In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a {\displaystyle a} and b {\displaystyle b} with a > b > 0 {\displaystyle a>b>0} , a {\displaystyle a} is in a golden ratio to b {\displaystyle b} if a + b a = a b = φ , {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi ,} where the Greek letter phi ( φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } , or Φ {\displaystyle \Phi } ) denotes the golden ratio. The constant φ {\displaystyle \varphi } satisfies the quadratic equation φ 2 = φ + 1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} and is an irrational number with a value of
φ = 1 + 5 2 =
{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}={}} 1.618033988749....
The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of φ {\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. 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 non-zero quantities a {\displaystyle a} and b {\displaystyle b} are in the golden ratio φ {\displaystyle \varphi } if
To determine φ {\displaystyle \varphi } as a number, we can divide the numerator and denominator of the fraction on the left-hand side by b {\displaystyle b} ,
and then substitute a b = φ {\displaystyle {\tfrac {a}{b}}=\varphi } , to obtain
Multiplying both sides by φ {\displaystyle \varphi } gives
which can be rearranged to
The quadratic formula yields two solutions:
The positive root, φ {\displaystyle \varphi } , is the golden ratio. The negative root is its negative inverse − 1 / φ {\displaystyle -1/\varphi } , with which it shares many properties.
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. Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:
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.
The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers. Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.
German mathematician Simon Jacob (d. 1564) noted that ratios of consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about 0.6180340 {\displaystyle 0.6180340} " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these: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. Eighteenth-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". In 1789, Johann Samuel Traugott Gehler made the earliest known use of the term 'golden section’ in his popular dictionary of physical sciences, Physikalisches Wörterbuch, referring to it as ‘güldnen Schnitt (media et extrema ratione, sectione aurea s[ive] divina)’. James Sully used the equivalent English term in 1875. By 1910, inventor Mark Barr began using the Greek letter phi ( φ {\displaystyle \varphi } ) as a symbol for the golden ratio. It has also been represented by tau ( τ {\displaystyle \tau } ), the first letter of the ancient Greek τομή ('cut' or 'section').
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. This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling.
Mathematics
Irrationality The golden ratio is an irrational number. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms
This is a proof by infinite descent. Recall that:
If we call the whole n {\displaystyle n} and the longer part m {\displaystyle m} , then the second statement above becomes
To say that the golden ratio φ {\displaystyle \varphi } is rational means that φ {\displaystyle \varphi } is a fraction n / m {\displaystyle n/m} where n {\displaystyle n} and m {\displaystyle m} are integers. We may take n / m {\displaystyle n/m} to be in lowest terms and n {\displaystyle n} and m {\displaystyle m} to be positive. But if n / m {\displaystyle n/m} is in lowest terms, then the equally valued m / ( n − m ) {\displaystyle m/(n-m)} is in still lower terms. That is a contradiction that follows from the assumption that φ {\displaystyle \varphi } is rational.
By irrationality of the square root 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 φ = 1 2 ( 1 + 5 ) {\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}} is assumed to be rational, then 2 φ − 1 = 5 {\displaystyle 2\varphi -1={\sqrt {5}}} , the square root of 5 {\displaystyle 5} , must also be rational. This is a contradiction, as the square roots of all non-square natural numbers are irrational.
Minimal polynomial
Since the golden ratio is a root of a polynomial with rational coefficients, it is an algebraic number. Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is x 2 − x − 1. {\displaystyle x^{2}-x-1.} This quadratic polynomial has two roots, φ {\displaystyle \varphi } and − φ − 1 {\displaystyle \textstyle -\varphi ^{-1}} . Because the leading coefficient of this polynomial is 1, both roots are algebraic integers. The golden ratio is also closely related to the polynomial x 2 + x − 1 {\displaystyle \textstyle x^{2}+x-1} , which has roots − φ {\displaystyle -\varphi } and φ − 1 {\displaystyle \textstyle \varphi ^{-1}} . The golden ratio φ {\displaystyle \varphi } is a fundamental unit of the quadratic field Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} , sometimes called the golden field. In this field, any element can be written in the form r + s φ {\displaystyle r+s\varphi } , with rational coefficients r {\displaystyle r} and s {\displaystyle s} ; such a number has norm r 2 + r s − s 2 {\displaystyle \textstyle r^{2}+rs-s^{2}} . Other units, with norm ± 1 {\displaystyle \pm 1} , are the positive and negative powers of φ {\displaystyle \varphi } . The quadratic integers in this field, which form a ring, are all numbers of the form a + b φ {\displaystyle a+b\varphi } where a {\displaystyle a} and b {\displaystyle b} are integers. As the root of a quadratic polynomial, the golden ratio is a constructible number.
Golden ratio conjugate and powers The conjugate root to the minimal polynomial x 2 − x − 1 {\displaystyle \textstyle x^{2}-x-1} is
− 1 φ = 1 − φ = 1 − 5 2 = − 0.618033 … . {\displaystyle -{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .}
The absolute value of this quantity ( 0.618 … {\displaystyle 0.618\ldots } ) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b / a {\displaystyle b/a} ). This illustrates the unique property of the golden ratio among positive numbers, that
1 φ = φ − 1 , {\displaystyle {\frac {1}{\varphi }}=\varphi -1,}
or its inverse,
1 1 / φ = 1 φ + 1. {\displaystyle {\frac {1}{1/\varphi }}={\frac {1}{\varphi }}+1.}
The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with φ {\displaystyle \varphi } :
φ 2 = φ + 1 = 2.618033 … , 1 φ = φ − 1 = 0.618033 … . {\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 φ {\displaystyle \varphi } contains these values 0.618033 … {\displaystyle 0.618033\ldots } , 1.0 {\displaystyle 1.0} , 1.618033 … {\displaystyle 1.618033\ldots } , 2.618033 … {\displaystyle 2.618033\ldots } ; more generally, any power of φ {\displaystyle \varphi } is equal to the sum of the two immediately preceding powers:
φ n = φ n − 1 + φ n − 2 = φ ⋅ F n + F n − 1 . {\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 φ {\displaystyle \varphi } into a multiple of φ {\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 φ {\displaystyle \varphi } : If ⌊ 1 2 n − 1 ⌋ = m {\displaystyle {\bigl \lfloor }{\tfrac {1}{2}}n-1{\bigr \rfloor }=m} , then:
φ n = φ n − 1 + φ n − 3 + ⋯ + φ n − 1 − 2 m + φ n − 2 − 2 m φ n − φ n − 1 = φ n − 2 . {\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
The formula φ = 1 + 1 / φ {\displaystyle \varphi =1+1/\varphi } can be expanded recursively to obtain a simple continued fraction for the golden ratio:
φ = [ 1 ; 1 , 1 , 1 , … ] = 1 + 1 1 + 1 1 + 1 1 + 1 ⋱ {\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}
It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
φ − 1 = [ 0 ; 1 , 1 , 1 , … ] = 0 + 1 1 + 1 1 + 1 1 + 1 ⋱ {\displaystyle \varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}
The convergents of these continued fractions, 1 1 {\displaystyle {\tfrac {1}{1}}} , 2 1 {\displaystyle {\tfrac {2}{1}}} , 3 2 {\displaystyle {\tfrac {3}{2}}} , 5 3 {\displaystyle {\tfrac {5}{3}}} , 8 5 {\displaystyle {\tfrac {8}{5}}} , 13 8 {\displaystyle {\tfrac {13}{8}}} , ... or 1 1 {\displaystyle {\tfrac {1}{1}}} , 1 2 {\displaystyle {\tfrac {1}{2}}} , 2 3 {\displaystyle {\tfrac {2}{3}}} , 3 5 {\displaystyle {\tfrac {3}{5}}} , 5 8 {\displaystyle {\tfrac {5}{8}}} , 8 13 {\displaystyle {\tfrac {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 ξ {\displaystyle \xi } , there are infinitely many distinct fractions p / q {\displaystyle p/q} such that,
| ξ − p q | < 1 5 q 2 . {\displaystyle \left|\xi -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.}
This means that the constant 5 {\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. A continued square root form for φ {\displaystyle \varphi } can be obtained from φ 2 = 1 + φ {\displaystyle \textstyle \varphi ^{2}=1+\varphi } , yielding:
φ = 1 + 1 + 1 + ⋯ ) . {\displaystyle \varphi ={\sqrt {1+{\sqrt {\textstyle 1+{\sqrt {1+\cdots {\vphantom {)}}}}}}}}.}
Relationship to Fibonacci and Lucas numbers
Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term F n {\displaystyle F_{n}} is equal to the sum of the preceding two terms F n − 1 {\displaystyle F_{n-1}} and F n − 2 {\displaystyle F_{n-2}} , starting with the base sequence 0 , 1 {\displaystyle 0,1} as the 0th and 1st terms F 0 {\displaystyle F_{0}} and F 1 {\displaystyle F_{1}} :
The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in that each term L n {\displaystyle L_{n}} is the sum of the previous two terms L n − 1 {\displaystyle L_{n-1}} and L n − 2 {\displaystyle L_{n-2}} , however instead starts with 2 , 1 {\displaystyle 2,1} as the 0th and 1st terms L 0 {\displaystyle L_{0}} and L 1 {\displaystyle L_{1}} :
Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:
lim n → ∞ F n + 1 F n = lim n → ∞ L n + 1 L n = φ . {\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 φ {\displaystyle \varphi } . For example,
These approximations are alternately lower and higher than φ {\displaystyle \varphi } , and converge to φ {\displaystyle \varphi } as the Fibonacci and Lucas numbers increase. Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:
F ( n ) = φ n − ( − φ ) − n 5 = φ n − ( 1 − φ ) n 5 = 1 5 [ ( 1 + 5 2 ) n − ( 1 − 5 2 ) n ] , {\displaystyle F\left(n\right)={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}={\frac {1}{\sqrt {5}}}\left[\left({1+{\sqrt {5}} \over 2}\right)^{n}-\left({1-{\sqrt {5}} \over 2}\right)^{n}\right],}
L ( n ) = φ n + ( − φ ) − n = φ n + ( 1 − φ ) n = ( 1 + 5 2 ) n + ( 1 − 5