Golden ratio

For other uses, see Golden ratio (disambiguation) and Golden number (disambiguation).

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 ${\displaystyle a}$ and ${\displaystyle b}$ with ${\displaystyle a>b>0}$,

Golden ratio

Line segments in the golden ratio
Representations
Decimal	1.618033988749894...^[1]
Algebraic form	${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$
Continued fraction	${\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}$
Binary	1.10011110001101110111...
Hexadecimal	1.9E3779B97F4A7C15...

A golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left) combine to form a similar golden rectangle with long side a + b and short side a. This illustrates the relationship ${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi .}$

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi }$

where the Greek letter phi (${\displaystyle \varphi }$ or ${\displaystyle \phi }$) denotes the golden ratio.^[a] The constant ${\displaystyle \varphi }$ satisfies the quadratic equation ${\displaystyle \varphi ^{2}=\varphi +1}$ and is an irrational number with a value of^[1]

${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=}$1.618033988749....

The golden ratio was called the extreme and mean ratio by Euclid,^[2] and the divine proportion by Luca Pacioli,^[3] and also goes by several other names.^[b]

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.^[7] A golden rectangle—that is, a rectangle with an aspect ratio of ${\displaystyle \varphi }$—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.^[8] 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.

Calculation

Two quantities ${\displaystyle a}$  and ${\displaystyle b}$  are in the golden ratio ${\displaystyle \varphi }$  if^[9]

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}$ 

One method for finding ${\displaystyle \varphi }$ 's closed form starts with the left fraction. Simplifying the fraction and substituting the reciprocal ${\displaystyle b/a=1/\varphi }$ ,

${\displaystyle {\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}$ 

Therefore,

${\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}$ 

Multiplying by ${\displaystyle \varphi }$  gives

${\displaystyle \varphi +1=\varphi ^{2}}$ 

which can be rearranged to

${\displaystyle {\varphi }^{2}-\varphi -1=0.}$ 

The quadratic formula yields two solutions:

${\displaystyle {\frac {1+{\sqrt {5}}}{2}}=1.618033\dots }$  and ${\displaystyle {\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .}$ 

Because ${\displaystyle \varphi }$  is a ratio between positive quantities, ${\displaystyle \varphi }$  is necessarily the positive root.^[10] The negative root is in fact the negative inverse ${\displaystyle -{\frac {1}{\varphi }}}$ , which shares many properties with the golden ratio.

History

See also: Mathematics and art and Fibonacci number § History

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. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.^[11]

— 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;^[12] the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.^[13] According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans.^[14] Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,^[15][c] and contains its first known definition which proceeds as follows:^[16]

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.^[17][d]

 

Michael Maestlin, the first to write a decimal approximation of the ratio

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.^[19]

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.^[20][21] Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section').^[22] 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.^[23] 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.^[24]

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;^[25] this was rediscovered by Johannes Kepler in 1608.^[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about ${\displaystyle 0.6180340}$ " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.^[27] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.^[28]

18th-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler 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".^[29] Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.^[30] James Sully used the equivalent English term in 1875.^[31]

By 1910, inventor Mark Barr began using the Greek letter Phi (${\displaystyle {\boldsymbol {\varphi }}}$ ) as a symbol for the golden ratio.^[32][e] It has also been represented by tau (${\displaystyle {\boldsymbol {\tau }}}$ ), the first letter of the ancient Greek τομή ('cut' or 'section').^[35]

 

Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.

The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, 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.^[36] This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.^[37]

Mathematics

Irrationality

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

Contradiction from an expression in lowest terms

 

If ${\displaystyle \varphi }$  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 positive integers have a lower bound, so ${\displaystyle \varphi }$  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 ${\displaystyle n}$  and the longer part ${\displaystyle m,}$  then the second statement above becomes

${\displaystyle n}$  is to ${\displaystyle m}$  as ${\displaystyle m}$  is to ${\displaystyle n-m.}$ 

To say that the golden ratio ${\displaystyle \varphi }$  is rational means that ${\displaystyle \varphi }$  is a fraction ${\displaystyle n/m}$  where ${\displaystyle n}$  and ${\displaystyle m}$  are integers. We may take ${\displaystyle n/m}$  to be in lowest terms and ${\displaystyle n}$  and ${\displaystyle m}$  to be positive. But if ${\displaystyle n/m}$  is in lowest terms, then the equally valued ${\displaystyle m/(n-m)}$  is in still lower terms. That is a contradiction that follows from the assumption that ${\displaystyle \varphi }$  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 ${\displaystyle \varphi ={\tfrac {1}{2}}(1+{\sqrt {5}})}$  is rational, then ${\displaystyle 2\varphi -1={\sqrt {5}}}$  is also rational, which is a contradiction if it is already known that the square root of all non-square natural numbers are irrational.

