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. Expressed algebraically, for quantities a and b with a>b>0,
where the Greek letter phi ( {displaystyle varphi } or {displaystyle phi } ) represents the golden ratio. Its value is:
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.[6][7][8]
Some twentieth-century artists and architects, including Le Corbusier and Dal, have proportioned their works to approximate the golden ratioespecially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratiobelieving this proportion to be aesthetically pleasing. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.
Mathematicians since Euclid have studied the properties of the golden ratio, including its 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 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]
Two quantities a and b are said to be in the golden ratio if
One method for finding the value of is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/,
Therefore,
Multiplying by gives
which can be rearranged to
Using the quadratic formula, two solutions are obtained:
and
Because is the ratio between positive quantities is necessarily positive:
This derivation can also be found with a compass-and-straightedge construction:
The golden ratio has been claimed to have held a special fascination for at least 2,400 years, though without reliable evidence.[11] 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.[12]
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. Euclid's Elements (Greek: ) 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 lesser."[13] Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e., the golden ratio.[14] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[15]
The golden ratio is explored in Luca Pacioli's book De divina proportione of 1509.
The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340", was written in 1597 by Michael Maestlin of the University of Tbingen in a letter to his former student Johannes Kepler.[16]
Since the 20th century, the golden ratio has been represented by the Greek letter (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by (tau, the first letter of the ancient Greek root meaning cut).[1][17]
Timeline according to Priya Hemenway:[18]
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 the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[1] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. De Divina Proportione contains illustrations of regular solids by Leonardo da Vinci, Pacioli's longtime friend and collaborator.
The Parthenon's faade as well as elements of its faade and elsewhere are said by some to be circumscribed by golden rectangles.[25] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazal 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 (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."[26] 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."[27] Later sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
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.[28] 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."[29]
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.[30]
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.[31]
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.[32]
Patrice Foutakis examined the measurements of 15 temples, 18 monumental tombs, 8 sarcophagi and 58 grave stelae from the fifth century BC to the second century AD. The temples were the main place for communication between the humans and Gods, while the tombs, sarcophagi and grave stelae were connected with the mortals' passage from the material life to the eternal one. Should the golden ratio imply any divine, mystical or aesthetic property, then, according to the author, most of these constructions would be characterized by a golden-section rule. The result of this original research is that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Four extremely rare and therefore valuable examples of golden-mean proportions were identified in an ancient tower in Modon (Peloponnese, Greece), in the Great Altar of Pergamon (Pergamon Museum, Berlin), in a grave stele from Edessa (Greece), and in a monumental tomb at Pella (Greece). Although these cases Foutakis claims to be evidence about a golden-section application in constructions of ancient Greece, he concludes that it was a marginal application indicating that the ancient Greeks did not pay any particular attention to the golden ratio as far as their architecture was concerned.[33]
The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.[2]
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.[34] But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings.[35] Similarly, although the Vitruvian Man is often[36] shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.[37]
Salvador Dal, influenced by the works of Matila Ghyka,[38] 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.[1][39]
Mondrian has been said to have used the golden section extensively in his geometrical paintings,[40] though other experts (including critic Yve-Alain Bois) have disputed this claim.[1]
A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[41] 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.[42]
According to Jan Tschichold,[44]
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.
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.[45][46][47][48][49]
Ern Lendvai analyzes Bla Bartk's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[50] though other music scholars reject that analysis.[1] 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."[51]
The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.[52] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[53]
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.[54]
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 (Play(helpinfo)).[55]
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of parts such as leaves and 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 patterns in nature he saw the golden ratio operating as a universal law.[56][57] 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."[58]
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.[59]
Since 1991, several researchers have proposed connections between the golden ratio and human genome DNA.[60][61][62]
However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[63]
The golden ratio is key to the golden section search.
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.[1][64]
The golden ratio is an irrational number. Below are two short proofs of irrationality:
Recall that:
If we call the whole n and the longer part m, then the second statement above becomes
or, algebraically
To say that is rational means that 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/(nm) is in still lower terms. That is a contradiction that follows from the assumption that is rational.
