Brave New World

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Brave New World

First edition
Author	Aldous Huxley
Cover artist	Leslie Holland
Genre	Science fiction, dystopian fiction
Publisher	Chatto & Windus
Publication date	4 February 1932
Publication place	United Kingdom
Pages	311 (1932 ed.) 63,766 words
Awards	Le Monde's 100 Books of the Century
OCLC	20156268
Text	Brave New World online

Brave New World is a dystopian novel by English author Aldous Huxley, written in 1931, and published in 1932. Largely set in a futuristic World State, whose citizens are environmentally engineered into an intelligence-based social hierarchy, the novel anticipates huge scientific advancements in reproductive technology, sleep-learning, psychological manipulation and classical conditioning that are combined to make a dystopian society which is challenged by the story's protagonist. Huxley followed this book with a reassessment in essay form, Brave New World Revisited (1958), and with his final novel, Island (1962), the utopian counterpart. This novel is often used as a companion piece, or inversion counterpart to George Orwell's Nineteen Eighty-Four (1949).

In 1998 and 1999, the Modern Library ranked Brave New World at number 5 on its list of the 100 Best Novels in English of the 20th century. In 2003, Robert McCrum, writing for The Observer, included Brave New World chronologically at number 53 in "the top 100 greatest novels of all time", and the novel was listed at number 87 on The Big Read survey by the BBC. Brave New World has frequently been banned and challenged since its original publication. It has landed on the American Library Association list of top 100 banned and challenged books of the decade since the association began the list in 1990.

Title

The title Brave New World derives from William Shakespeare's The Tempest, Act V, Scene I, Miranda's speech:

O wonder!
How many goodly creatures are there here!
How beauteous mankind is! O brave new world,
That has such people in 't.

— William Shakespeare, The Tempest, Act V, Scene I, ll. 203–206

Shakespeare's use of the phrase is intended ironically, as the speaker is failing to recognise the evil nature of the island's visitors because of her innocence. Indeed, the next speaker—Miranda's father Prospero—replies to her innocent observation with the statement "'Tis new to thee".

Translations of the title often allude to similar expressions used in domestic works of literature: the French edition of the work is entitled Le Meilleur des mondes (The Best of All Worlds), an allusion to an expression used by the philosopher Gottfried Leibniz and satirised in Candide, Ou l'Optimisme by Voltaire (1759). The first Standard Chinese translation, done by novelist Lily Hsueh and Aaron Jen-wang Hsueh in 1974, is entitled "美麗新世界" (Pinyin: Měilì Xīn Shìjiè, literally "Beautiful New World").

History

Huxley wrote Brave New World while living in Sanary-sur-Mer, France, in the four months from May to August 1931. By this time, Huxley had established himself as a writer and social satirist. He was a contributor to Vanity Fair and Vogue magazines and had published a collection of his poetry (The Burning Wheel, 1916) and four satirical novels, Crome Yellow (1921), Antic Hay (1923), Those Barren Leaves (1925) and Point Counter Point (1928). Brave New World was Huxley's fifth novel and first dystopian work.

A short passage in Crome Yellow foreshadows Brave New World, showing that Huxley had such a future in mind already in 1921. Mr Scogan, one of the earlier book's characters, describes an "impersonal generation" of the future that will "take the place of Nature's hideous system. In vast state incubators, rows upon rows of gravid bottles will supply the world with the population it requires. The family system will disappear; society, sapped at its very base, will have to find new foundations; and Eros, beautifully and irresponsibly free, will flit like a gay butterfly from flower to flower through a sunlit world".

Huxley said that Brave New World was inspired by the utopian novels of H. G. Wells, including A Modern Utopia (1905), and as a parody of Men Like Gods (1923). Wells' hopeful vision of the future gave Huxley the idea to begin writing a parody of the novels, which became Brave New World. He wrote in a letter to Mrs. Arthur Goldsmith, an American acquaintance, that he had "been having a little fun pulling the leg of H. G. Wells" but then he "got caught up in the excitement of [his] own ideas". Unlike the most popular optimistic utopian novels of the time, Huxley sought to provide a frightening vision of the future. Huxley referred to Brave New World as a "negative utopia", somewhat influenced by Wells's own The Sleeper Awakes (dealing with subjects like corporate tyranny and behavioural conditioning) and the works of D. H. Lawrence.