Minimal polynomial

 

The golden ratio ${\displaystyle \varphi }$  and its negative reciprocal ${\displaystyle -\varphi ^{-1}}$  are the two roots of the quadratic polynomial ${\displaystyle x^{2}-x-1}$ . The golden ratio's negative ${\displaystyle -\varphi }$  and reciprocal ${\displaystyle \varphi ^{-1}}$  are the two roots of the quadratic polynomial ${\displaystyle x^{2}+x-1}$ .

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

${\displaystyle x^{2}-x-1.}$ 

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

The golden ratio is also closely related to the polynomial

${\displaystyle x^{2}+x-1,}$ 

which has roots ${\displaystyle -\varphi }$  and ${\displaystyle \varphi ^{-1}.}$  As the root of a quadratic polynomial, the golden ratio is a constructible number.^[38]

Golden ratio conjugate and powers

The conjugate root to the minimal polynomial ${\displaystyle x^{2}-x-1}$  is

${\displaystyle -{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .}$ 

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

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

${\displaystyle {\frac {1}{\varphi }}=\varphi -1,}$ 

or its inverse:

${\displaystyle {\frac {1}{1/\varphi }}=(1/\varphi )+1.}$ 

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

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

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

${\displaystyle \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.}$ 

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

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

${\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}\\[5mu]\varphi ^{n}-\varphi ^{n-1}&=\varphi ^{n-2}.\end{aligned}}}$ 

Continued fraction and square root

See also: Lucas number § Continued fractions for powers of the golden ratio

 

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

The formula ${\displaystyle \varphi =1+1/\varphi }$  can be expanded recursively to obtain a continued fraction for the golden ratio:^[39]

${\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}$ 

It is in fact the simplest form of a continued fraction, alongside its reciprocal form:

${\displaystyle \varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}$ 

The convergents of these continued fractions (${\displaystyle 1/1,}$  ${\displaystyle 2/1,}$  ${\displaystyle 2/1,}$  ${\displaystyle 3/2,}$  ${\displaystyle 5/3,}$  ${\displaystyle 8/5,}$  ${\displaystyle 13/8,}$  ... or ${\displaystyle 1/1,}$  ${\displaystyle 1/2,}$  ${\displaystyle 2/3,}$  ${\displaystyle 3/5,}$  ${\displaystyle 5/8,}$  ${\displaystyle 8/13,}$  ...) 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 ${\displaystyle \xi }$ , there are infinitely many distinct fractions ${\displaystyle p/q}$  such that,

${\displaystyle \left|\xi -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.}$ 

This means that the constant ${\displaystyle {\sqrt {5}}}$  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.^[40]

A continued square root form for ${\displaystyle \varphi }$  can be obtained from ${\displaystyle \varphi ^{2}=1+\varphi }$ , yielding:

${\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}.}$ 

Relationship to Fibonacci and Lucas numbers

Further information: Fibonacci number § Relation to the golden ratio

See also: Lucas number § Relationship to Fibonacci numbers

 

 

A Fibonacci spiral (top) which approximates the golden spiral, using Fibonacci sequence square sizes up to ${\displaystyle 21.}$  A golden spiral is also generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of Lucas numbers, here up to ${\displaystyle 76.}$ 

Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence ${\displaystyle 0,1}$ :

${\displaystyle 0,}$  ${\displaystyle 1,}$  ${\displaystyle 1,}$  ${\displaystyle 2,}$  ${\displaystyle 3,}$  ${\displaystyle 5,}$  ${\displaystyle 8,}$  ${\displaystyle 13,}$  ${\displaystyle 21,}$  ${\displaystyle 34,}$  ${\displaystyle 55,}$  ${\displaystyle 89,}$  ${\displaystyle \ldots }$ (OEIS: A000045).