Another short proofperhaps more commonly knownof the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If 1 + 5 2 {displaystyle textstyle {frac {1+{sqrt {5}}}{2}}} is rational, then 2 ( 1 + 5 2 ) 1 = 5 {displaystyle textstyle 2left({frac {1+{sqrt {5}}}{2}}right)-1={sqrt {5}}} is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.
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.
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.[10] It is denoted here by the capital Phi ( {displaystyle Phi } ):
Alternatively, {displaystyle Phi } 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....
The formula = 1 + 1/ can be expanded recursively to obtain a continued fraction for the golden ratio:[65]
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 2 = 1 + likewise produces the continued square root, or infinite surd, form:
An infinite series can be derived to express phi:[66]
Also:
These correspond to the fact that the length of the diagonal of a regular pentagon is times the length of its side, and similar relations in a pentagram.
The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is 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/ 222.5. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[67]
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.
The both above displayed different algorithms produce geometric constructions that divides a line segment into two line segments where the ratio of the longer to the shorter line segment is the golden ratio.
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 = , then XCA = because of the bisection, and CAB = because of the similar triangles; ABC = 2 from the original isosceles symmetry, and BXC = 2 by similarity. The angles in a triangle add up to 180, so 5 = 180, giving = 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 . Because of the isosceles triangles XC=XA and BC=XC, so these are also length. Length AC=AB, therefore equals +1. But triangle ABC is similar to triangle CXB, so AC/BC=BC/BX, AC/=/1, and so AC also equals 2. Thus 2 =+1, confirming that is indeed the golden ratio.
Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to , while the inverse ratio is1.
In a regular pentagon the ratio between a side and a diagonal is {displaystyle Phi } (i.e. 1/), while intersecting diagonals section each other in the golden ratio.[8]
George Odom has given a remarkably simple construction for 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!" [68]
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 , 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 . The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.
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
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 and can be made as close as desired to .[69]
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 1 < r < , 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 = then the shorter two sides are 1 and but their sum is 2, thus r < . A similar calculation shows that r > 1. A triangle whose sides are in the ratio 1: : is a right triangle (because 1 + = 2) known as a Kepler triangle.[70]
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.[71]
The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:
The closed-form expression for the Fibonacci sequence involves the golden ratio:
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:[20]
Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates ; e.g., 987/6101.6180327868852. These approximations are alternately lower and higher than , and converge on as the Fibonacci numbers increase, and:
More generally:
where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when a = 1 {displaystyle a=1} .
Furthermore, the successive powers of obey the Fibonacci recurrence:
This identity allows any polynomial in 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 F k {displaystyle F_{k}} is the kth Fibonacci number.
However, this is no special property of , 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 is a root of an irreducible nth-degree polynomial, then Q ( ) {displaystyle mathbb {Q} (alpha )} has degree n over Q {displaystyle mathbb {Q} } , with basis { 1 , , , n 1 } {displaystyle {1,alpha ,dots ,alpha ^{n-1}}} .
The golden ratio and inverse golden ratio = ( 1 5 ) / 2 {displaystyle varphi _{pm }=(1pm {sqrt {5}})/2} have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations x , 1 / ( 1 x ) , ( x 1 ) / x , {displaystyle x,1/(1-x),(x-1)/x,} this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps 1 / x , 1 x , x / ( x 1 ) {displaystyle 1/x,1-x,x/(x-1)} they are reciprocals, symmetric about 1 / 2 {displaystyle 1/2} , and (projectively) symmetric about 2.
More deeply, these maps form a subgroup of the modular group PSL ( 2 , Z ) {displaystyle operatorname {PSL} (2,mathbf {Z} )} isomorphic to the symmetric group on 3 letters, S 3 , {displaystyle S_{3},} corresponding to the stabilizer of the set { 0 , 1 , } {displaystyle {0,1,infty }} of 3 standard points on the projective line, and the symmetries correspond to the quotient map S 3 S 2 {displaystyle S_{3}to S_{2}} the subgroup C 3 < S 3 {displaystyle C_{3} 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).[72] The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with : More here: