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Golden ratio ( Euclidean Plane Geometry )

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In , two quantities are in the golden ratio if their is the same as the ratio of their to the larger of the two quantities. Expressed algebraically, for quantities $a$ and $b$ with $a > b > 0,$

where the Greek letter phi or $\phi$) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation $x^2 - x - 1 = 0,$ with a value of

The golden ratio is also called the golden mean or golden section (: sectio aurea).Dunlap, Richard A., The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997 Other names include extreme and mean ratio,Euclid, Elements, Book 6, Definition 3. medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut,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 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." and golden number.Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.

since have studied the properties of the golden ratio, including its appearance in the dimensions of a and in a , which may be cut into a square and a smaller rectangle with the same . The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as , in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the and other plant parts.

Some twentieth-century and , including and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be pleasing. These often appear in the form of the , in which the ratio of the longer side to the shorter is the golden ratio.

Calculation
Two quantities $a$ and $b$ are said to be in the golden ratio $\varphi$ if

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

Therefore,

Multiplying by $\varphi$ gives

which can be rearranged to

Using the quadratic formula, two solutions are obtained:

Because $\varphi$ is the ratio between positive quantities, $\varphi$ is necessarily the positive one. The negative root is $-\frac\left\{1\right\}\left\{\varphi\right\}$.

History
According to ,

mathematicians first studied what we now call the golden ratio, because of its frequent appearance in ; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and . According to one story, 5th-century BC mathematician discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising . 's Elements () provides several and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:

(2022). 9781402735226, Sterling.

The golden ratio was studied peripherally over the next millennium. (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems although he never connected it to the .

named his book Divina proportione (1509) after the ratio, and explored its properties including its appearance in some of the . Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section'). 16th-century mathematicians such as solved geometric problems using the ratio.

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

18th-century mathematicians Abraham de Moivre, , and used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.

(1987). 9780889201521, Wilfrid Laurier University Press. .
used the equivalent English term in 1875.
(2022). 9781616144241, . .

By 1910, mathematician began using the as a for the golden ratio.

(2022). 9781616144241, . .
It has also been represented by the first letter of the τομή ('cut' or 'section').

The construction system, developed by in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, developed , a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.

(2022). 9780393020236, W.W. Norton & Company. .
This led to 's early 1980s discovery of , some of which exhibit icosahedral symmetry.

Applications and observations

Architecture
The Swiss , famous for his contributions to the 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, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, Frings, Marcus, The Golden Section in Architectural Theory, Nexus Network Journal vol. 4 no. 1 (Winter 2002).

Le Corbusier explicitly used the golden ratio in his system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of , Leonardo da Vinci's "", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of .

In addition to the golden ratio, Le Corbusier based the system on , 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 system. Le Corbusier's 1927 Villa Stein in exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. : "Both the paintings and the architectural designs make use of the golden section".

Another Swiss architect, , 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 , the golden ratio is the proportion between the central section and the side sections of the house.Urwin, Simon. Analysing Architecture (2003) pp. 154–155,

Art
Divina proportione ( Divine proportion), a three-volume work by , was published in 1509. Pacioli, a , was known mostly as a mathematician, but he was also trained and keenly interested in art. 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 system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.

Leonardo da Vinci's illustrations of in Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his , for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although the 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 , 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 and dominates the composition.Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47,

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).Olariu, Agata, Golden Section and the Art of Painting Available online 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 $6.$Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, pp. 134–144

Books and design
According to ,

There was a time when deviations from the truly beautiful page proportions $2\mathbin\left\{:\right\}3,$ $1\mathbin\left\{:\right\}\sqrt3,$ and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.
(1991). 9780881791167, Hartley & Marks.

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.

(1999). 9781563084461, Libraries Unlimited. .
(2022). 9780760759318, Barnes & Noble Books. .

Flags
The (height to width ratio) of the flag of Togo is in the golden ratio.