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-which each term is the sum of the previous two, however instead starts with ${\displaystyle 2,1}$ :

${\displaystyle 2,}$  ${\displaystyle 1,}$  ${\displaystyle 3,}$  ${\displaystyle 4,}$  ${\displaystyle 7,}$  ${\displaystyle 11,}$  ${\displaystyle 18,}$  ${\displaystyle 29,}$  ${\displaystyle 47,}$  ${\displaystyle 76,}$  ${\displaystyle 123,}$  ${\displaystyle 199,}$  ${\displaystyle \ldots }$ (OEIS: A000032).

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:^[41]

${\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\varphi .}$ 

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates ${\displaystyle \varphi }$ .

For example, ${\displaystyle {\frac {F_{16}}{F_{15}}}={\frac {987}{610}}=1.6180327\ldots ,}$  and ${\displaystyle {\frac {L_{16}}{L_{15}}}={\frac {2207}{1364}}=1.6180351\ldots .}$ 

These approximations are alternately lower and higher than ${\displaystyle \varphi ,}$  and converge to ${\displaystyle \varphi }$  as the Fibonacci and Lucas numbers increase.

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

${\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}={{\varphi ^{n}-(-\varphi )^{-n}} \over {\sqrt {5}}},}$ 

${\displaystyle L\left(n\right)=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}\,.}$ 

Combining both formulas above, one obtains a formula for ${\displaystyle \varphi ^{n}}$  that involves both Fibonacci and Lucas numbers:

${\displaystyle \varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}\,.}$ 

Between Fibonacci and Lucas numbers one can deduce ${\displaystyle L_{2n}=5F_{n}^{2}+2(-1)^{n}=L_{n}^{2}-2(-1)^{n},}$  which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:

${\displaystyle \lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}.}$ 

Indeed, much stronger statements are true:

${\displaystyle \vert L_{n}-{\sqrt {5}}F_{n}\vert ={\frac {2}{\varphi ^{n}}}\to 0}$ ,

${\displaystyle (L_{3n}/2)^{2}=5(F_{3n}/2)^{2}+(-1)^{n}}$ .

These values describe ${\displaystyle \varphi }$  as a fundamental unit of the algebraic number field ${\displaystyle \mathbb {Q} ({\sqrt {5}})}$ .

Successive powers of the golden ratio obey the Fibonacci recurrence, i.e. ${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}$ 

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

${\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}$ 

This identity allows any polynomial in ${\displaystyle \varphi }$  to be reduced to a linear expression, as in:

${\displaystyle {\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\[5mu]&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\[5mu]&=\varphi +2\approx 3.618033.\end{aligned}}}$ 

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:

${\displaystyle \sum _{n=1}^{\infty }|F_{n}\varphi -F_{n+1}|=\varphi .}$ 

In particular, the powers of ${\displaystyle \varphi }$  themselves round to Lucas numbers (in order, except for the first two powers, ${\displaystyle \varphi ^{0}}$  and ${\displaystyle \varphi }$ , are in reverse order):

${\displaystyle {\begin{aligned}\varphi ^{0}&=1,\\[5mu]\varphi ^{1}&=1.618033989...\approx 2,\\[5mu]\varphi ^{2}&=2.618033989...\approx 3,\\[5mu]\varphi ^{3}&=4.236067978...\approx 4,\\[5mu]\varphi ^{4}&=6.854101967...\approx 7,\end{aligned}}}$ 

and so forth.^[42] The Lucas numbers also directly generate powers of the golden ratio; for ${\displaystyle n\geq 2}$ :

${\displaystyle \varphi ^{n}=L_{n}-(-\varphi )^{-n}.}$ 

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is ${\displaystyle L_{n}=F_{n-1}+F_{n+1}}$ ; and, importantly, that ${\displaystyle {L_{n}}={\frac {F_{2n}}{F_{n}}}}$ .

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.

Geometry

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 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction

 

 

Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.

Dividing by interior division

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

Dividing by exterior division

1. Draw a line segment ${\displaystyle AS}$  and construct off the point ${\displaystyle S}$  a segment ${\displaystyle SC}$  perpendicular to ${\displaystyle AS}$  and with the same length as ${\displaystyle AS.}$ 
2. Do bisect the line segment ${\displaystyle AS}$  with ${\displaystyle M.}$ 
3. A circular arc around ${\displaystyle M}$  with radius ${\displaystyle MC}$  intersects in point ${\displaystyle B}$  the straight line through points ${\displaystyle A}$  and ${\displaystyle S}$  (also known as the extension of ${\displaystyle AS}$ ). The ratio of ${\displaystyle AS}$  to the constructed segment ${\displaystyle SB}$  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 algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden angle

Main article: Golden angle

 

${\displaystyle g\approx 137.508^{\circ }}$ 

When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure ${\textstyle g\colon }$ 

$${\displaystyle {\begin{aligned}{\frac {2\pi -g}{g}}&={\frac {2\pi }{2\pi -g}}=\varphi ,\\[8mu]2\pi -g&={\frac {2\pi }{\varphi }}\approx 222.5^{\circ },\\[8mu]g&={\frac {2\pi }{\varphi ^{2}}}\approx 137.5^{\circ }.\end{aligned}}}$$

 

This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.^[43]

Golden spiral

Main article: Golden spiral

 

A golden logarithmic spiral swirls around a golden triangle, touching its three vertices, moving inwardly inside similar fractal golden triangles.

Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. Importantly, isosceles golden triangles can be encased by a golden logarithmic spiral, such that successive turns of a spiral generate new golden triangles inside. This special case of logarithmic spirals is called the golden spiral, and it exhibits continuous growth in golden ratio. That is, for every ${\displaystyle 90^{\circ }}$  turn, there is a growth factor of ${\displaystyle \varphi }$ . As mentioned above, these golden spirals can be approximated by quarter-circles generated from Fibonacci and Lucas number-sized squares that are tiled together. In their exact form, they can be described by the polar equation with ${\displaystyle (r,\theta )}$ :

${\displaystyle r=\varphi ^{2\theta /\pi }.}$ 

As with any logarithmic spiral, for ${\displaystyle r=ae^{b\theta }}$  with ${\displaystyle e^{b\theta _{\mathrm {right} }}=\varphi }$  at right angles:

${\displaystyle |b|={\ln {\varphi } \over \theta _{\mathrm {right} }}\doteq 0.0053468^{\circ }\doteq 0.3063489{\text{ rad.}}}$ 

Its polar slope ${\displaystyle \alpha }$  can be calculated using ${\displaystyle \tan \alpha =b}$  alongside ${\displaystyle |b|}$  from above,

${\displaystyle \alpha =\arctan(|b|)=\arctan \left({\ln {\varphi } \over \pi /2}\right)\doteq 17.03239113^{\circ }\doteq 0.2972713047{\text{ rad.}}}$ 

It has a complementary angle, ${\displaystyle \beta }$ :

${\displaystyle \beta =\pi /2-\alpha \doteq 72.96760887^{\circ }\doteq 1.273525022{\text{ rad.}}}$ 

Golden spirals can be symmetrically placed inside pentagons and pentagrams as well, such that fractal copies of the underlying geometry are reproduced at all scales.

In triangles, quadrilaterals, and pentagons

Odom's construction

 

Odom's construction: ${\displaystyle {\tfrac {|AB|}{|BC|}}={\tfrac {|AC|}{|AB|}}=\varphi }$ 

George Odom found a construction for ${\displaystyle \varphi }$  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.^[44]

Kepler triangle

Main article: Kepler triangle

 

A Kepler triangle has sides shared by squares that have areas in geometric progression: ${\displaystyle 1\mathbin {:} \varphi \mathbin {:} \varphi ^{2}}$ .

The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression:

${\displaystyle 1\mathbin {:} {\sqrt {\varphi }}\mathbin {:} \varphi }$ .

The Kepler triangle can also be understood as the right triangle formed by three squares whose areas are also in golden geometric progression ${\displaystyle 1\mathbin {:} \varphi \mathbin {:} \varphi ^{2}}$ .

