninths for the bottom margin. Type area and paper size are of equal proportions. ... What I uncovered as the canon of the manuscript writers, Raul Rosarivo proved to have been Gutenberg's canon as well. He finds the size and position of the type area by dividing the page diagonal into ninths.[9] The figures he refers to are reproduced in combination here.

John Man's interpretation of Gutenberg

Medieval manuscript framework according to Tschichold, in which a text area proportioned near the golden ratio is constructed. "Page proportion is 2:3, text area proportioned in the [9] Golden Section."

Historian John Man suggests that Gutenberg's Bible page was based on the golden ratio (commonly approximated as the decimal 0.618 or the ratio 5:8), and that the printed area also had that shape.[15] He quotes the dimensions of Gutenberg'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 to the golden ratio 1.618.

Canons of page construction

47

Tschichold and the golden section Building on Rosarivo's work, contemporary experts in book design such as Tschichold and Richard Hendel assert as well that the page proportion of the golden section (21:34) has been used in book design, in manuscripts, and incunabula, mostly in those produced between 1550 and 1770. Hendel writes that since Gutenberg's time, books have been most often printed in an upright position, that conform loosely, if not precisely, to the golden ratio.[16] These page proportions based on the golden section or golden ratio, are usually described through its convergents such as 2:3, 5:8, and 21:34. Tschichold says that common ratios for page proportion used in book design include as 2:3, 1:Ç3, and the golden section. The image with circular arcs depicts the proportions in a medieval manuscript, that according to Tschichold feature a "Page proportion 2:3. Margin proportions 1:1:2:3. Text area in accord with the Golden Section. The lower 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's drawing of an octavo-format page proportioned in the golden ratio or golden section "34:21". The text area and margin proportions are determined by the starting page proportions.

Tschichold refers to a construction equivalent to van de Graaf's or Rosarivo's with a 2:3 page ratio as "the Golden Canon of book page construction as it was used during late Gothic times by the finest of scribes." For the canon with the arc construction, which yields a text area ratio closer to the golden ratio, he says "I abstracted 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 many proportions, i.e., relationships between width and height. Everybody knows, at least from hearsay, the proportion of the Golden Section, exactly 1:1.618. A ratio of 5:8 is no more than an approximation of the Golden Section. It would be 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 proportions like 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 5:9 I call clear, intentional and definite. All others are unclear and accidental ratios. The difference between a clear and an unclear ratio, though frequently slight, is noticeable. ... Many books show none of the clear proportions, but accidental ones."[20] John Man's quoted Gutenberg page sizes are in a proportion not very close to the golden ratio,[15] but Rosarivo's or van de Graaf's construction is applied by Tschichold to make a pleasing text area on pages of arbitrary proportions, even such accidental ones.

Current applications Richard Hendel, associate director of the University of North Carolina Press, describes book design as a craft with its own 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 his views 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 same proportions as the page: he preferred to trust visual judgment in assessing the placement of the type area on 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 antiphonal geometry, which has the capability to bind the reader to the book, or conversely put the reader's nerve on edge or drive 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 by web 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 of Villard's figures: fol. 18r, two figures at the bottom; and fol. 19r, rightmost figure in the second row from the top. While the geometric design itself is unquestionably that generated from the Pythagorian monochord, Kayser does not convince the reader that Villard understood its musical basis. Kayser apparently worked from photographs of the original folios, and the significance of Kayser's claim may be summarized in his own ission (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 specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8 (footnote: The ratio is 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. The lower 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 in accord 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

48

Canons of page construction

References í 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 1947 í 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 reading í 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 links í "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 Book í "Rosarivo - Divina proporciûn tipogr´fica" (http://rosarivo.com.ar/ladivina.htm) (in Spanish).