For his part, Wells published, two years after Brave New World, his utopian Shape of Things to Come. Seeking to rebut the argument of Huxley's Mustapha Mond—that moronic underclasses were a necessary "social gyroscope" and that a society composed solely of intelligent, assertive "Alphas" would inevitably disintegrate in internecine struggle—Wells depicted a stable egalitarian society emerging after several generations of a reforming elite having complete control of education throughout the world. In the future depicted in Wells's book, posterity remembers Huxley as "a reactionary writer". The scientific futurism in Brave New World is believed to be appropriated from Daedalus by J. B. S. Haldane.

The events of the Great Depression in Great Britain in 1931, with its mass unemployment and the abandonment of the gold standard, persuaded Huxley to assert that stability was the "primal and ultimate need" if civilisation was to survive the present crisis. The Brave New World character Mustapha Mond, Resident World Controller of Western Europe, is named after Sir Alfred Mond. Shortly before writing the novel, Huxley visited the Billingham Manufacturing Plant, Mond's technologically advanced factory near Billingham, north-east England, and it made a great impression on him.

Huxley used the setting and characters in his science fiction novel to express widely felt anxieties, particularly the fear of losing individual identity in the fast-paced world of the future. An early trip to the United States gave Brave New World much of its character. Huxley was outraged by the culture of youth, commercial cheeriness, sexual promiscuity and the inward-looking nature of many Americans; he had also found the book My Life and Work by Henry Ford on the boat to North America and he saw the book's principles applied in everything he encountered after leaving San Francisco.

Plot

The novel opens in the World State city of London in AF (After Ford) 632 (AD 2540 in the Gregorian calendar), where citizens are engineered through artificial wombs and childhood indoctrination programmes into predetermined classes (or castes) based on intelligence and labour. Embryos in different bottles are treated with chemicals to suit them for their planned roles; those for the higher classes get chemicals to optimise them, and those of the lower classes are made increasingly imperfect. The classes are Alpha (planned leaders), Beta, Gamma, Delta, and Epsilon (menial labourers of limited intelligence). Each caste is indoctrinated to prefer their own class—epsilons are happy that they do not have the intellectual burden of alphas—and wears a uniform colour of clothing for easy identification.

Lenina Crowne, a hatchery worker, is popular and sexually desirable, but Bernard Marx, a psychologist, is not. He is shorter in stature than the average member of his high alpha caste, which gives him an inferiority complex. His work with sleep-learning allows him to understand, and disapprove of, his society's methods of keeping its citizens peaceful, which includes their constant consumption of a soothing, happiness-producing drug called "soma." Courting disaster, Bernard is vocal and arrogant about his criticisms, and his boss contemplates exiling him to Iceland because of his nonconformity. His only friend is Helmholtz Watson, a gifted writer who finds it difficult to use his talents creatively in their pain-free society. Bernard takes a holiday with Lenina outside the World State to a "Savage Reservation" in New Mexico, in which the two observe natural-born people, disease, the ageing process, other languages, and religious lifestyles for the first time. The culture of the village folk resembles the contemporary Native American groups of the region, descendants of the Anasazi, including the Puebloan peoples of Hopi and Zuni. Bernard and Lenina witness a violent public ritual and then encounter Linda, a woman originally from the World State who is living on the reservation with her son John, now a young man. She, too, visited the reservation on a holiday many years ago, but became separated from her group and was left behind. She had meanwhile become pregnant by a fellow holidaymaker (who is revealed to be Bernard's boss, the Director of Hatcheries and Conditioning). She did not try to return to the World State, because of her shame at her pregnancy. Despite spending his whole life in the reservation, John has never been accepted by the villagers, and his and Linda's lives have been hard and unpleasant. Linda has taught John to read, although from the only book in her possession—a scientific manual—and another book found nearby by Popé: the complete works of Shakespeare. Ostracised by the villagers, John is able to articulate his feelings only in terms of Shakespearean drama, quoting often from The Tempest, King Lear, Othello, Romeo and Juliet and Hamlet. Linda now wants to return to London, and John, too, wants to see this "brave new world" that his mother so often praised. Bernard sees an opportunity to thwart plans to exile him, and gets permission to take Linda and John back. On their return to London, John meets the Director and calls him his "father", a vulgarity which causes a roar of laughter. The humiliated Director resigns in shame before he can follow through with exiling Bernard.

