The golden ratio, or phi, is the unique ratio in which the ratio of the larger portion is equal to the ratio of the smaller portion. The golden ratio is another irrational number. It is usually rounded to 1.618. The golden ratio has been used since ancient time in architecture of buildings.

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagram, decagon and dodecahedron. It is denoted abbreviation of the Greek “tome,” meaning “to cut”. It is denoted by $\phi$ or sometimes $\tau$ (which is an abbreviation of the Greek “tome”, meaning “to cut”.)
The term “golden section” (goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley’s 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol (“phi”, $\phi$) was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6).

[Note] However the elegance of $\phi$ could have been appreciated before it gained the name that we are familiar with. In other cultures, there could have been a different way of expressing the concept. Diameter and circumference are two concepts that were familiar to the Greeks, and from that it would be a small step to $\phi$

$\phi$ has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers. It is also a so-called Pisot Number.
![[img/sg_img-038.png]]

Given a rectangle having sides in the ratio, is defined such that partitioning the origin rectangle into a square and new rectangle results in a new rectangle having sides with a golden ratio. Such a rectangle is called a golden rectangle, and successive points dividing a golden rectangle into squares lie on a logarithmic spiral. This figure is known as a whirling square. The legs of a golden triangle (an isosceles triangle with a vertex angle of $\phi$) are in a golden ratio to its base and, in fact, this was the method used by Pythagoras to construct $\phi$.
The ratio of the circumradius to the length of the side of a decagon is also

$\frac{R}{s} = \frac{1}{2}\csc (\frac{\pi}{10}) = \frac{1}{2}(1 + \sqrt{ 5}) = \phi$

Bisecting a (schematic) Gaullist cross also gives a golden ratio (Gardner 1961, p. 102).

![[img/sg_img-040.png]]

Euclid ca. 300 BC defined the “extreme and mean ratios” on a line segment as the lengths such that

$\phi = \frac{AC}{CB} = \frac{AB}{AC}$

(Livio 2002, pp. 3-4). Plugging in

$\frac{\phi +1}{\phi} = \phi$

and clearing denominators gives

$\phi_{2} - \phi -1 = 0$

(Incidentally, this means that is a algebraic number of degree 2.) So, using the quadratic equation and taking the positive sign (since the figure is defined so that $\phi > 1$),

\begin{array}{rc1} \phi = \frac{1}{2}(1 + \sqrt{5}) \\ = 1.6180339849894848204586834365638117720\dots \end{array}

$$

(Sloane’s A001622).

Exact trigonometric formulas for $\phi$ include

\begin{array}{rc1} \phi = 2 \cos (\frac{\pi}{5}) \\ = \frac{1}{2} \sec(\frac{2\pi}{5}) \\ = \frac{1}{2} \csc(\frac{\pi}{10}) \end{array}

The golden ratio is given by the infinite series

$\phi = \frac{13}{8} + \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \text{ }(2n + 1)!} {(n+2)! \text{ } n! \text{ } 4^{2n+3}}$

(B. Roselle). Another fascinating connection with the Fibonacci numbers is given by the infinite series

$\phi = 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{F_{n} \text{ }F_{n+1}}$

A representation in terms of a nested radical is

$\phi = \sqrt{ 1 + {\sqrt{ 1 + {\sqrt{ 1 + \sqrt{ 1 + \dots }}}}}}$

(Livio 2002, p. 83).

$\phi$ is the “most” irrational number because it has a continued fraction representation

$\phi = [1,1,1, \dots] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1+ \dots}}}$

(Sloane’s A000012; Williams 1979, p. 52; Steinhaus 1999, p. 45; Livio 2002, p. 84). This means that the convergents are given by the quadratic recurrence equation $$ x_{n} = 1 + \frac{1}{x_{n-1}} $$ with $x_1=1$, which has solution

$x_n = \frac{F_{n+1}}{F_{n}}$

where $F_{}2$ is the $n^th$ Fibonacci number. As a result,

$\phi = \lim_{ n \to \infty }x_{n} \lim_{ n \to \infty } \begin{array}{cc} F_{n} \\ F_{n-1} \end{array} ,$

as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62; Livio 2002, p. 101).

The golden ratio has Engel expansion 1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, … (Sloane’s A028259).

The golden ratio also satisfies the recurrence relation

$\phi^n = \phi^{n-1} + \phi^{n-2}$

Taking $n =1$ gives the special case

$\phi = \phi^{-1} + 1$

Treating ($\phi$) as a linear recurrence equation

$\phi(n) = \phi(n-1) + \phi(n-2)$

in $\phi(n) = \phi^n$, setting $\phi(0) = 1$ and $\phi(1) = \phi$, and solving gives $$\phi(n) = \phi^n$$as expected. The powers of the golden ratio also satisfy

$\phi(n) = F_{n}\phi + F_{n-1}$

where is a Fibonacci number (Wells 1986, p. 39).

The sine of certain complex numbers involving gives particularly simple answers, for example:

$\begin{array}{lc} \sin(i \ln \phi) & = &\frac{1}{2}i \\ \sin(\frac{1}{2}\pi - i \ln \phi) & = &\frac{1}{2} \sqrt{ 5 } \end{array}$

(D. Hoey, pers. comm.).

![[img/sg_img-076.png]]

In the figure above, three triangles can be inscribed in the rectangle $ABCD$ of arbitrary aspect ratio $1:r$ such that the three right triangles have equal areas by dividing AB and BC in the golden ratio. Then

$\begin{array}{lc} K_{ \triangle ADE } = \frac{1}{2} \cdot r(1+\phi) \cdot 1 = \frac{1}{2}r \phi^2 \\ K_{\triangle BEF} = \frac{1}{2} \cdot r \phi \cdot \phi = \frac{1}{2} r \phi^2 \\ K_{\triangle CDF} = \frac{1}{2}(1 + \phi)r = \frac{1}{2} r \phi^2 \end{array}$

which are all equal.

The substitution map

$\begin{array}{cc} 0 \to 01 \\ 1 \to 0 \end{array}$

gives $$ 0 \to 01 \to 010 \to 01001 \to \dots,$$giving rise to the sequence $$0100101001001010010100100101$$(Sloane’s A003849). Here, the zeros occur at positions 1, 3, 4, 6, 8, 9, 11, 12, … (Sloane’s A000201), and the ones occur at positions 2, 5, 7, 10, 13, 15, 18, … (Sloane’s A001950). The sequence also has many connections with the Fibonacci numbers.

![[img/sg_img-087.png|600]]
Steinhaus (1983, pp. 48-49) considers the distribution of the fractional parts of $\phi$ of $?$ in the intervals bounded by $0, 1 /h, 2/h, …, (h-1)/h, 1, and notes that they are much more
uniformly distributed than would be expected due to chance (i.e., is close to an
equidistributed sequence). In particular, the number of empty intervals for h=1, 2, …, are
mere 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, … (Sloane’s A036414). The values of for
which no bins are left blank are then given by 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89,
144, … (Sloane’s A036415). Steinhaus (1983) remarks that the highly uniform distribution
has its roots in the continued fraction for $\phi$.

The sequence ${frac (x^n)}$, of power fractional parts, where is the fractional part, is
equidistributed for almost all real numbers $x>1$, with the golden ratio being one exception.

Salem showed that the set of Pisot numbers is closed, with the smallest accumulation point of the set (Le Lionnais 1983).

SEE ALSO: Beraha Constants, Decagon, Equidistributed Sequence, Euclidean Algorithm, Five Disks Problem, Golden Angle, Golden Gnomon, Golden Ratio Conjugate, Golden Rectangle, Golden Triangle, Icosidodecahedron, Noble Number, Pentagon, Pentagram, Phi Number System, Phyllotaxis, Pisot Number, Power Fractional Parts, Ramanujan Continued Fractions, Rogers-Ramanujan Continued Fraction, Secant Method.

According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart. (Modern music theory calls a 5:4 ratio a “major third” and an 8:5 ratio a “major sixth”.) In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers.

Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.

The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.

![[img/sg_img-089.png]]
The Greeks knew it as the Golden Section and used it for beauty and balance in the design of architecture. They based the entire design of the Parthenon on this proportion.

Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied phi and
applied it to the design of sculptures for the Parthenon.
![[img/sg_img-090.png|Porch of Maidens, Acropolis, Athens]]

Euclid proved that the diagonals of the regular pentagon cut each other in “extreme and mean ratio”, now more commonly known as the golden ratio. Here we represent the golden ratio by phi. $F_n$ is the $n^{th}$ Fibonacci number.
![[img/sg_img-091.png]]

\begin{array} 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1+ \dots}}} & \phi = \displaystyle\lim_{ n \to \infty } \frac{F_{n+1}}{F_{n}} \\ \phi = \sqrt{ 1 + {\sqrt{ 1 + {\sqrt{ 1 + \sqrt{ 1 + \dots }}}}}} & \phi^2 = \phi + 1 \end{array}