49

List of works designed with the golden ratio

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

Early history It 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 geometric construction that generates it can only be surmised.[2] Various authors discern golden ratio proportions in Egyptian, Summerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from the late Bronze Age, which predates by about 1,000 years the Greek mathematicians who were first known to have studied the golden ratio.[2] [4] However, the historical sources are 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 various pyramidologists, including Charles Funck-Hellet.[4] [5] John F. Pile, interior design professor and historian, has claimed that Egyptian designers sought the golden proportions without mathematical techniques and that it is common 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 proposed alternative theories for the pyramid designs that are not related to any use of the golden ratio, and are instead based on purely rational slopes that only approximate the golden ratio.[7] The Egyptians of those times apparently did not know 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 its secondary 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 the golden ratio.[9] Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion.[10] Building the Acropolis is calculated to have been started around 600 BC, but the works said to exhibit the golden ratio proportions were created from 468 BC to 430 BC. The Parthenon (447Å432 BC), was a temple built on the Acropolis in the 5th century BC for the Greek goddess Athena. It is the most The Parthenon west fa≠ade important surviving building of Classical Greece. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by a progression of golden rectangles.[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

50

List of works designed with the golden ratio 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, to denote the golden ratio.[15] According to Lothar Haselberger,[4] the temple of Apollo in Didyma (c. 334 BC), designed by Daphnis of Mileto and Paionios of Efesus, have golden proportions.

Prehispanic Mesoamerican architecture Olmos claims the presence of the golden ratio in a series of olmec heads, the Aztec calendar stone, and a series of Aztec permission house plans. In the fifties, Manuel Amabilis applied some of the analysis methods of Frederik Macody Lund and Jay Hambidge to several plans and sections of prehispanic buildings, such as El Toloc and La Iglesia of Las Monjas, a notable complex of Terminal Classic buildings constructed in the Puuc architectural style at Chichen Itza. According to his studies, their proportions derived from a series of successively inscribed pentagons, circles and pentagrams, just as the Gothic churches Lund studied do. Amabilis published his studies along with several self-explanatory images of various 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 San Fernando" (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 and 13th centuries AD to serve as a temple to the god Kukulcan, has golden proportions in its interior layout with walls placed so that the outer spaces relate to the center chamber as 0.618:1.[17]

Islamic architecture A geometrical analysis of the Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D.) reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz, who say it is found in the overall 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)

51

List of works designed with the golden ratio

Buddhist architecture The 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 to Pile.[19]

Gothic era In his 1919 book Ad Quadratum, Frederik Macody Lund, a historian who studied the geometry of several gothic structures, claims that the Cathedral of Chartres (begun in the 12th century), the Notre-Dame of Laon (1157Å1205), and the Notre Dame de Paris (1160) are designed according to the golden ratio.[4] Other scholars argue that until Pacioli's 1509 publication (see next section), the golden ratio was unknown to artists and architects.[10] A 2003 conference on medieval architecture resulted in the book Ad Quadratum: The Application of Geometry to Medieval Architecture. According to a summary by one reviewer: Most of the contributors consider that the setting out was done ad quadratum, using the sides of a square and its diagonal. This gave an incommensurate ratio of [square root of (2)] by striking a circular arc (which could easily Illustration of the Notre-Dame of Laon cathedral. be done with a rope rotating around a peg). Most also According to Macody Lund, the superimposed argued that setting out was done geometrically rather than regulator lines show that the cathedral has golden arithmetically (with a measuring rod). Some considered proportions. that setting out also involved the use of equilateral or Pythagorean triangles, pentagons, and octagons. Two authors believe the Golden Section (or at least its approximation) was used, but its use in medieval times is not ed by most architectural historians.[20]

52

List of works designed with the golden ratio

53

Renaissance De divina proportione, written by Luca Pacioli in Milan in 1496Å1498, published in Venice in 1509,[21] features 60 drawings by Leonardo da Vinci, some of which illustrate the appearance of the golden ratio in geometric figures. Starting with part of the work of Leonardo Da Vinci, this architectural treatise was a major influence on generations of 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 its name, 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 the golden ratio in Vitruvian Man;[23] however, Olmos[4] (1991) observes otherwise through geometrical analysis. He also proposes Leonardo da Vinci's self portrait, Michelangelo's David (1501Å1504), Albrecht Dèrer's Melencolia and the classic violin design by the Masters of Cremona, as having similar regulator lines related to the golden ratio.