Bernard, as "custodian" of the "savage" John who is now treated as a celebrity, is fawned on by the highest members of society and revels in attention he once scorned. Bernard's popularity is fleeting, though, and he becomes envious that John only really bonds with the literary-minded Helmholtz. Considered hideous and friendless, Linda spends all her time using soma, which she craved for so long, while John refuses to attend social events organised by Bernard, appalled by what he perceives to be an empty society. Lenina and John are physically attracted to each other, but John's view of courtship and romance, based on Shakespeare's writings, is utterly incompatible with Lenina's freewheeling attitude to sex. She tries to seduce him, but he attacks her, before suddenly being informed that his mother is on her deathbed. He rushes to Linda's bedside, causing a scandal, as this is not the "correct" attitude to death. Some children who enter the ward for "death-conditioning" come across as disrespectful to John, and he attacks one physically. He then tries to break up a distribution of soma to a lower-caste group, telling them that he is freeing them. Helmholtz and Bernard rush in to stop the ensuing riot, which the police quell by spraying soma vapour into the crowd.

Bernard, Helmholtz, and John are all brought before Mustapha Mond, the "Resident World Controller for Western Europe", who tells Bernard and Helmholtz that they are to be exiled to islands for antisocial activity. Bernard pleads for a second chance, but Helmholtz welcomes the opportunity to be a true individual, and chooses the Falkland Islands as his destination, believing that their bad weather will inspire his writing. Mond tells Helmholtz that exile is actually a reward. The islands are full of the most interesting people in the world, individuals who did not fit into the social model of the World State. Mond outlines for John the events that led to the present society and his arguments for a caste system and social control. John rejects Mond's arguments, and Mond sums up John's views by claiming that John demands "the right to be unhappy". John asks if he may go to the islands as well, but Mond refuses, saying he wishes to see what happens to John next.

Jaded with his new life, John moves to an abandoned hilltop lighthouse, near the village of Puttenham, where he intends to adopt a solitary ascetic lifestyle in order to purify himself of civilisation, practising self-flagellation. This draws reporters and eventually hundreds of amazed sightseers, hoping to witness his bizarre behaviour.

For a while, it seems that John might be left alone, after the public's attention is drawn to other diversions, but a documentary-maker has secretly filmed John's self-flagellation from a distance, and when released, the documentary causes an international sensation. Helicopters arrive with more journalists. Crowds of people descend on John's retreat, demanding that he perform his whipping ritual for them. From one helicopter a young woman emerges who is implied to be Lenina. John, at the sight of a woman he both adores and loathes, whips at her in a fury and then turns the whip on himself, exciting the crowd, whose wild behaviour transforms into a soma-fuelled orgy. The next morning, John awakes on the ground and is consumed by remorse over his participation in the orgy.

That evening, a swarm of helicopters appear on the horizon, with the story of last night's orgy having been in all the newspapers. The first onlookers and reporters to arrive find that John is dead, having hanged himself.