Fittingly, the Pythagorean means for ${\displaystyle \varphi \pm 1}$  are precisely ${\displaystyle 1}$ , ${\displaystyle \varphi }$ , and ${\displaystyle \varphi ^{2}}$ . It is from these ratios that we are able to geometrically express the fundamental defining quadratic polynomial for ${\displaystyle \varphi }$  with the Pythagorean theorem; that is, ${\displaystyle \varphi ^{2}=\varphi +1}$ .

The inradius of an isosceles triangle is greatest when the triangle is composed of two mirror Kepler triangles, such that their bases lie on the same line.^[45] Also, the isosceles triangle of given perimeter with the largest possible semicircle is one from two mirrored Kepler triangles.^[46]

For a Kepler triangle with smallest side length ${\displaystyle s}$ , the area and acute internal angles are:

${\displaystyle {\begin{aligned}A&={\tfrac {s^{2}}{2}}{\sqrt {\varphi }},\\[5mu]\theta &=\sin ^{-1}{\frac {1}{\varphi }}\approx 38.1727^{\circ },\\[5mu]\theta &=\cos ^{-1}{\frac {1}{\varphi }}\approx 51.8273^{\circ }.\end{aligned}}}$ 

Golden triangle and golden gnomon

Main article: Golden triangle (mathematics)

 

A golden triangle subdivided by an angle bisector into a smaller golden triangle and a golden gnomon

The golden triangle or sublime triangle is a acute isosceles triangle with apex angle 36° and base angles 72°.^[47] Its two equal sides are in the golden ratio to its base. Another isosceles triangle, obtuse, with apex angle 108° and base angle 36°, is called the golden gnomon. The golden ratio is both the ratio of side length to base in the golden triangle, and the ratio of base to side length in the golden gnomon.^[48]

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.^[48] Golden triangles that are decomposed further like this into pairs of isosceles and obtuse golden triangles are known as Robinson triangles.^[49][50]

If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.^[48]

The triangles formed by two sides and a diagonal of a regular pentagon are golden gnomons; the pentagon can be subdivided in this way into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles,^[48] as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.^[51]

Golden rectangle

Main article: Golden rectangle

 

To construct a golden rectangle with only a straightedge and compass in four simple steps:

Draw a square.

Draw a line from the midpoint of one side of the square to an opposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

Complete the golden rectangle.

The golden ratio proportions the adjacent side lengths of a golden rectangle in ${\displaystyle 1:\varphi }$  ratio.^[52] Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ${\displaystyle \varphi }$  ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).^[53]

Golden rhombus

Main article: Golden rhombus

A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly ${\displaystyle 1:\varphi }$ .^[54] For a rhombus of such proportions, its acute angle and obtuse angles are:

${\displaystyle {\begin{aligned}\alpha &=2\arctan {1 \over \varphi }\approx 63.43495^{\circ },\\[5mu]\beta &=2\arctan \varphi =\pi -\arctan 2=\arctan 1+\arctan 3\approx 116.56505^{\circ }.\end{aligned}}}$ 

The lengths of its short and long diagonals ${\displaystyle d}$  and ${\displaystyle D}$ , in terms of side length ${\displaystyle a}$  are:

${\displaystyle {\begin{aligned}d&={2a \over {\sqrt {2+\varphi }}}=2{\sqrt {{3-\varphi } \over 5}}a\approx 1.05146a,\\[5mu]D&=2{\sqrt {{2+\varphi } \over 5}}a\approx 1.70130a.\end{aligned}}}$ 

Its area, in terms of ${\displaystyle a}$ ,and ${\displaystyle d}$ :

${\displaystyle {\begin{aligned}A&=(\sin(\arctan 2))~a^{2}={2 \over {\sqrt {5}}}~a^{2}\approx 0.89443a^{2},\\[5mu]A&={{\varphi } \over 2}d^{2}\approx 0.80902d^{2}.\end{aligned}}}$ 

Its inradius, in terms of side ${\displaystyle a}$ :

${\displaystyle r={\frac {a}{\sqrt {5}}}.}$ 

Golden rhombi feature in the rhombic triacontahedron (see section below). They also are found in the golden rhombohedron, the Bilinski dodecahedron,^[55] and the rhombic hexecontahedron.^[54]