Music
Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the ,Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill. though other music scholars reject that analysis. French composer 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 '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 and and the main climax sits at the phi position".Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) p. 83,

The musicologist has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.

(1983). 9780521311458, Cambridge University Press. .
Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.
(1994). 9780521446563, Cambridge University Press. .

Though proposed the non-octave-repeating cents scale based on , the tuning features relations based on the golden ratio. As a musical interval the ratio is cents ()." An 833 Cents Scale: An experiment on harmony", Huygens-Fokker.org. Accessed December 1, 2012.

Nature
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 noted that the golden ratio appeared in and argued from these patterns in nature that the golden ratio was a universal law.

(1999). 9780419227809, Taylor & Francis. .
Zeising wrote in 1854 of a universal 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.Pommersheim, James E., Tim K. Marks, and , eds. 2010. "Number Theory: A Lively Introduction with Proofs, Applications, and Stories". John Wiley and Sons: 82.

Optimization
The golden ratio is a critical element to golden-section search.

Mathematics

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

Contradiction from an expression in lowest terms
Recall that:

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

or, algebraically

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

By irrationality of $\sqrt5$
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 $\varphi = \tfrac12\left(1 + \sqrt5\right)$ is rational, then $2\varphi - 1 = \sqrt5$ is also rational, which is a contradiction if it is already known that the square root of a non- is irrational.

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

This quadratic polynomial has two roots, $\varphi$ and $-\varphi^\left\{-1\right\}.$

The golden ratio is also closely related to the polynomial

which has roots $-\varphi$ and $\varphi^\left\{-1\right\}.$

Golden ratio conjugate
The conjugate root to the minimal polynomial $x^2-x-1$ is

The absolute value of this quantity 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 or silver ratio.Weisstein, Eric W. (2002). "Golden Ratio Conjugate". CRC Concise Encyclopedia of Mathematics, Second Edition, pp. 1207–1208. CRC Press. . It is denoted here by the capital Phi

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

or its inverse:

Alternative forms
The formula $\varphi = 1 + 1/\varphi$ can be expanded recursively to obtain a continued fraction for the golden ratio:
(1998). 9780534952112, Brooks/Cole Pub. Co. .

and its reciprocal:

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

The equation $\varphi^2 = 1 + \varphi$ likewise produces the :

An infinite series can be derived to express $\varphi$:Brian Roselle, "Golden Mean Series"

Also:

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

Geometry
The number $\varphi$ turns up frequently in , particularly in figures with pentagonal . The length of a regular 's is $\varphi$ times its side. The vertices of a regular are those of three mutually golden rectangles.

There is no known general to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, or ). However, a useful approximation results from dividing the sphere into parallel bands of equal and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. $360^\circ/\varphi \approx 222.5^\circ.$ This method was used to arrange the 1500 mirrors of the student-participatory satellite .

Dividing a line segment by interior division
1. Having a line segment $AB,$ construct a perpendicular $BC$ at point $B,$ with $BC$ half the length of $AB.$ Draw the $AC.$
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
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 of the above displayed different produce geometric constructions that determine two aligned where the ratio of the longer one to the shorter one is the golden ratio.

Golden triangle, pentagon and pentagram

Golden triangle
The golden triangle can be characterized as an isosceles triangle $ABC$ with the property that the angle $C$ produces a new $CXB$ which is a to the original.

If angle $BCX = \alpha,$ then $XCA = \alpha$ because of the bisection, and $CAB = \alpha$ because of the similar triangles; $ABC = 2\alpha$ from the original isosceles symmetry, and $BXC = 2\alpha$ by similarity. The angles in a triangle add up to $180^\circ,$ so $5\alpha = 180^\circ,$ giving $\alpha = 36^\circ.$ So the angles of the golden triangle are thus $36^\circ$$72^\circ$$72^\circ.$ The angles of the remaining obtuse isosceles triangle $AXC$ (sometimes called the golden gnomon) are $36^\circ$$36^\circ$$108^\circ.$

Suppose $XB$ has length $1,$ and we call $BC$ length $\varphi.$ Because of the isosceles triangles $XC = XA$ and $BC = XC,$ so these are also length $\varphi.$ Length $AC = AB,$ therefore equals $\varphi + 1.$ But triangle $ABC$ is similar to triangle $CXB,$ so $AC/BC = BC/BX,$ $AC/\varphi = \varphi/1,$ and so $AC$ also equals $\varphi^2.$ Thus $\varphi^2 = \varphi + 1,$ confirming that $\varphi$ is indeed the golden ratio.