Characters

Bernard Marx, a sleep-learning specialist at the Central London Hatchery and Conditioning Centre. Although Bernard is an Alpha-Plus (the upper class of the society), he is a misfit. He is unusually short for an Alpha; an alleged accident with alcohol in Bernard's blood-surrogate before his decanting has left him slightly stunted. Unlike his fellow utopians, Bernard is often angry, resentful, and jealous. At times, he is also cowardly and hypocritical. His conditioning is clearly incomplete. He does not enjoy communal sports, solidarity services, or promiscuous sex. He does not particularly enjoy soma. Bernard is in love with Lenina and does not like her sleeping with other men, even though "everyone belongs to everyone else". Bernard's triumphant return to utopian civilisation with John the Savage from the Reservation precipitates the downfall of the Director, who had been planning to exile him. Bernard's triumph is short-lived; he is ultimately banished to an island for his non-conformist behaviour.

John, the illicit son of the Director and Linda, born and reared on the Savage Reservation ("Malpais") after Linda was unwittingly left behind by her errant lover. John ("the Savage" or "Mr Savage", as he is often called) is an outsider both on the Reservation—where the natives still practise marriage, natural birth, family life and religion—and the ostensibly civilised World State, based on principles of stability and happiness. He has read nothing but the complete works of William Shakespeare, which he quotes extensively, and, for the most part, aptly, though his allusion to the "Brave New World" (Miranda's words in The Tempest) takes on a darker and bitterly ironic resonance as the novel unfolds. John is intensely moral according to a code that he has been taught by Shakespeare and life in Malpais but is also naïve: his views are as imported into his own consciousness as are the hypnopedic messages of World State citizens. The admonishments of the men of Malpais taught him to regard his mother as a whore; but he cannot grasp that these were the same men who continually sought her out despite their supposedly sacred pledges of monogamy. Because he is unwanted in Malpais, he accepts the invitation to travel back to London and is initially astonished by the comforts of the World State. He remains committed to values that exist only in his poetry. He first spurns Lenina for failing to live up to his Shakespearean ideal and then the entire utopian society: he asserts that its technological wonders and consumerism are poor substitutes for individual freedom, human dignity and personal integrity. After his mother's death, he becomes deeply distressed with grief, surprising onlookers in the hospital. He then withdraws himself from society and attempts to purify himself of "sin" (desire), but is unable to do so. His unusual behaviour eventually attracts the attention of reporters and, later, huge amounts of people, who arrive in helicopters and make John furious with their behaviour. Excited by his fury, people start an orgy, which he cannot resist joining. After waking up the next morning, John is horrified by his actions and hangs himself.

Helmholtz Watson, a handsome and successful Alpha-Plus lecturer at the College of Emotional Engineering and a friend of Bernard. He feels unfulfilled writing endless propaganda doggerel, and the stifling conformism and philistinism of the World State make him restive. Helmholtz is ultimately exiled to the Falkland Islands—a cold asylum for disaffected Alpha-Plus non-conformists—after reading a heretical poem to his students on the virtues of solitude and helping John destroy some Deltas' rations of soma following Linda's death. Unlike Bernard, he takes his exile in his stride and comes to view it as an opportunity for inspiration in his writing. His first name derives from the German physicist Hermann von Helmholtz.

Lenina Crowne, a young, beautiful foetus technician at the Central London Hatchery and Conditioning Centre. Lenina Crowne is a Beta who enjoys being a Beta. She is a vaccination worker with beliefs and values that are in line with a citizen of the World State. She is part of the 30% of the female population that are not freemartins (sterile women). Lenina is promiscuous and popular but somewhat quirky in her society: she had a four-month relation with Henry Foster, choosing not to have sex with anyone but him for a period of time. She is basically happy and well-conditioned, using soma to suppress unwelcome emotions, as is expected. Lenina has a date with Bernard, to whom she feels ambivalently attracted, and she goes to the Reservation with him. On returning to civilisation, she tries and fails to seduce John the Savage. John loves and desires Lenina but he is repelled by her forwardness and the prospect of pre-marital sex, rejecting her as an "impudent strumpet". Lenina visits John at the lighthouse but he attacks her with a whip, unwittingly inciting onlookers to do the same. Her exact fate is left unspecified.

