Page semi-protected

Golden ratio

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

Line segments in the golden ratio
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 .
The golden spiral is calculated by tiling the square of Fibonacci numbers, which are related by the golden ratio.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number (not to be confused with the related Fibonacci numbers).[6][7][8]

Mathematicians since Euclid—and perhaps earlier—have studied the properties of the golden ratio, including its self-similarity and appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9]

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

Expressed algebraically, for quantities a and b with a > b > 0,

where the Greek letter phi ( or ) represents the golden ratio.[a] It is an irrational number with a value of:

[10]

History

The Great Pyramid of Giza (c. 2560 BC) is a golden pyramid according to various measurements. The Babylonian Tablet of Shamash (c. 888–855 BC) can be superimposed with two orders of golden rectangles.[11] Various attempts have been made to link other ancient objects to the golden ratio, but many of these claims lack specificity or evidence to back them.[12]

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

Though ancient Greek philosopher Pythagoras (c. 570 BC–c. 495 BC) left behind no writing, he is generally credited with knowledge of the golden ratio and the related dodecahedron, and imparting this knowledge to his followers; their symbol was the pentagram.[13][14] According to one story, the 5th-century Pythagorean Hippasus discovered that the golden ratio was an irrational number, surprising other Pythagoreans.[15] Hippasus was also the first to write of the dodecahedron,[16] with Theaetetus (c. 417–c. 369 BC) being the first to describe all five possible regular solids.[17] In the Republic (c. 380 BC), Plato expresses the concept of self-similarity in his Analogy of the Divided Line.[18] Euclid in his Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[b] using it in the construction of the pentagon, icosahedron and dodecahedron.[19] It also contains the first known definition:[20][21]

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 lesser.[4]

Hero of Alexandria developed formulas for the area of a pentagon and the volumes of icosahedrons and dodecahedrons in the first century AD.[22] Fellow Alexandrians Ptolemy calculated trigonometric chords for angles relating to the golden ratio in the second century, and Pappus made calculations for the comparison of Platonic solids related to the golden ratio c. 340 AD.[23]

Abū Kāmil Shujāʿ ibn Aslam (c. 850–930) employed the golden ratio in his calculations of pentagons; some of his work influenced that of Fibonacci.[24] Luca Pacioli discusses the golden ratio in his book On the Divine Proportion (1509).[20] The first known approximation of the (inverse) golden ratio by a decimal fraction was stated as "about 0.6180340" in 1597 by Michael Maestlin in a letter to his former student Johannes Kepler (1571–1630).[20][25] Kepler proved that consecutive Fibonacci numbers approach the golden ratio asymptotically,[26] and combined two of his favorite mathematical concepts in the triangle named for him,[20] saying:

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.[27]

In 1754, Charles Bonnet pointed out that the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently expressed in two successive golden ratio series.[28][20] Martin Ohm is believed to have been the first to use the term goldener Schnitt (golden section) to describe the ratio in 1835.[20][29] In the mid-19th century, Röber theorized that the Great Pyramid of Giza was based on a Kepler triangle.[30]

Phi (pronounced 'fee'[31] or 'fai')[32]

By 1914, mathematician Mark Barr suggested Greek letter Phi (φ) as a symbol for the golden ratio.[c] Previously, it was represented by τ (tau, the first letter of the ancient Greek root τομή—meaning cut or section).[35][36] In 1966, Fibonacci Quarterly published φ calculated to 4,600 digits; thirty years later, it was calculated to 10 million decimal places.[37] In 1974, Roger Penrose discovered a form of tiling derived from the diagonal division of pentagons which is related to the golden ratio both in the relationship of areas of its two rhombic tiles and their relative frequency within the pattern.[38][39] This in turn led to new discoveries about quasicrystals.[20][40]

Applications

Architecture

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 are of particular interest in relation to Egyptian pyramids.