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

Da Vinci's Mona Lisa (c. 1503Å1506) "has been the subject of so many volumes of contradicting scholarly and popular speculations that it virtually 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 the golden ratio in its elevation and interior lines.[24]

The Baroque and the Spanish empire Jose Villagran Garcia has claimed[25] that the golden ratio is an important element in the design of the Mexico City Metropolitan 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 Crucificado by Diego Vel´zquez (1639) and La Madona de Media Luna of Bartolomõ Esteban Murillo.[4]

Neoclassicism and romanticism Leonid Sabaneyev hypothesizes that the separate time intervals of the musical pieces connected by the "culmination event", as a rule, are in the ratio of the golden section.[26] However the author attributes this incidence to the instinct of the mucisians: "All such events are timed by author's instinct to such points of the whole length that they divide temporary 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 of classical music: í An article of American Scientist[28] (Did Mozart use the Golden mean?, March/April 1996), reports that John Putz found that there was considerable deviation from ratio section division in many of Mozart's sonatas and claimed that 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 387.

Impressionism Matila Ghyka[30] and others[31] contend that Georges Seurat used golden ratio proportions in paintings like La Parade, Le Pont de Courbevoie and Bathers at Asniéres. However, there is no direct evidence to these claims.[23]

Neogothic According 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]

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

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

De Stijl Some works in the Dutch artistic movement called De Stijl, or neoplasticism, exhibit golden ratio proportions. Piet Mondrian 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 or mathematical methods.[36] Juan Gris also used golden ratio proportions.[31]

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

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

54

List of works designed with the golden ratio function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system.[41] In The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics Le Corbusier reveals he used his system in the Marseilles Unite D'Habitation (in the general plan and section, the front elevation, plan and section of the apartment, in the woodwork, the wall, the roof and some prefabricated furniture), a small office in 35 rue de SÆvres, a factory in Saint-Die and the United Nations Headquarters building in New York City.[42] Many authors claim that the shape of the facade of the second is the result of three golden rectangles;[43] however, each of the three rectangles that can actually be appreciated have different heights.

Post-modern architecture Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[44]

Contemporary music James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced. Ernù Lendvai analyzes Bõla Bartûk's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[45] though other music scholars reject that analysis.[10] In Bartûk's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[46] The French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry. 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 the main climax sits at the phi position."[46] The musicologist Roy Howat has observed that the formal boundaries of La mer correspond exactly to the golden section.[47] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[48] This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled "1.618" in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video [49] (link broken) displays various animated versions of the golden mean. Pearl Drums positions the air vents on its Masters models based on the golden ratio. The company claims that 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 have involved Fibonacci numbers and the related golden ratio."[51]

55

List of works designed with the golden ratio

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), Princeton Univ. 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. [18] Kenza Boussora and Said Mazouz, " The Use of the Golden Section in the Great Mosque of Kairouan (http:/ / www. nexusjournal. com/ BouMaz. html)", Nexus Network Journal, vol. 6 no. 1 (Spring 2004), pp. 7-16. DOI 10.1007/s00004-004-0002-y [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& lang=en) [33] BALMORI, Santos, Aurea mesura, UNAM, 1978, 189 p. P. 23-24. [34] Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X

56

List of works designed with the golden ratio [35] Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp. 247-48, Harcourt, Brace & World, ISBN 0-87817-259-9 [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 Courts and 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 and Francis, 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•, 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). Courier Dover 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 " (http:/ / www. pearldrum. com/ -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 links í Nexux Network Journal Å Architecture and Mathematics Online. (http://www.emis.de/journals/NNJ/Frings. html) Kim Williams Books

57

Article Sources and Contributors

Article Sources and Contributors Golden ratio üSource: http://en.wikipedia.org/w/index.php?oldid=422471442 üContributors: (jarbarf), 11quintanarq, 123987456, 20-dude, 22over7, 28bytes, 2over0, 5 albert square, 5theye, 7, 8bits, A2-computist, A8UDI, AAAAA, ABF, ACupOfCoffee, AVRS, AVX, Academic Challenger, Adamfinmo, Addshore, AdultSwim, Afa86, Agrofe, Aheyfromhome, Ahoerstemeier, Alan De Smet, Alanbly, Alansohn, Ale jrb, Alethiophile, Alex S, [email protected], Amazing Steve, AndrewKepert, Androl, Antandrus, Arthur Rubin, AstroWiki, Athenean, Atif.t2, AustinKnight, AxelBoldt, Az1568, B4hand, BL, Barneca, Bdesham, Beach drifter, Ben Standeven, Beremiz, Bettername, 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, tmurdok, 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, Loaster, Loonymonkey, LordFoom, Luke-Jr, MacMed, Macy, Math789, Martin451, Masgatotkaca, Maverick starstrider, MaybeJesusMaybeNot, Mdd, Melchoir, Michael Hardy, Mild Bill Hiccup, Minesweeper, Mjb, Momo san, Mouse is 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, Googol, 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, CanadianLinux, Canton Viaduct, Celestianpower, Charles Matthews,