Similarly, the ratio of the area of the larger triangle $AXC$ to the smaller $CXB$ is equal to $\varphi,$ while the inverse ratio is $\varphi - 1.$

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.

Odom's construction
George Odom has given a remarkably simple construction for $\varphi$ 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!"

Pentagram
[[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.]] The golden ratio plays an important role in the geometry of . 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 $\varphi,$ as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five and five isosceles triangles. In all of them, the ratio of the longer side to the shorter side is $\varphi.$ The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

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 $b^2 = a^2 + ab$ which yields

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

(2022). 9780521850148, Cambridge University Press. .

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

More generally

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when $a = 1.$

Furthermore, the successive powers of $\varphi$ obey the Fibonacci recurrence

This identity allows any polynomial in $\varphi$ 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 $F_k$ is the $k$th Fibonacci number.

However, this is no special property of $\varphi,$ 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 $n$th-degree polynomial over the rationals can be reduced to a polynomial of degree $n-1.$ Phrased in terms of field theory, if $\alpha$ is a root of an irreducible $n$th-degree polynomial, then $\mathbb\left\{Q\right\}\left(\alpha\right)$ has degree $n$ over $\mathbb\left\{Q\right\},$ with basis $\\left\{ 1, \alpha, \ldots, \alpha^\left\{n-1\right\} \\right\}.$

Symmetries
The golden ratio and inverse golden ratio $\varphi_\pm = \tfrac12\bigl\left(1 \pm \sqrt5\bigr\right)$ have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations $x, 1/\left(1-x\right), \left(x-1\right)/x$ – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps $1/x, 1-x, x/\left(x-1\right)$ – they are reciprocals, symmetric about $\tfrac12,$ and (projectively) symmetric about $2.$

More deeply, these maps form a subgroup of the $\operatorname\left\{PSL\right\}\left(2, \mathbb\left\{Z\right\}\right)$ isomorphic to the on $3$ letters, $S_3,$ corresponding to the stabilizer of the set $\\left\{0, 1, \infty\\right\}$ of $3$ standard points on the , and the symmetries correspond to the quotient map $S_3 \to S_2$ – the subgroup $C_3 < S_3$ consisting of the identity and the $3$-cycles, in cycle notation $\\left\{\left(1\right), \left(0\,1\,\infty\right), \left(0\,\infty\, 1\right)\\right\},$ fixes the two numbers, while the $2$-cycles $\\left\{\left(0\,1\right), \left(0\,\infty\right), \left(1\,\infty\right)\\right\}$ 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 (the growth of plants). 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.

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

The sequence of powers of $\varphi$ contains these values $0.618033\ldots,$ $1.0,$ $1.618033\ldots,$ $2.618033\ldots;$ more generally, any power of $\varphi$ is equal to the sum of the two immediately preceding powers:

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

If $\lfloor n/2 - 1 \rfloor = m,$ then:

The golden ratio is a fundamental unit of the algebraic number field $\mathbb\left\{Q\right\}\left(\sqrt5\right)$ and is a Pisot–Vijayaraghavan number. In the field $\mathbb\left\{Q\right\}\left(\sqrt5\right)$ we have $\varphi^n = \tfrac12\bigl\left(L_n + F_n\sqrt5\bigr\right),$ where $L_n$ is the $n$-th .