According to the dimensions of the Great Pyramid of Giza (c. 2560 BC) given by Herodotus c. 440 BC, its perimeter/height ratio is 2π. The pyramid's slant height divided by half its base width produces φ. Together with its height, the three lengths form a Kepler triangle, making it a golden pyramid.[41] This was first theorized by Röber in the mid-19th century.[30] The Great Pyramid's face angle of 51.854° approximates the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship . In 1859, the pyramidologist John Taylor claimed that the golden ratio is represented in the Great Pyramid by the ratio of its slant height, inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ.[42] Those 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 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.[43]

Modern scholars have long debated whether the Egyptians had knowledge of the golden ratio,[44][45] with the many contending that its appearance in an ancient Egyptian architecture is the result of chance.[46][47][48] Eric Temple Bell argued in 1950 that ancient Egyptians would not have had the ability to calculate a pyramid's slant height or its ratio to the height, except in one simple case:[49] a pyramid design based on the 3:4:5 triangle as described in the Rhind Mathematical Papyrus.[50] Ancient Egyptian mathematics did not include the notion of irrational numbers,[51] and the rational inverse slope (multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of some pyramids.[50][52]

It has been speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.[53] A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[54] They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction.

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."[55][56] 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 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.[57]

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.[58]

Art

Da Vinci's illustration of a dodecahedron

De divina proportione (On the Divine Proportion), 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 the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[59] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.

De divina proportione contains illustrations of polyhedra by Leonardo da Vinci;[60] this collaboration has led some to speculate that Leonardo incorporated the golden ratio in his work, but this is not supported by any of his writings.[61] Similarly, although the Vitruvian Man is often connected with the golden ratio,[62] its proportions do not actually match it, and the text only mentions whole number ratios.[63] The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a more direct relationship to the golden ratio.[2]

The Section d'Or was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism.[64] Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated art-form, and in homage to the mathematical harmony associated with Georges Seurat.[65][66] The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature, an absolute scientific clarity of conception.[67] However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio itself in their compositions is more difficult to determine. Mario Livio, for example, claims that they did not,[68] and Marcel Duchamp said as much in an interview with art historian William A. Camfield.[69] On the other hand, an analysis by Camfield suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.[70][71] (Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been deeply involved.)[72]

Salvador Dalí, influenced by the works of Matila Ghyka,[73] 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.[61][74]

Mondrian has been said to have used the golden section extensively in his geometrical paintings,[75] though other experts (including critic Yve-Alain Bois) have disputed this claim.[61]

A statistical study performed in 1999 on 565 works of art by different painters found that they did not use the golden ratio in their canvas sizes. 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 to 1.46.[76] 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.[77]

Books

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."[78]

According to Jan Tschichold,[79]

There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

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, light switch plates and cars.[80][81] [82][83][84]

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[85] though other music scholars reject that analysis.[86] 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".[87]

The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.[88] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[89]

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[90]

Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents (About this soundPlay ).[91]

Nature

Detail of Aeonium tabuliforme showing the multiple spiral arrangement (parastichy)

In 1754, Charles Bonnet pointed out that the spiral phyllotaxis of plants were frequently expressed in successive golden ratio series.[20][28][92] Adolf Zeising (1810–1876) also found the golden ratio expressed in the arrangement of parts such as leaves and branches along the stems of plants and veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, the proportions of chemical compounds, and the geometry of crystals. In 1854, he wrote 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."[93][94][95] However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[96]

In 1989, Paul Davies proved that rotating black holes transition between hot and cold when the square of their mass equals ϕ times the square of their speed.[97][98][99] 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.[100]

Perception

Studies by psychologists, starting with Gustav 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 test such a hypothesis have been inconclusive.[101][61]

Disputed observations

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

  • Historian John Man states that the pages of the Gutenberg Bible were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.[102]
  • Some specific proportions in the bodies of many animals (including humans)[103][104] and parts of the shells of mollusks[3] are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[103] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[104] 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;[105] however, measurements of nautilus shells do not support this claim.[106]
  • In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[107] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[108]

The Parthenon

Many of the proportions of the Parthenon have been theorized to exhibit the golden ratio, but this has largely been discredited.