58

Article Sources and Contributors Chocolatemeow, Clarince63, CyberSkull, D-Notice, DaveWF, David Eppstein, Dekaptein, Dicklyon, Discospinster, Dragon76, EVula, El C, Epbr123, Erik9, Error792, Fieldday-sunday, Fiftytwo thirty, Fingerz, Fischer.sebastian, Fram, Galoubet, GianniG46, Giftlite, Glax2, Goodvac, Hadal, HarlandQPitt, Hasanisawi, Hephaestos, Hu, Humpty Dumpty secretary, Hunt286, Igor Yalovecky, Inbetweener, JCO312, Jackol, Jasper Deng, Jengelh, Jleedev, JocK, Joelholdsworth, Joeyszy, John.d.page, John254, JohnClarknew, JohnOwens, Johnuniq, Jossi, Jswayze, Junglecat, Justinpwilsonadvocate, K. Annoyomous, Kapgains, Kaspazes, Kbreedlo, Kittenono, Lambiam, Lampica, Ldesign, Linas, Madhero88, Marek69, Michael Hardy, Michaelas10, Mindmatrix, Monkeygirlsboys, Nae'blis, NawlinWiki, NickRinger, Noe, Oleg Alexandrov, Palica, Personline, Philg88, Philip Trueman, Pinethicket, Pjacklam, Pmcnett, Pokerstud001, RadioFan2 (usurped), RaseaC, Redrvm, Rettetast, Reywas92, Romanm, Ronhjones, Rrburke, Sean.hoyland, Sebleblanc, Sir Intellegence, Smaffy, Snowolf, StephanCom, Stephenb, Supercriminal, Swift1357, Technopat, TheArtificer, TheBlueWizard, TigerShark, Tikiwont, Tom harrison, Toughpigs, Traal, Trevor Burnham, Underedge1, Vaughan Pratt, Vivio Testarossa, W3stfa11, Wimt, Wine Guy, WingsGoesWiki, Wtmitchell, Wu1v3rine, Xmarxthespot, 218 anonymous edits 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 anonymous edits Golden angle üSource: http://en.wikipedia.org/w/index.php?oldid=400818230 üContributors: Alexwcovington, Ambarsande, Captain Disdain, CecilBlade, Charles Matthews, D-Notice, David Schaich, Dicklyon, Duncan., Gene Nygaard, Icairns, Jamrb, Jonathan.s.kt, Linas, Michael Hardy, Michael Tiemann, Nekura, Noe, Palica, Postdlf, Salix alba, Starie17, Superm401, The Anome, 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 anonymous edits 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, Sotho Tal Ker, Speight, Svick, Tosha, Unyoyega, William Evans, WilliamKF, Xiutwel, Xplat, Yghwtrrl, Yosha, 58 anonymous edits Canons of page construction üSource: http://en.wikipedia.org/w/index.php?oldid=417592858 üContributors: Agaffss, Alansohn, Arielgodwin, Charles Matthews, Circeus, Crispmuncher, Dan337, Danny, DeadEyeArrow, Dicklyon, Ealdgyth, Epbr123, Fabiancarreras, Fainites, Gogo Dodo, Gun Powder Ma, H1nkles, Jossi, Juliancolton, Kasyapa, Kozuch, Laureapuella, Lightmouse, Ling.Nut, Lotje, Markwalters79, Max Naylor, Michael Devore, Milton Stanley, Oicumayberight, Quiddity, RobertG, Selket, Slashme, TreasuryTag, Welsh, 30 anonymous edits 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

