Golden Mean (Golden Section, Golden Ratio)

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

 \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,

where the Greek letter phi (φ) represents the golden ratio. Its value is:

\varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.

The golden ratio is also called the golden section (Latin: sectio aurea) or golden mean.[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number.[6][7][8]

Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9]

Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship.

Calculation

Binary 1.1001111000110111011…
Decimal 1.6180339887498948482…
Hexadecimal 1.9E3779B97F4A7C15F39…
Continued fraction 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}
Algebraic form \frac{1 + \sqrt{5}}{2}
Infinite series \frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}

 

Two quantities a and b are said to be in the golden ratio φ if

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

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

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

Therefore,

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

Multiplying by φ gives

\varphi + 1 = \varphi^2

which can be rearranged to

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

Using the quadratic formula, two solutions are obtained:

\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots

and

\varphi = \frac{1 - \sqrt{5}}{2} = -0.6180\,339887\dots

Because φ is the ratio between positive quantities φ is necessarily positive:

\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots .

History

Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the reciprocal of the golden ratio, 1/φ.[10]

Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597

The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. 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. But the fascination with the Golden Ratio is not confined just to mathematicians. 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]

Ancient Greek mathematicians first studied what we now call 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. Euclid‘s Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: “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.”[12] Euclid explains a construction for cutting (sectioning) a line “in extreme and mean ratio”, i.e., the golden ratio.[13] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[14]

The golden ratio is explored in Luca Pacioli‘s book De divina proportione of 1509.

The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as “about 0.6180340”, was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.[15]

Since the 20th century, the golden ratio has been represented by the Greek letter φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).[1][16]

Timeline

Timeline according to Priya Hemenway:[17]

  • Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
  • Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), some of which are related to the golden ratio.[18]
  • Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, “extreme and mean ratio” (Greek: ἄκρος καὶ μέσος λόγος).[4]
  • Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.
  • Luca Pacioli (1445–1517) defines the golden ratio as the “divine proportion” in his Divina Proportione.
  • Michael Maestlin (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a decimal fraction.
  • Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,[19] and describes the golden ratio as a “precious jewel”: “Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.” These two treasures are combined in the Kepler triangle.
  • Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
  • Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.[20]
  • Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
  • Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias’s name, as a symbol for the golden ratio.[21]
  • Roger Penrose (b. 1931) discovered in 1974 the Penrose tiling, a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[22] This in turn led to new discoveries about quasicrystals.[23]
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