The Parthenon's façade has been said by some to contain golden ratio properties,[109] but this has been discredited by other scholars.[110] For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 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 pentagons and decagons was duly observed, as well as in the dodecahedron".[111] And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value."[112] Later sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions. One researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries, based on measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae.[113]

Mathematics

Usually, the lowercase form of phi (φ or φ) is used to symbolize the golden ratio. Sometimes the uppercase form () is used for the reciprocal of the golden ratio, 1/φ.[114]

Calculation

Binary 1.1001111000110111011...
Decimal 1.6180339887498948482...[10]
Hexadecimal 1.9E3779B97F4A7C15F39...
Continued fraction
Algebraic form
Infinite series

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

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

Therefore,

Multiplying by φ gives

which can be rearranged to

Using the quadratic formula, two solutions are obtained:

and

Because φ is the ratio between positive quantities φ is necessarily positive:

.

Irrationality

The golden ratio is an irrational number. Below are two short proofs of irrationality:

Contradiction from an expression in lowest terms

If φ were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the integers have a lower bound, so φ cannot be rational.

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 n − m,

or, algebraically

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

By irrationality of 5

Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If 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.

Minimal polynomial

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.

Golden ratio conjugate

The conjugate root to the minimal polynomial x2 − x − 1 is

The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[114] It is denoted here by the capital Phi ():

Alternatively, 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....

Alternative forms

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:[115]

and its reciprocal:

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

The equation φ2 = 1 + φ likewise produces the continued square root, or infinite surd, form:

An infinite series can be derived to express phi:[116]

Also:

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

Geometry

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 one square divided by that of the next smaller square is the golden ratio.

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

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 surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[117]

Dividing a line segment by interior division

Dividing a line segment by interior division according to the golden ratio
  1. Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC.
  2. Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.
  3. Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original line segment AB into line segments AS and SB with lengths in the golden ratio.

Dividing a line segment by exterior division

Dividing a line segment by exterior division according to the golden ratio
  1. Draw a line segment AS and construct off the point S a segment SC perpendicular to AS and with the same length as AS.
  2. Do bisect the line segment AS with M.
  3. A circular arc around M with radius MC intersects in point B the straight line through points A and S (also known as the extension of AS). The ratio of AS to the constructed segment SB is the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.

Both the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer to the shorter one is the golden ratio.

Golden triangle, pentagon and pentagram

Golden triangle. The double-red-arched angle is 36 degrees, or radians.
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 = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.

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

Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse ratio is φ − 1.

Pentagon

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio.[8]

Odom's construction
Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C.

George Odom has given a remarkably simple construction for φ involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom's name as a diagram in the American Mathematical Monthly accompanied by the single word "Behold!" [118]

Pentagram

The golden ratio plays an important role in the geometry of 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 two intersecting edges (a side of the pentagon in the pentagram's center) is φ, 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 φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

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

The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b2 = a2 + ab which yields

Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.[119]

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 φ−1 < r < φ, 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 = φ then the shorter two sides are 1 and φ but their sum is φ2, thus r < φ. A similar calculation shows that r > φ−1. A triangle whose sides are in the ratio 1 : φ : φ is a right triangle (because 1 + φ = φ2) known as a Kepler triangle.[30]

Golden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron's rhombi
All of the faces of the rhombic triacontahedron are golden rhombi

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.[120]

Golden pyramid

A square pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is 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 apothem by semi-base). The height of this pyramid is times the semi-base, and the medial right triangle of the pyramid is a Kepler triangle, with sides .[121] This is the only right triangle proportion with edge lengths in geometric progression.[30] The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").[122]

According to various measurements, the Great Pyramid of Giza (c. 2560 BC) closely approximates a golden pyramid.