When the golden ratio is used as the base of a (see golden ratio base, sometimes dubbed phinary or $\varphi$ -nary), quadratic integers in the ring $\mathbb\left\{Z\right\}\varphi$ – that is, numbers of the form $a + b\varphi$ for $a, b \in \mathbb\left\{Z\right\}$ – have terminating representations, but rational fractions have non-terminating representations.

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an 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 $4\mathbin\left\{:\right\}\log\left(\varphi\right).$ Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.

The golden ratio appears in the theory of as well. Let Then Also for positive real numbers $a, b \in \mathbb\left\{R\right\}^+$ and $ab = \pi^2,$ thenBrendt, B. et al. "The Rogers–Ramanujan Continued Fraction"

and

For the , the only solutions to the equation are and .

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

with $\sqrt5 =$ . The square root of $5$ can be calculated via the Babylonian method, starting with an initial estimate such as $x_0 = 2$ and

for $n = 0, 1, 2, 3, \ldots,$ until the difference between $x_n$ and $x_\left\{n-1\right\}$ becomes zero to the desired number of digits. Then $\varphi \approx \tfrac12\left(1 + x_n\right).$

The Babylonian algorithm for $\sqrt5$ is equivalent to Newton's method for solving the equation $x^2 - 5 = 0,$ and it converges quadratically, meaning that the number of correct digits is roughly doubled each iteration.

To avoid the computationally expensive division operation, Newton's method can instead be used to solve the equation $5x^\left\{-2\right\} - 4 = 0$ for the root $\tfrac12\sqrt5.$ Then $\varphi \approx \tfrac12 + x_n$ and the update step is

Alternately Newton's method can be applied directly to any equation that has the golden ratio as a solution, such as $x^2 - x - 1 = 0.$ In this case, $\varphi \approx x_n$ and the update step is

Halley's method has cubic convergence (roughly tripling the number of correct digits with each iteration), but may be slower for practical computation because each step takes more work. To solve $x^2 - x - 1 = 0,$ the update step is

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 $\pi$ 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 $F_\left\{25001\right\}$ and $F_\left\{25000\right\},$ each over $5000$ digits, yields over $10\left\{,\right\}000$ significant digits of the golden ratio.

The decimal expansion of the golden ratio $\varphi$ has been calculated to an accuracy of ten trillion digits. Two independent computations done by Clifford Spielman.

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

Mathematical pyramids
A pyramid in which the apothem (slant height along the bisector of a face) is equal to $\varphi$ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is $\sqrt\varphi$ times the semi-base (that is, the slope of the face is $\sqrt\varphi$); the square of the height is equal to the area of a face, $\varphi$ times the square of the semi-base. The medial of this "golden" pyramid (see diagram), with sides $1\mathbin\left\{:\right\}\sqrt\varphi\mathbin\left\{:\right\}\varphi$ is interesting in its own right, demonstrating via the Pythagorean theorem the relationship $\sqrt\varphi = \sqrt{\varphi^2 - 1}$ or $\varphi = \sqrt\left\{1 + \varphi\right\}.$ This
is the only right triangle proportion with edge lengths in geometric progression,
(1977). 9780486235424, Dover. .
(2022). 9780889203242, Wilfrid Laurier University Press.
just as the $3\mathbin\left\{:\right\}4\mathbin\left\{:\right\}5$ triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent $\sqrt\varphi$ corresponds to the angle that the side of the pyramid makes with respect to the ground, $51.827^\circ$ Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the $3\mathbin\left\{:\right\}4\mathbin\left\{:\right\}5$ triangle; the face slope corresponding to the angle with tangent $4/3$ is, to two decimal places, $53.13^\circ$ The slant height or apothem is $5/3 = 1.666\ldots$ times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by , Lost Discoveries: The Ancient Roots of Modern Science – from the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56 and the rational inverse slope (run/rise, multiplied by a factor of $7$ to convert to their conventional units of palms per cubit) was used in the building of pyramids., Trigonometric Delights, Princeton Univ. Press, 2000

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

Egyptian pyramids
One Egyptian pyramid that is close to a "golden pyramid" is the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of $51^\circ\;52\text{'}$ is close to the "golden" pyramid inclination of $51^\circ\;50\text{'}$ – and even closer to the $\pi$-based pyramid inclination of $51^\circ\;51\text{'}.$ However, several other mathematical theories of the shape of the great pyramid, based on rational slopes, have been found to be both more accurate and more plausible explanations for the $51^\circ\;52\text{'}$ slope.

