Golden ratio

Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes; Golden numbers, an indicator of years in astronomy and calendar studies; or the Golden Rule.

The golden section is a line segment sectioned into two according to the golden ratio. The total length a + b is to the longer segment a as a is to the shorter segment b.

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

At least since the Renaissance, many artists and architects 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. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio is often denoted by the Greek letter ϕ (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has as its unique positive solution the algebraic irrational number

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi (ϕ).^[2][3][4] Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias.

Construction of a golden rectangle:
1. Construct a unit square (red).
2. Draw a line from the midpoint of one side to an opposite corner.
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

Calculation

List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
Binary	1.1001111000110111011…
Decimal	1.6180339887498948482…
Hexadecimal	1.9E3779B97F4A7C15F39…
Continued fraction
Algebraic form

Two quantities (positive numbers) a and b are said to be in the golden ratio ϕ if:

This equation unambiguously defines ϕ.

The right equation shows that a = bϕ, which can be substituted in the left part, giving

Cancelling b yields

Multiplying both sides by ϕ and rearranging terms leads to:

The only positive solution to this quadratic equation is

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.

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

The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa (Fibonacci) 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.

—Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number

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. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

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 less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number.

The name "extreme and mean ratio" was the principal term used from the 3rd century BC. until about the 18th century.

The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

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

Since the twentieth 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).