Relationship to Fibonacci sequence

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

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

A closed-form expression for the Fibonacci sequence involves the golden ratio:

Golden squares with T-branching
Golden square fractal

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler:[26]

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

More generally:

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

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

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

The reduction to a linear expression can be accomplished in one step by using the relationship

where is the kth Fibonacci number.

However, this is no special property of φ, 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 ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then has degree n over , with basis

Symmetries

The golden ratio and inverse golden ratio have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps – they are reciprocals, symmetric about , and (projectively) symmetric about 2.

More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on 3 letters, corresponding to the stabilizer of the set of 3 standard points on the projective line, and the symmetries correspond to the quotient map – the subgroup consisting of the 3-cycles and the identity 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 and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).[123]

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

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

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

If , then:

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

The golden ratio is a fundamental unit of the algebraic number field and is a Pisot–Vijayaraghavan number.[124] In the field we have , where is the -th Lucas 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 .[125]

Decimal expansion

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

with 5 ≈ 2.2360679774997896964 OEISA002163. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as xφ = 2 and iterating

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

The Babylonian algorithm for 5 is equivalent to Newton's method for solving the equation x2 − 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation, including the equation x2 − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

for an appropriate initial estimate xφ such as xφ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/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 π 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 F 25001 and F 25000, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

The decimal expansion of the golden ratio φ[10] has been calculated to an accuracy of two trillion (2×1012 = 2,000,000,000,000) digits.[126]

See also

References

Footnotes

  1. ^ If the constraint on a and b each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. ϕ is defined as the positive solution. The negative solution can be written as . The sum of the two solutions is one, and the product of the two solutions is negative one.
  2. ^ Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
  3. ^ Barr chose Phi because it is the first letter of the name of Classical Greek sculptor Phidias (c. 490–430 BC),[33] but later wrote that he thought it unlikely that Phidias had actually used the golden proportion.[34]