Mustapha Mond, Resident World Controller of Western Europe, "His Fordship" Mustapha Mond presides over one of the ten zones of the World State, the global government set up after the cataclysmic Nine Years' War and great Economic Collapse. Sophisticated and good-natured, Mond is an urbane and hyperintelligent advocate of the World State and its ethos of "Community, Identity, Stability". Among the novel's characters, he is uniquely aware of the precise nature of the society he oversees and what it has given up to accomplish its gains. Mond argues that art, literature, and scientific freedom must be sacrificed to secure the ultimate utilitarian goal of maximising societal happiness. He defends the caste system, behavioural conditioning, and the lack of personal freedom in the World State: these, he says, are a price worth paying for achieving social stability, the highest social virtue because it leads to lasting happiness.

Fanny Crowne, Lenina Crowne's friend (they have the same last name because only ten thousand last names are in use in a World State comprising two billion people). Fanny voices the conventional values of her caste and society, particularly the importance of promiscuity: she advises Lenina that she should have more than one man in her life because it is unseemly to concentrate on just one. Fanny then warns Lenina away from a new lover whom she considers undeserving, yet she is ultimately supportive of the young woman's attraction to the savage John.

Henry Foster, one of Lenina's many lovers, is a perfectly conventional Alpha male, casually discussing Lenina's body with his coworkers. His success with Lenina, and his casual attitude about it, infuriate the jealous Bernard. Henry ultimately proves himself every bit the ideal World State citizen, finding no courage to defend Lenina from John's assaults despite having maintained an uncommonly longstanding sexual relationship with her.

Benito Hoover, another of Lenina's lovers. She remembers that he is particularly hairy when he takes his clothes off.

The Director of Hatcheries and Conditioning (DHC), also known as Thomas "Tomakin", is the administrator of the Central London Hatchery and Conditioning Centre, where he is a threatening figure who intends to exile Bernard to Iceland. His plans take an unexpected turn when Bernard returns from the Reservation with Linda (see below) and John, a child they both realise is actually his. This fact, scandalous and obscene in the World State, not because it was extramarital (which all sexual acts are), but because it was procreative, leads the Director to resign his post in shame.

Linda, John's mother, decanted as a Beta-Minus in the World State, originally worked in the DHC's Fertilizing Room, and subsequently lost during a storm while visiting the New Mexico Savage Reservation with the Director many years before the events of the novel. Despite following her usual precautions, Linda became pregnant with the Director's son during their time together and was therefore unable to return to the World State by the time that she found her way to Malpais. Having been conditioned to the promiscuous social norms of the World State, Linda finds herself at once popular with every man in the pueblo (because she is open to all sexual advances) and also reviled for the same reason, seen as a whore by the wives of the men who visit her and by the men themselves (who come to her nonetheless). Her only comforts there are mescal brought by Popé as well as peyotl. Linda is desperate to return to the World State and to soma, wanting nothing more from her remaining life than comfort until death.

The Arch-Community-Songster, the secular equivalent of the Archbishop of Canterbury in the World State society. He takes personal offense when John refuses to attend Bernard's party.

The Director of Crematoria and Phosphorus Reclamation, one of the many disappointed, important figures to attend Bernard's party.

The Warden, an Alpha-Minus, the talkative chief administrator for the New Mexico Savage Reservation. He is blond, short, broad-shouldered, and has a booming voice.

Darwin Bonaparte, a "big game photographer" (i.e., filmmaker) who films John flogging himself. Darwin Bonaparte became known for two works: "feely of the gorillas' wedding", and "Sperm Whale's Love-life". He had already made a name for himself but still seeks more. He renews his fame by filming the savage, John, in his newest release "The Savage of Surrey". His name alludes to Charles Darwin and Napoleon Bonaparte.

Dr. Shaw, Bernard Marx's physician who consequently becomes the physician of both Linda and John. He prescribes a lethal dose of soma to Linda, which will stop her respiratory system from functioning in a span of one to two months, at her own behest but not without protest from John. Ultimately, they all agree that it is for the best, since denying her this request would cause more trouble for Society and Linda herself.