In the mid-nineteenth century, Friedrich Röber studied various Egyptian pyramids including those of Khafre, Menkaure, and some of the Giza, , and groups. He did not apply the golden ratio to the Great Pyramid of Giza, but instead agreed with John Shae Perring that its side-to-height ratio is $8\mathbin\left\{:\right\}5.$ For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or half-side lengths are related to their heights by the golden ratio.

(2022). 9780889203242, Wilfrid Laurier University Press. .

In 1859, the John Taylor misinterpreted () as indicating that the Great Pyramid's height squared equals the area of one of its face triangles. This led Taylor to claim that, in the Great Pyramid, the golden ratio is represented by the ratio of the length of the face (the slope height, inclined at an angle $\theta$ to the ground) to half the length of the side of the square base (equivalent to the of the angle $\theta$).Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859 The above two lengths are about and , 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 , and half-base , yielding $1.6189$ for the ratio of slant height to half-base, again more accurate than the data variability.

Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the $3\mathbin\left\{:\right\}4\mathbin\left\{:\right\}5$ pyramid, since the $3\mathbin\left\{:\right\}4\mathbin\left\{:\right\}5$ triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as $\pi$ or $\varphi.$

(2022). 9780486272399, Dover. .
Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes.

Michael RiceRice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C. p. 24 Routledge, 2003, asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of the mathematics of the pyramids, citing Giedon (1957).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. p. 24 Routledge, 2003 Historians of science have long debated whether the Egyptians had any such knowledge, contending that its appearance in the Great Pyramid is the result of chance.

Disputed observations
Examples of disputed observations of the golden ratio include the following:
• Some specific proportions in the bodies of many animals (including humans)
(1998). 9780748400676, Taylor & Francis.
and parts of the shells of mollusks 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. The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio. The 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 golden-proportioned relative to the previous one., Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004 However, measurements of nautilus shells do not support this claim.
• Historian John Man states that both the pages and text area of the were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is $1.45.$Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166–167, Wiley, . "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."
• Studies by psychologists, starting with c. 1876, have been devised to test the idea that the golden ratio plays a role in human perception of . 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.
• 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.For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in 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. 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", , Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.

The Parthenon
The 's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.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. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, 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."
(2022). 9781560256724, . .
Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."Gazalé, Midhat J., Gnomon: From Pharaohs to Fractals, Princeton University Press, 1999, p. 125.

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, 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.Patrice Foutakis, "Did the Greeks Build According to the Golden Ratio?", Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, pp. 71–86. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

Modern art
The Section d'Or ('Golden Section') was a collective of , sculptors, poets and critics associated with and Orphism. Le Salon de la Section d'Or, October 1912, Mediation Centre Pompidou 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 movement, and in homage to the mathematical harmony associated with . 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 The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature, an absolute scientific clarity of conception.Herbert, Robert, Neo-Impressionism, New York: The Solomon R. Guggenheim Foundation, 1968 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 in any compositions is more difficult to determine. Livio, for example, claims that they did not, and said as much in an interview. Camfield, William A., Juan Gris and the Golden Section, Art Bulletin, 47, no. 1, March 1965, 128–134. 68 On the other hand, an analysis suggests that made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.Green, Christopher, 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, Cottington, David, 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, 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 and other former members of the Abbaye de Créteil had been involved.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.

has been said to have used the golden section extensively in his geometrical paintings,Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp. 247–248, Harcourt, Brace & World, though other experts (including critic ) have discredited these claims.

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