Citations

  1. ^ Livio 2003, pp. 3, 81.
  2. ^ a b Piotr Sadowski (1996). The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight. University of Delaware Press. p. 124. ISBN 978-0-87413-580-0.
  3. ^ a b Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
  4. ^ a b Euclid, Elements, Book 6, Definition 3.
  5. ^ 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."
  6. ^ Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920
  7. ^ William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003
  8. ^ a b Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
  9. ^ Strogatz, Steven (September 24, 2012). "Me, Myself, and Math: Proportion Control". The New York Times.
  10. ^ a b c OEISA001622
  11. ^ Olsen 2006, p. 3.
  12. ^ Livio 2003, pp. 46, 61.
  13. ^ Livio 2003, pp. 6, 25, 34, 36.
  14. ^ Constantine J. Vamvacas (2009). The Founders of Western Thought – The Presocratics. Springer Science & Business Media. p. 65. ISBN 9781402097911.
  15. ^ Livio 2003, pp. 4–5.
  16. ^ Livio 2003, p. 36.
  17. ^ Livio 2003, p. 67.
  18. ^ Plato, The Republic, Book 6, translated by Benjamin Jowett, online Archived 18 April 2009 at the Wayback Machine.
  19. ^ Livio 2003, p. 78.
  20. ^ a b c d e f g h Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 1-4027-3522-7.
  21. ^ Livio 2003, p. 3.
  22. ^ Livio 2003, pp. 86–87.
  23. ^ Livio 2003, p. 87.
  24. ^ Livio 2003, pp. 89–90.
  25. ^ "The Golden Ratio". The MacTutor History of Mathematics archive. Retrieved 18 September 2007.
  26. ^ a b James Joseph Tattersall (2005). Elementary number theory in nine chapters (2nd ed.). Cambridge University Press. p. 28. ISBN 978-0-521-85014-8.
  27. ^ Karl Fink; Wooster Woodruff Beman; David Eugene Smith (1903). A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik (2nd ed.). Chicago: Open Court Publishing Co. p. 223.
  28. ^ a b Olsen 2006, p. 12.
  29. ^ Underwood Dudley (1999). Die Macht der Zahl: Was die Numerologie uns weismachen will. Springer. p. 245. ISBN 3-7643-5978-1.
  30. ^ a b c d Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. p. 81. ISBN 0-88920-324-5.
  31. ^ Livio 2003, cover.
  32. ^ Oxford English Dictionary, 3rd ed. "phi, n." Oxford University Press (Oxford), 2005.
  33. ^ Cook, Theodore Andrea (1914). The Curves of Life. London: Constable and Company Ltd. p. 420.
  34. ^ Barr, Mark (1929). "Parameters of beauty". Architecture (NY). Vol. 60. p. 325. Reprinted: "Parameters of beauty". Think. Vol. 10–11. International Business Machines Corporation. 1944.
  35. ^ Livio 2003, p. 5.
  36. ^ Weisstein, Eric W. "Golden Ratio". MathWorld.
  37. ^ Livio 2003, p. 81.
  38. ^ Olsen 2006, p. 48.
  39. ^ Gardner, Martin (2001), The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics, W. W. Norton & Company, p. 88, ISBN 9780393020236.
  40. ^ Jaric, Marko V. (2012), Introduction to the Mathematics of Quasicrystals, Elsevier, p. x, ISBN 9780323159470, Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
  41. ^ Alison, Jim (2006). Nixon, Steve, ed. The Best of Astraea: 17 Articles on Science, History and Philosophy. Astrea Web Radio. pp. 92–93. ISBN 1-4259-7040-0.
  42. ^ Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859
  43. ^ Ghyka, Matila. The Geometry of Art and Life, New York: Dover, 1977
  44. ^ Rice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp. 24 Routledge, 2003, ISBN 0-415-26876-1
  45. ^ 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
  46. ^ Markowsky, George (January 1992). "Misconceptions about the Golden Ratio" (PDF). College Mathematics Journal. Mathematical Association of America. 23 (1): 2–19. doi:10.2307/2686193. JSTOR 2686193.
  47. ^ Livio 2003, p. 61.
  48. ^ Burton, David M. (1999). The history of mathematics: an introduction (4 ed.). WCB McGraw-Hill. p. 56. ISBN 0-07-009468-3.
  49. ^ Bell, Eric Temple (1940). The Development of Mathematics. New York: Dover. p. 40.
  50. ^ a b Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000
  51. ^ Lancelot Hogben, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56
  52. ^ "The Great Pyramid, The Great Discovery, and The Great Coincidence". Archived from the original on 2014-01-02. Retrieved 2007-11-25.
  53. ^ Jason Elliot (2006). Mirrors of the Unseen: Journeys in Iran. Macmillan. pp. 277, 284. ISBN 978-0-312-30191-0.
