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. Expressed algebraically, for quantities ⁠\(a\)⁠ and ⁠\(b\)⁠ with ⁠\(a > b > 0\)⁠, ⁠\(a\)⁠ is in a golden ratio to ⁠\(b\)⁠ if \[\frac{a+b}{a} = \frac{a}{b} = \varphi,\] where the Greek letter phi (⁠\(\varphi\)⁠ or ⁠\(\phi\)⁠, or ⁠\(\Phi\)⁠) denotes the golden ratio. The constant ⁠\(\varphi\)⁠ satisfies the quadratic equation ⁠\(\textstyle \varphi^2 = \varphi + 1\)⁠ and is an irrational number with a value of \(\varphi = \frac{1+\sqrt5}{2} = {}\)1.618033988749.... The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle, that is, a rectangle with an aspect ratio of ⁠\(\varphi\)⁠, may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

Two non-zero quantities ⁠\(a\)⁠ and ⁠\(b\)⁠ are in the golden ratio ⁠\(\varphi\)⁠ if

To determine ⁠\(\varphi\)⁠ as a number, we can divide the numerator and denominator of the fraction on the left-hand side by ⁠\(b\)⁠,

and then substitute ⁠\(\tfrac{a}{b} = \varphi\)⁠, to obtain

Multiplying both sides by ⁠\(\varphi\)⁠ gives

which can be rearranged to

The quadratic formula yields two solutions:

The positive root, ⁠\(\varphi\)⁠, is the golden ratio. The negative root is its negative inverse ⁠\(-1/\varphi\)⁠, with which it shares many properties.

According to Mario Livio,

, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

Ancient Greek mathematicians first studied 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. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:

The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850-930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170-1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.

Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). 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. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.

German mathematician Simon Jacob (d. 1564) noted that ratios of consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about ⁠\(0.6180340\)⁠" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Condensed: the full section is in Wikipedia.

Since the golden ratio is a root of a polynomial with rational coefficients, it is an algebraic number. Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is \[x^2 - x - 1.\] This quadratic polynomial has two roots, ⁠\(\varphi\)⁠ and ⁠\(\textstyle -\varphi^{-1}\)⁠. Because the leading coefficient of this polynomial is 1, both roots are algebraic integers. The golden ratio is also closely related to the polynomial ⁠\(\textstyle x^2 + x - 1\)⁠, which has roots ⁠\(-\varphi\)⁠ and ⁠\(\textstyle \varphi^{-1}\)⁠.

The golden ratio ⁠\(\varphi\)⁠ is a fundamental unit of the quadratic field ⁠\(\mathbb{Q}\bigl(\sqrt5~\!\bigr)\)⁠, sometimes called the golden field. In this field, any element can be written in the form ⁠\(r + s\varphi\)⁠, with rational coefficients ⁠\(r\)⁠ and ⁠\(s\)⁠; such a number has norm ⁠\(\textstyle r^2 + rs - s^2\)⁠. Other units, with norm ⁠\(\pm1\)⁠, are the positive and negative powers of ⁠\(\varphi\)⁠. The quadratic integers in this field, which form a ring, are all numbers of the form ⁠\(a + b\varphi\)⁠ where ⁠\(a\)⁠ and ⁠\(b\)⁠ are integers.

As the root of a quadratic polynomial, the golden ratio is a constructible number.

The conjugate root to the minimal polynomial ⁠\(\textstyle x^2-x-1\)⁠ is

\[-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt5}{2} = -0.618033\dots.\]

The absolute value of this quantity (⁠\(0.618\ldots\)⁠) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ⁠\(b/a\)⁠).

This illustrates the unique property of the golden ratio among positive numbers, that \[\frac1\varphi = \varphi - 1,\]

or its inverse, \[\frac1{1/\varphi} = \frac1\varphi + 1.\]

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

\[\begin{aligned} \varphi^2 &= \varphi + 1 = 2.618033\dots, \\[5mu] \frac1\varphi &= \varphi - 1 = 0.618033\dots. \end{aligned}\]

Condensed: the full section is in Wikipedia.

The formula ⁠\(\varphi = 1 + 1/\varphi\)⁠ can be expanded recursively to obtain a simple continued fraction for the golden ratio: \[\varphi = [1; 1, 1, 1, \dots] = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}\]

It is in fact the simplest form of a continued fraction, alongside its reciprocal form: \[\varphi^{-1} = [0; 1, 1, 1, \dots] = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}\]

The convergents of these continued fractions, ⁠\(\tfrac11\)⁠, ⁠\(\tfrac21\)⁠, ⁠\(\tfrac32\)⁠, ⁠\(\tfrac53\)⁠, ⁠\(\tfrac85\)⁠, ⁠\(\tfrac{13}8\)⁠, ... or ⁠\(\tfrac11\)⁠, ⁠\(\tfrac12\)⁠, ⁠\(\tfrac23\)⁠, ⁠\(\tfrac35\)⁠, ⁠\(\tfrac58\)⁠, ⁠\(\tfrac8{13}\)⁠, ..., are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational ⁠\(\xi\)⁠, there are infinitely many distinct fractions ⁠\(p/q\)⁠ such that, \[\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}.\]

This means that the constant ⁠\(\sqrt5\)⁠ cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.