Pentagon and pentagram

 

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

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. 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 ${\displaystyle b,}$  and short edges are ${\displaystyle a,}$  then Ptolemy's theorem gives ${\displaystyle b^{2}=a^{2}+ab}$  which yields,

${\displaystyle {b \over a}={{1+{\sqrt {5}}} \over 2}={\varphi }.}$ 

The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by ${\displaystyle \varphi }$ . Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is ${\displaystyle \varphi ,}$  as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for ${\displaystyle \varphi }$ :

${\displaystyle {\begin{aligned}\varphi &=1+2\sin(\pi /10)=1+2\sin 18^{\circ },\\[5mu]\varphi &={\tfrac {1}{2}}\csc(\pi /10)={\tfrac {1}{2}}\csc 18^{\circ },\\[5mu]\varphi &=2\cos(\pi /5)=2\cos 36^{\circ },\\[5mu]\varphi &=2\sin(3\pi /10)=2\sin 54^{\circ }.\end{aligned}}}$ 

Penrose tilings

Main article: Penrose tiling

 

The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.

The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.^[49] Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:

• Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.^[56]
• The kite and dart Penrose tiling uses kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.^[49]
• The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. Again, these rhombi can be decomposed into golden Robinson triangles. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals ${\displaystyle 1:\varphi }$ , as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of these two tiles are in the golden ratio to each other.^[49]

 

Original four-tile Penrose tiling

 

Rhombic Penrose tiling

In the dodecahedron and icosahedron

 

Cartesian coordinates of the dodecahedron :
(±1, ±1, ±1)
(0, ±φ, ±1/φ)
(±1/φ, 0, ±φ)
(±φ, ±1/φ, 0)
A nested cube inside the dodecahedron is represented with dotted lines.

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. An icosahedron is made of ${\displaystyle 12}$  regular pentagonal faces, whereas the icosahedron is made of ${\displaystyle 20}$  equilateral triangles; both with ${\displaystyle 30}$  edges.^[57]

For a dodecahedron of side ${\displaystyle a}$ , the radius of a circumscribed and inscribed sphere, and midradius are (${\displaystyle r_{u}}$ , ${\displaystyle r_{i}}$  and ${\displaystyle r_{m}}$ , respectively):

${\displaystyle r_{u}=a\,{\frac {{\sqrt {3}}\varphi }{2}}}$ , ${\displaystyle r_{i}=a\,{\frac {\varphi ^{2}}{2{\sqrt {3-\varphi }}}}}$ , and ${\displaystyle r_{m}=a\,{\frac {\varphi ^{2}}{2}}}$ .

While for an icosahedron of side ${\displaystyle a}$ , the radius of a circumscribed and inscribed sphere, and midradius are:

${\displaystyle r_{u}=a{\frac {\sqrt {\varphi {\sqrt {5}}}}{2}}}$ , ${\displaystyle r_{i}=a{\frac {\varphi ^{2}}{2{\sqrt {3}}}}}$ , and ${\displaystyle r_{m}=a{\frac {\varphi }{2}}}$ .

The volume and surface area of the dodecahedron can be expressed in terms of ${\displaystyle \varphi }$ :

${\displaystyle A_{d}={\frac {15\varphi }{\sqrt {3-\varphi }}}}$  and ${\displaystyle V_{d}={\frac {5\varphi ^{3}}{6-2\varphi }}}$ .

As well as for the icosahedron:

${\displaystyle A_{i}=20{\frac {\varphi ^{2}}{2}}}$  and ${\displaystyle V_{i}={\frac {5}{6}}(1+\varphi ).}$ 

 

Three golden rectangles touch all of the ${\displaystyle 12}$  vertices of a regular icosahedron.

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving ${\displaystyle \varphi }$ . The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:

${\displaystyle (0,\pm 1,\pm \varphi )}$ , ${\displaystyle (\pm 1,\pm \varphi ,0)}$ , ${\displaystyle (\pm \varphi ,0,\pm 1).}$ 

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.^[58][53] In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain ${\displaystyle 12}$  vertices of the icosahedron, or equivalently, intersect the centers of ${\displaystyle 12}$  of the dodecahedron's faces.^[57]

A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is ${\displaystyle {\tfrac {2}{2+\varphi }}}$  times that of the dodecahedron's.^[59] In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in ${\displaystyle \varphi :\varphi ^{2}}$  ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's ${\displaystyle 12}$  vertices touch the ${\displaystyle 12}$  edges of an octahedron at points that divide its edges in golden ratio.^[60]

Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio.