  54. ^ 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), [1]
  55. ^ 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
  56. ^ Marcus Frings: The Golden Section in Architectural Theory, Nexus Network Journal vol. 4 no. 1 (Winter 2002), available online [2]
  57. ^ 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".
  58. ^ Urwin, Simon. Analysing Architecture (2003) pp. 154–5, ISBN 0-415-30685-X
  59. ^ Livio 2003, pp. 134–35.
  60. ^ Leonardo da Vinci's Polyhedra, by George W. Hart[3]
  61. ^ a b c d Livio, Mario (November 1, 2002). "The golden ratio and aesthetics". Plus Magazine. Retrieved November 26, 2018.
  62. ^ Keith Devlin (May 2007). "The Myth That Will Not Go Away". Retrieved September 26, 2013. Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
  63. ^ Donald E. Simanek. "Fibonacci Flim-Flam". Archived from the original on March 17, 2016. Retrieved November 27, 2018.
  64. ^ Le Salon de la Section d'Or, October 1912, Mediation Centre Pompidou
  65. ^ Jeunes Peintres ne vous frappez pas !, La Section d’Or: Numéro spécial consacré à l'Exposition de la "Section d’Or", première année, n° 1, 9 octobre 1912, pp. 1-2.
  66. ^ Jeunes Peintres ne vous frappez pas !, La Section d’Or: Numéro spécial consacré à l’Exposition de la "Section d’Or", première année, n° 1, 9 octobre 1912, pp. 1-7, Bibliothèque Kandinsky
  67. ^ Robert Herbert, Neo-Impressionism, New York: The Solomon R. Guggenheim Foundation, 1968
  68. ^ Livio 2003, p. 169.
  69. ^ William A. Camfield, Juan Gris and the Golden Section, Art Bulletin, 47, no. 1, March 1965, 128-34. 68
  70. ^ Christopher Green, Juan Gris, Whitechapel Art Gallery, London, 18 September - 29 November 1992 ; Staatsgalerie Stuttgart 18 December 1992-14 February 1993 ; Rijksmuseum Kröller-Müller, Otterlo, 6 March - 2 May 1993, Yale University Press, 1992, pp. 37-38, ISBN 0300053746
  71. ^ David Cottington, Cubism and Its Histories, Barber Institute's critical perspectives in art history series, Critical Perspectives in Art History, Manchester University Press, 2004, pp. 112, 142, ISBN 0719050049
  72. ^ Roger Allard, Sur quelques peintre, Les Marches du Sud-Ouest, June 1911, pp. 57–64. In Mark Antliff and Patricia Leighten, A Cubism Reader, Documents and Criticism, 1906-1914, The University of Chicago Press, 2008, pp. 178–191, 330.
  73. ^ Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.
  74. ^ Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X
  75. ^ 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
  76. ^ Olariu, Agata, Golden Section and the Art of Painting Available online
  77. ^ Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, p. 134–144
  78. ^ Tschichold, Jan. 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."
  79. ^ Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.
  80. ^ Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist. 11: 44–52. Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
  81. ^ Art Johnson (1999). Famous problems and their mathematicians. Libraries Unlimited. p. 45. ISBN 978-1-56308-446-1. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
  82. ^ Stakhov & Olsen 2006, p. 21. "A credit card has a form of the golden rectangle."
  83. ^ Simon Cox (2004). Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel. Barnes & Noble Books. ISBN 978-0-7607-5931-8. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
  84. ^ "The new Rapide S : Design". The ‘Golden Ratio’ sits at the heart of every Aston Martin.
  85. ^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
  86. ^ Livio 2003, p. 190.
  87. ^ Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) p.83, ISBN 0-415-30010-X
  88. ^ Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 0-521-31145-4.
  89. ^ Simon Trezise (1994). Debussy: La Mer. Cambridge University Press. p. 53. ISBN 0-521-44656-2.
  90. ^ "Pearl Masters Premium". Pearl Corporation. Archived from the original on December 19, 2007. Retrieved December 2, 2007.
  91. ^ "An 833 Cents Scale: An experiment on harmony", Huygens-Fokker.org. Accessed December 1, 2012.
  92. ^ "The Secret of the Fibonacci Sequence in Trees". American Museum of Natural History. 2011. Archived from the original on 4 May 2013. Retrieved 9 December 2018.
  93. ^ Zeising, Adolf (1854). Neue Lehre van den Proportionen des meschlischen Körpers. preface.