59

Image Sources, Licenses and Contributors

Image Sources, Licenses and Contributors Image:Golden ratio line.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_ratio_line.svg üLicense: Public Domain üContributors: :Stannered Image:Golden Rectangle Construction.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_Rectangle_Construction.svg üLicense: Public Domain üContributors: :Joelholdsworth Image:SimilarGoldenRectangles.png üSource: http://en.wikipedia.org/w/index.php?title=File:SimilarGoldenRectangles.png üLicense: Creative Commons Zero üContributors: :Ahecht Image:Phi uc lc.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Phi_uc_lc.svg üLicense: Public Domain üContributors: :Dcoetzee, :F l a n k e r Image:Michael Maestlin.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Michael_Maestlin.jpg üLicense: Public Domain üContributors: original er File:Acropolis of Athens 01361.JPG üSource: http://en.wikipedia.org/w/index.php?title=File:Acropolis_of_Athens_01361.JPG üLicense: unknown üContributors: Fred J, Itsmine, Jkelly, LX, Medea, 1 anonymous edits File:Pentagram and human body (Agrippa).jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Pentagram_and_human_body_(Agrippa).jpg üLicense: Public Domain üContributors: Heinrich Cornelius Agrippa Image:Divina proportione.png üSource: http://en.wikipedia.org/w/index.php?title=File:Divina_proportione.png üLicense: Public Domain üContributors: HenkvD, Homonihilis, Jossifresco, Thuresson, Warburg Image:Medieval manuscript framework.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Medieval_manuscript_framework.svg üLicense: Public Domain üContributors: :Jossifresco Image:Golden triangle (math).svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_triangle_(math).svg üLicense: Public Domain üContributors: :Jheald Image:Pentagram-phi.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Pentagram-phi.svg üLicense: Public Domain üContributors: Original er was Jamiemichelle at en.wikipedia Image:Ptolemy Pentagon.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Ptolemy_Pentagon.svg üLicense: unknown üContributors: AnonMoos, Athaenara, Stotr Image:GoldenRhombus.png üSource: http://en.wikipedia.org/w/index.php?title=File:GoldenRhombus.png üLicense: Creative Commons Attribution-Sharealike 3.0 üContributors: Zom-B File:Icosidodecahedron.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Icosidodecahedron.svg üLicense: GNU Free Documentation License üContributors: atropos235 on en.wikipedia Image:Fibonacci spiral 34.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Fibonacci_spiral_34.svg üLicense: Public Domain üContributors: Darapti, Dicklyon Image:Mathematical Pyramid.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Mathematical_Pyramid.svg üLicense: Public Domain üContributors: Dicklyon Image:1868 Lawrence Alma-Tadema - Phidias Showing the Frieze of the Parthenon to his Friends.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:1868_Lawrence_Alma-Tadema_-_Phidias_Showing_the_Frieze_of_the_Parthenon_to_his_Friends.jpg üLicense: Public Domain üContributors: DenghiΩComm, Juanpdp, Mattes, Oxxo, Raeky, Shakko, 2 anonymous edits Image:Acropolis3.JPG üSource: http://en.wikipedia.org/w/index.php?title=File:Acropolis3.JPG üLicense: Public Domain üContributors: Itsmine, Jkelly, Ronaldino Image:NAMA Athâna Varvakeion.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:NAMA_Athõna_Varvakeion.jpg üLicense: GNU Free Documentation License üContributors: G.dallorto, Jastrow, Marsyas, Sailko, Tetraktys Image:Statue of Zeus.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Statue_of_Zeus.jpg üLicense: Public Domain üContributors: Aseditor, G.dallorto, Gryffindor, Jastrow, Larry Yuma, Macion96, Man vyi, Shakko, Talmoryair, Warburg, 2 anonymous edits File:Aphrodite Sappho Chiaramonti Inv1459.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Aphrodite_Sappho_Chiaramonti_Inv1459.jpg üLicense: Public Domain üContributors: :Jastrow File:Zeus.in.Olympia.representation.on.coin.drawing.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Zeus.in.Olympia.representation.on.coin.drawing.jpg üLicense: Public Domain üContributors: Carlomorino, Jastrow, Longbow4u, Mogelzahn, Shakko, Umherirrender, 1 anonymous edits File:NAMA Athâna du Pnyx.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:NAMA_Athõna_du_Pnyx.jpg üLicense: Creative Commons Attribution 2.5 üContributors: Bibi Saint-Pol, ChristosV, G.dallorto, Jastrow, Marsyas, Shakko File:Lemnia torso04 pushkin.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Lemnia_torso04_pushkin.jpg üLicense: Creative Commons Attribution 3.0 üContributors: shakko File:Testa di Minerva S.Giulia Brescia.JPG üSource: http://en.wikipedia.org/w/index.php?title=File:Testa_di_Minerva_S.Giulia_Brescia.JPG üLicense: Attribution üContributors: :Bolo77 File:Amazzone ferita - Wounded Amazon - Phidias - Musei Capitolini Roma.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Amazzone_ferita_-_Wounded_Amazon_-_Phidias_-_Musei_Capitolini_Roma.jpg üLicense: Public Domain üContributors: Gradiva File:PD-icon.svg üSource: http://en.wikipedia.org/w/index.php?title=File:PD-icon.svg üLicense: Public Domain üContributors: :Duesentrieb, :Rfl Image:Square root of 2 triangle.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Square_root_of_2_triangle.svg üLicense: unknown üContributors: :Pbroks13 Image:FakeRealLogSprial.svg üSource: http://en.wikipedia.org/w/index.php?title=File:FakeRealLogSprial.svg üLicense: GNU Free Documentation License üContributors: FakeRealLogSpiral.png: Pau derivative work: silverhammermba Image:History of Gold. Rersum 2007.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:History_of_Gold._Rersum_2007.jpg üLicense: Public Domain üContributors: :arz Image:Golden Angle.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_Angle.svg üLicense: Public Domain üContributors: Dicklyon, 1 anonymous edits Image:Goldener Schnitt Blattstand.png üSource: http://en.wikipedia.org/w/index.php?title=File:Goldener_Schnitt_Blattstand.png üLicense: GNU Free Documentation License üContributors: Original er was Wolfgangbeyer at de.wikipedia Image:Logarithmic Spiral Pylab.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Logarithmic_Spiral_Pylab.svg üLicense: Public Domain üContributors: :Morn the Gorn Image:NautilusCutawayLogarithmicSpiral.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:NautilusCutawayLogarithmicSpiral.jpg üLicense: Attribution üContributors: :Chris 73 File:Brassica romanesco.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Brassica_romanesco.jpg üLicense: unknown üContributors: :Dino Image:Mandel zoom 04 seehorse tail.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Mandel_zoom_04_seehorse_tail.jpg üLicense: Creative Commons Attribution-Sharealike 2.5 üContributors: :Wolfgangbeyer Image:Low pressure system over Iceland.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Low_pressure_system_over_Iceland.jpg üLicense: Public Domain üContributors: NASA File:Messier51 sRGB.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Messier51_sRGB.jpg üLicense: unknown üContributors: NASA and European Space Agency Image:Polygon spiral.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Polygon_spiral.svg üLicense: Creative Commons Attribution-Sharealike 2.5 üContributors: :Sbrools Image:Incunabulum.JPG üSource: http://en.wikipedia.org/w/index.php?title=File:Incunabulum.JPG üLicense: Public Domain üContributors: Original er was GeorgeStepanek at en.wikipedia Image:Van de Graaf canon in book design.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Van_de_Graaf_canon_in_book_design.svg üLicense: Public Domain üContributors: :jossifresco Image:Golden canon of page construction.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_canon_of_page_construction.svg üLicense: Public Domain üContributors: Original er was Dicklyon at en.wikipedia Image:Tschichold medieval canon.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Tschichold_medieval_canon.svg üLicense: Public Domain üContributors: Dicklyon, Nehrams2020 Image:Golden section page Tschichold.svg üSource: http://en.wikipedia.org/w/index.php?title=File:Golden_section_page_Tschichold.svg üLicense: Public Domain üContributors: Original er was Dicklyon at en.wikipedia Image:Parthenon from west.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Parthenon_from_west.jpg üLicense: Public Domain üContributors: :Mountain Image:Kairuan.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Kairuan.jpg üLicense: Creative Commons Attribution-Sharealike 2.5 üContributors: Mathiasrex, Moumou82, 1 anonymous edits Image:Laon Cathedral's regulator lines.jpg üSource: http://en.wikipedia.org/w/index.php?title=File:Laon_Cathedral's_regulator_lines.jpg üLicense: Public Domain üContributors: 20-dude, Dicklyon

60

License

License Creative Commons Attribution-Share Alike 3.0 Unported http:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/

61