Dr. Gaffney, Provost of Eton, an Upper School for high-caste individuals. He shows Bernard and John around the classrooms, and the Hypnopaedic Control Room (used for behavioural conditioning through sleep learning). John asks if the students read Shakespeare but the Provost says the library contains only reference books because solitary activities, such as reading, are discouraged.

Miss Keate, Head Mistress of Eton Upper School. Bernard fancies her, and arranges an assignation with her.

Others

• Freemartins, women who have been deliberately made sterile by exposure to male hormones during foetal development but are still physically normal except for "the slightest tendency to grow beards". In the book, government policy requires freemartins to form 70% of the female population.

Of Malpais

• Popé, a native of Malpais. Although he reinforces the behaviour that causes hatred for Linda in Malpais by sleeping with her and bringing her mescal, he still holds the traditional beliefs of his tribe. In his early years John attempted to kill him, but Popé brushed off his attempt and sent him fleeing. He gave Linda a copy of the Complete Works of Shakespeare. (Historically, Popé or Po'pay was a Tewa religious leader who led the Pueblo Revolt in 1680 against Spanish colonial rule.)
• Mitsima, an elder tribal shaman who also teaches John survival skills such as rudimentary ceramics (specifically coil pots, which were traditional to Native American tribes) and bow-making.
• Kiakimé, a native girl whom John fell for, but is instead eventually wed to another boy from Malpais.
• Kothlu, a native boy with whom Kiakimé is wed.

Background figures

These are non-fictional and factual characters who lived before the events in this book, but are of note in the novel:

• Henry Ford, who has become a messianic figure to the World State. "Our Ford" is used in place of "Our Lord", as a credit to popularising the use of the assembly line.
• Sigmund Freud, "Our Freud" is sometimes said in place of "Our Ford" because Freud's psychoanalytic method depends implicitly upon the rules of classical conditioning,^{[citation needed]} and because Freud popularised the idea that sexual activity is essential to human happiness. (It is also strongly implied that citizens of the World State believe Freud and Ford to be the same person.)
• H. G. Wells, "Dr. Wells", British writer and utopian socialist, whose book Men Like Gods was a motivation for Brave New World. "All's well that ends Wells", wrote Huxley in his letters, criticising Wells for anthropological assumptions Huxley found unrealistic.
• Ivan Pavlov, whose conditioning techniques are used to train infants.
• William Shakespeare, whose banned works are quoted throughout the novel by John, "the Savage". The plays quoted include Macbeth, The Tempest, Romeo and Juliet, Hamlet, King Lear, Troilus and Cressida, Measure for Measure and Othello. Mustapha Mond also knows them because as a World Controller he has access to a selection of books from throughout history, including the Bible.
• Thomas Robert Malthus, 19th century British economist, believed the people of the Earth would eventually be threatened by their inability to raise enough food to feed the population. In the novel, the eponymous character devises the contraceptive techniques (Malthusian belt) that are practiced by women of the World State.
• Reuben Rabinovitch, the Polish-Jew character on whom the effects of sleep-learning, hypnopædia, are first observed.
• John Henry Newman, 19th century Catholic theologian and educator, believed university education the critical element in advancing post-industrial Western civilization. Mustapha Mond and The Savage discuss a passage from one of Newman's books.
• Alfred Mond, British industrialist, financier and politician. He is the namesake of Mustapha Mond.
• Mustafa Kemal Atatürk, the founder and first President of Republic of Turkey. Naming Mond after Atatürk links up with their characteristics; he reigned during the time Brave New World was written and revolutionised the 'old' Ottoman state into a new nation.

Sources of names and references

The limited number of names that the World State assigned to its bottle-grown citizens can be traced to political and cultural figures who contributed to the bureaucratic, economic, and technological systems of Huxley's age, and presumably those systems in Brave New World.