  94. ^ Richard Padovan (1999). Proportion. Taylor & Francis. pp. 305–6. ISBN 978-0-419-22780-9.
  95. ^ Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.
  96. ^ Pommersheim, James E., Tim K. Marks, and Erica L. Flapan, eds. 2010. "Number Theory: A Lively Introduction with Proofs, Applications, and Stories". John Wiley and Sons: 82.
  97. ^ Olsen 2006, p. 46.
  98. ^ Livio, Mario (22 August 2012). "The Golden Ratio and Astronomy". HuffPost. Retrieved 4 December 2018.
  99. ^ "Black Holes and the Golden Ratio". Azimuth. 28 February 2013. Retrieved 4 December 2018.
  100. ^ "Golden ratio discovered in a quantum world". Eurekalert.org. 7 January 2010. Retrieved 31 October 2011.
  101. ^ Livio 2003, p. 7.
  102. ^ Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166–167, Wiley, ISBN 0-471-21823-5. "The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8."
  103. ^ a b Pheasant, Stephen (1998). Bodyspace. London: Taylor & Francis. ISBN 0-7484-0067-2.
  104. ^ a b van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach GmbH.
  105. ^ Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004
  106. ^ Peterson, Ivars. "Sea shell spirals". Science News.
  107. ^ For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68.
  108. ^ Roy Batchelor and Richard Ramyar, "Magic numbers in the Dow," 25th International Symposium on Forecasting, 2005, p. 13, 31. "Not since the 'big is beautiful' days have giants looked better", Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and "Technical failure", The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.
  109. ^ 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.
  110. ^ Livio 2003, pp. 74–5.
  111. ^ Midhat J. Gazalé , Gnomon, Princeton University Press, 1999. ISBN 0-691-00514-1
  112. ^ Keith J. Devlin The Math Instinct: Why You're A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs), p. 108. New York: Thunder's Mouth Press, 2005, ISBN 1-56025-672-9
  113. ^ Patrice Foutakis, "Did the Greeks Build According to the Golden Ratio?", Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, pp. 71–86.
  114. ^ a b Weisstein, Eric W. "Golden Ratio Conjugate". MathWorld.
  115. ^ Max. Hailperin; Barbara K. Kaiser; Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. ISBN 0-534-95211-9.
  116. ^ Brian Roselle, "Golden Mean Series"
  117. ^ "A Disco Ball in Space". NASA. 2001-10-09. Retrieved 2007-04-16.
  118. ^ Chris and Penny. "Quandaries and Queries". Math Central. Retrieved 23 October 2011.
  119. ^ American Mathematical Monthly, pp. 49–50, 1954.
  120. ^ Koca, Mehmet; Koca, Nazife Ozdes; Koç, Ramazan (2010), "Catalan solids derived from three-dimensional-root systems and quaternions", Journal of Mathematical Physics, 51: 043501, arXiv:0908.3272, Bibcode:2010JMP....51d3501K, doi:10.1063/1.3356985.
  121. ^ Alison, Jim (2006). Nixon, Steve, ed. The Best of Astraea: 17 Articles on Science, History and Philosophy. Astrea Web Radio. p. 93. ISBN 1-4259-7040-0.
  122. ^ Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999
  123. ^ Fibonacci Numbers and Nature – Part 2 : Why is the Golden section the "best" arrangement?, from Dr. Ron Knott's Fibonacci Numbers and the Golden Section, retrieved 2012-11-29.
  124. ^ Weisstein, Eric W. "Pisot Number". MathWorld.
  125. ^ Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.
  126. ^ Yee, Alexander J. (17 August 2015). "Golden Ratio". numberword.org. Independent computations done by Ron Watkins and Dustin Kirkland.

Works cited

Further reading

  • Doczi, György (2005) [1981]. The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications. ISBN 1-59030-259-1.
  • Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications. ISBN 0-486-22254-3.
  • Joseph, George G. (2000) [1991]. The Crest of the Peacock: The Non-European Roots of Mathematics (New ed.). Princeton, NJ: Princeton University Press. ISBN 0-691-00659-8.
  • Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design (3rd Rev. ed.). Charleston, SC: BookSurge. ISBN 1-4196-2157-2.
  • Schneider, Michael S. (1994). A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. ISBN 0-06-016939-7.
  • Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.
  • Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 0-88385-534-8.

External links