A continued square root form for ⁠\(\varphi\)⁠ can be obtained from ⁠\(\textstyle \varphi^2 = 1 + \varphi\)⁠, yielding: \[\varphi = \sqrt{1 + \sqrt{\textstyle 1 + \sqrt{ 1 + \cdots \vphantom)}}}.\]

Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term \(F_n\) is equal to the sum of the preceding two terms \(F_{n-1}\) and \(F_{n-2}\), starting with the base sequence ⁠\(0,1\)⁠ as the 0th and 1st terms \(F_0\) and \(F_1\):

The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in that each term \(L_n\) is the sum of the previous two terms \(L_{n-1}\) and \(L_{n-2}\), however instead starts with ⁠\(2,1\)⁠ as the 0th and 1st terms \(L_0\) and \(L_1\):

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: \[\lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \lim_{n\to\infty} \frac{L_{n+1}}{L_n} = \varphi.\]

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates ⁠\(\varphi\)⁠. For example,

These approximations are alternately lower and higher than ⁠\(\varphi\)⁠, and converge to ⁠\(\varphi\)⁠ as the Fibonacci and Lucas numbers increase.

Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:

\[F\left(n\right) = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5} = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt5} = \frac{1}{\sqrt5}\left[\left({ 1+ \sqrt{5} \over 2}\right)^n - \left({ 1- \sqrt{5} \over 2}\right)^n\right],\] \[L\left(n\right) = \varphi^n + (- \varphi)^{-n} = \varphi^n + (1 - \varphi)^n = \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n .\]

Condensed: the full section is in Wikipedia.

The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a regular icosahedron. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation ⁠\(\textstyle x^2-x-1=0\)⁠ or on ⁠\(\textstyle x^2-5=0\)⁠ (to compute ⁠\(\sqrt5\)⁠ first). The time needed to compute ⁠\(n\)⁠ digits of the golden ratio using Newton's method is essentially ⁠\(O(M(n))\)⁠, where ⁠\(M(n)\)⁠ is the time complexity of multiplying two ⁠\(n\)⁠-digit numbers. This is considerably faster than known algorithms for 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 ⁠\(\varphi\)⁠ has been calculated to an accuracy of twenty trillion (⁠\(\textstyle 2 \times 10^{13} = 20{,}000{,}000{,}000{,}000\)⁠) digits.

In the complex plane, the fifth roots of unity ⁠\(\textstyle z = e^{2\pi k i/5}\)⁠ (for an integer ⁠\(k\)⁠) satisfying ⁠\(\textstyle z^5 = 1\)⁠ are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, ⁠\(z + \bar z\)⁠, is a quadratic integer, an element of ⁠\(\Z[\varphi]\)⁠. Specifically,

\[\begin{aligned} e^{0} + e^{-0} &= 2, \\[5mu] e^{2\pi i / 5} + e^{-2\pi i / 5} &= \varphi^{-1} = -1 + \varphi, \\[5mu] e^{4\pi i / 5} + e^{-4\pi i / 5} &= -\varphi. \end{aligned}\]

This also holds for the remaining tenth roots of unity satisfying ⁠\(\textstyle z^{10} = 1\)⁠,

\[\begin{aligned} e^{\pi i} + e^{-\pi i} &= -2, \\[5mu] e^{\pi i / 5} + e^{-\pi i / 5} &= \varphi, \\[5mu] e^{3\pi i / 5} + e^{-3\pi i / 5} &= -\varphi^{-1} = 1 - \varphi. \end{aligned}\]

For the gamma function ⁠\(\Gamma\)⁠, the only solutions to the equation ⁠\(\Gamma(z-1) = \Gamma(z+1)\)⁠ are ⁠\(z = \varphi\)⁠ and ⁠\(\textstyle z = -\varphi^{-1}\)⁠.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or ⁠\(\varphi\)⁠-nary), quadratic integers in the ring ⁠\(\Z[\varphi]\)⁠, that is, numbers of the form ⁠\(a + b\varphi\)⁠ for ⁠\(a\)⁠ and ⁠\(b\)⁠ in ⁠\(\Z\)⁠, have terminating representations, but rational fractions have non-terminating representations.

Condensed: the full section is in Wikipedia.

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."

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.

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.

Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's Vitruvian Man is often 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.

Salvador Dalí, influenced by the works of Matila Ghyka, 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. The futurist artist Almada Negreiros openly used geometrical constructions involving the golden ratio in various artworks.

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). 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 ⁠\(\sqrt5\)⁠ proportions, and others with proportions like ⁠\(\sqrt2\)⁠, ⁠\(3\)⁠, ⁠\(4\)⁠, and ⁠\(6\)⁠.

According to Jan Tschichold,

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.

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

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

Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.

Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".

The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.