• Soma: Huxley took the name for the drug used by the state to control the population after the Vedic ritual drink Soma, inspired by his interest in Indian mysticism.
• Malthusian belt: A contraceptive device worn by women. When Huxley was writing Brave New World, organizations such as the Malthusian League had spread throughout Europe, advocating contraception. Although the controversial economic theory of Malthusianism was derived from an essay by Thomas Malthus about the economic effects of population growth, Malthus himself was an advocate of abstinence rather than contraception.
• Bokanovsky's Process: A scientific process used in the World State to mass-produce human beings. Specifically, the "Bokanovsky Process" is a method of producing multiple embryos from a single fertilized egg, creating up to 96 identical individuals. This technique is central to the society's efforts to maintain social stability and control, as it allows for the creation of a standardized, docile workforce. It's part of the larger theme in the novel of dehumanization and the reduction of individuality in the pursuit of a controlled, stable society. It is thought that the process's name is a reference to Maurice Bokanowski, a French Bureaucrat who believed strongly in the idea of governmental and social efficiency. Complementing this, Podsnap's Technique accelerates the maturation of human eggs, enabling the rapid production of thousands of nearly identical individuals. Together, these methods facilitate the creation of a large, standardized population, eliminating natural reproduction and traditional family structures, thereby reinforcing the World State's control over its citizens.

Reception

Upon its publication, Rebecca West praised Brave New World as "The most accomplished novel Huxley has yet written", Joseph Needham lauded it as "Mr Huxley's remarkable book",^[36] and Bertrand Russell also praised it, stating, "Mr Aldous Huxley has shown his usual masterly skill in Brave New World."^[37] Brave New World also received negative responses from other contemporary critics, although his work was later embraced.

In an article in the 4 May 1935 issue of the Illustrated London News, G. K. Chesterton explained that Huxley was revolting against the "Age of Utopias". Much of the discourse on man's future before 1914 was based on the thesis that humanity would solve all economic and social issues. In the decade following the war the discourse shifted to an examination of the causes of the catastrophe. The works of H. G. Wells and George Bernard Shaw on the promises of socialism and a World State were then viewed as the ideas of naive optimists. Chesterton wrote:

After the Age of Utopias came what we may call the American Age, lasting as long as the Boom. Men like Ford or Mond seemed to many to have solved the social riddle and made capitalism the common good. But it was not native to us; it went with a buoyant, not to say blatant optimism, which is not our negligent or negative optimism. Much more than Victorian righteousness, or even Victorian self-righteousness, that optimism has driven people into pessimism. For the Slump brought even more disillusionment than the War. A new bitterness, and a new bewilderment, ran through all social life, and was reflected in all literature and art. It was contemptuous, not only of the old Capitalism, but of the old Socialism. Brave New World is more of a revolution against Utopia than against Victoria.

Similarly, in 1944 economist Ludwig von Mises described Brave New World as a satire of utopian predictions of socialism: "Aldous Huxley was even courageous enough to make socialism's dreamed paradise the target of his sardonic irony."

Common misunderstandings

See also: Fetal alcohol spectrum disorder, Eugenics § In science fiction, and Island (Huxley novel)

Various authors assume that the book was first and foremost a cautionary tale regarding human genetic enhancement, indeed about–as an infamous report of Bush associate Leon Kass states–"producing improved [...] perfect or post-human" people. In fact, the title itself has become a mere stand-in used to "evoke the general idea of a futuristic dystopia". Geneticist Derek So suggests that this is a misunderstanding, however. According to him, a 'more careful reading of the text' shows that:

there does not seem to be any genetic testing in Brave New World, and most of the methods described involve hormones and chemicals rather than heritable interventions. Although Huxley wrote that "eugenics and dysgenics were practiced systematically", this seems to refer only to selective breeding and not to any kind of direct manipulation on the genetic level. (The Bokanovsky process does represent a form of cloning, but this is not ethically equivalent to germline genome editing, and references to Brave New World may lead some readers to confuse the two technologies.) [...] While it's true that the upper castes in Brave New World are smarter than the others, this is more because of the deliberate impairment of the lower castes than because the upper castes are "perfect". Rather than reducing the number of individuals born with genetic disorders or handicaps, Huxley's dystopia involves dramatically increasing their number. [...] Quite the opposite: Huxley thought that Brave New World might come about if we didn't start selecting better children.

