This article is about the number. For the pop music album, see The Golden Ratio (album). For calendar dates, see Golden number (time).
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,
where the Greek letter phi () represents the golden ratio. Its value is:
The golden ratio also is called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio,medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
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 may 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.
Two quantities a and b are said to be in the golden ratio φ if
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,
Multiplying by φ gives
which can be rearranged to
Using the quadratic formula, two solutions are obtained:
Because φ is the ratio between positive quantities φ is necessarily positive:
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.
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." 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.
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.
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).
Text from this biography licensed under creative commons license