Other properties

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 ${\displaystyle x^{2}-x-1=0}$  or on ${\displaystyle x^{2}-5=0}$  (to compute ${\displaystyle {\sqrt {5}}}$  first). The time needed to compute ${\displaystyle n}$  digits of the golden ratio using Newton's method is essentially ${\displaystyle O(M(n))}$ , where ${\displaystyle M(n)}$  is the time complexity of multiplying two ${\displaystyle n}$ -digit numbers.^[61] This is considerably faster than known algorithms for ${\displaystyle \pi }$  and ${\displaystyle e}$ . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers ${\displaystyle F_{25001}}$  and ${\displaystyle F_{25000},}$  each over ${\displaystyle 5000}$  digits, yields over ${\displaystyle 10{,}000}$  significant digits of the golden ratio. The decimal expansion of the golden ratio ${\displaystyle \varphi }$ ^[1] has been calculated to an accuracy of ten trillion (${\displaystyle 1\times 10^{13}=10{,}000{,}000{,}000{,}000}$ ) digits.^[62]

The golden ratio and inverse golden ratio ${\displaystyle \varphi _{\pm }={\tfrac {1}{2}}{\bigl (}1\pm {\sqrt {5}}{\bigr )}}$  have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations ${\displaystyle x,1/(1-x),(x-1)/x}$  – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps ${\displaystyle 1/x,1-x,x/(x-1)}$  – they are reciprocals, symmetric about ${\displaystyle {\tfrac {1}{2}},}$  and (projectively) symmetric about ${\displaystyle 2.}$  More deeply, these maps form a subgroup of the modular group ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$  isomorphic to the symmetric group on ${\displaystyle 3}$  letters, ${\displaystyle S_{3},}$  corresponding to the stabilizer of the set ${\displaystyle \{0,1,\infty \}}$  of ${\displaystyle 3}$  standard points on the projective line, and the symmetries correspond to the quotient map ${\displaystyle S_{3}\to S_{2}}$  – the subgroup ${\displaystyle C_{3}<S_{3}}$  consisting of the identity and the ${\displaystyle 3}$ -cycles, in cycle notation ${\displaystyle \{(1),(0\,1\,\infty ),(0\,\infty \,1)\},}$  fixes the two numbers, while the ${\displaystyle 2}$ -cycles ${\displaystyle \{(0\,1),(0\,\infty ),(1\,\infty )\}}$  interchange these, thus realizing the map.

In the complex plane, the fifth roots of unity ${\displaystyle z=e^{2\pi ki/5}}$  (for an integer ${\textstyle k}$ ) satisfying ${\displaystyle z^{5}=1}$  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, ${\displaystyle z+{\bar {z}},}$  is a quadratic integer, an element of ${\textstyle \mathbb {Z} [\varphi ].}$  Specifically,

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

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

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

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

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or ${\displaystyle \varphi }$ -nary), quadratic integers in the ring ${\displaystyle \mathbb {Z} [\varphi ]}$  – that is, numbers of the form ${\displaystyle a+b\varphi }$  for ${\displaystyle a,b\in \mathbb {Z} }$  – 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 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 ${\displaystyle 4\log(\varphi ).}$ ^[63]

The golden ratio appears in the theory of modular functions as well. For ${\displaystyle \left|q\right|<1}$ , let

${\displaystyle R(q)={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}.}$ 

Then

${\displaystyle R(e^{-2\pi })={\sqrt {\varphi {\sqrt {5}}}}-\varphi ,\quad R(-e^{-\pi })=\varphi ^{-1}-{\sqrt {2-\varphi ^{-1}}}}$ 

and

${\displaystyle R(e^{-2\pi i/\tau })={\frac {1-\varphi R(e^{2\pi i\tau })}{\varphi +R(e^{2\pi i\tau })}}}$ 

where ${\displaystyle \op$

January 10, 2023