Overall, Derek So notes that "Huxley was much more worried about totalitarianism than about the new biotechnologies per se that he alluded to in Brave New World." Despite claims to the contrary then, Huxley remained a committed eugenicist all throughout his life, much like his comparably famous brother Julian, and one just as keen on stressing its humanistic underpinnings.

The World State and Fordism

The World State is built upon the principles of Henry Ford's assembly line: mass production, homogeneity, predictability, and consumption of disposable consumer goods. While the World State lacks any supernatural-based religions, Ford himself is revered as the creator of their society but not as a deity, and characters celebrate Ford Day and swear oaths by his name (e.g., "By Ford!"). In this sense, some fragments of traditional religion are present, such as Christian crosses, which had their tops cut off to be changed to a "T", representing the Ford Model T. In England, there is an Arch-Community-Songster of Canterbury, obviously continuing the Archbishop of Canterbury, and in America The Christian Science Monitor continues publication as The Fordian Science Monitor. The World State calendar numbers years in the "AF" era—"After Ford"—with the calendar beginning in AD 1908, the year in which Ford's first Model T rolled off his assembly line. The novel's Gregorian calendar year is AD 2540, but it is referred to in the book as AF 632.

From birth, members of every class are indoctrinated by recorded voices repeating slogans while they sleep (called "hypnopædia" in the book) to believe that membership of their own class is preferable, but that the other classes perform needed functions. Any residual unhappiness is resolved by an antidepressant and hallucinogenic drug called soma.

The biological techniques used to control the populace in Brave New World do not include genetic engineering; Huxley wrote the book before the structure of DNA was known. However, Gregor Mendel's work with inheritance patterns in peas had been rediscovered in 1900 and the eugenics movement, based on artificial selection, was well established. Huxley's family included a number of prominent biologists including Thomas Huxley, half-brother and Nobel Laureate Andrew Huxley, and his brother Julian Huxley who was a biologist and involved in the eugenics movement. Nonetheless, Huxley emphasises conditioning over breeding (nurture versus nature); human embryos and fetuses are conditioned through a carefully designed regimen of chemical (such as exposure to hormones and toxins), thermal (exposure to intense heat or cold, as one's future career would dictate), and other environmental stimuli, although there is an element of selective breeding as well.

Comparisons with George Orwell's Nineteen Eighty-Four

Further information: Nineteen Eighty-Four § Brave New World comparisons

In a letter to George Orwell about Nineteen Eighty-Four, Huxley wrote "Whether in actual fact the policy of the boot-on-the-face can go on indefinitely seems doubtful. My own belief is that the ruling oligarchy will find less arduous and wasteful ways of governing and of satisfying its lust for power, and these ways will resemble those which I described in Brave New World." He went on to write "Within the next generation I believe that the world's rulers will discover that infant conditioning and narco-hypnosis are more efficient, as instruments of government, than clubs and prisons, and that the lust for power can be just as completely satisfied by suggesting people into loving their servitude as by flogging and kicking them into obedience."

Social critic Neil Postman contrasted the worlds of Nineteen Eighty-Four and Brave New World in the foreword of his 1985 book Amusing Ourselves to Death. He writes:

What Orwell feared were those who would ban books. What Huxley feared was that there would be no reason to ban a book, for there would be no one who wanted to read one. Orwell feared those who would deprive us of information. Huxley feared those who would give us so much that we would be reduced to passivity and egoism. Orwell feared that the truth would be concealed from us. Huxley feared the truth would be drowned in a sea of irrelevance. Orwell feared we would become a captive culture. Huxley feared we would become a trivial culture, preoccupied with some equivalent of the feelies, the orgy porgy, and the centrifugal bumblepuppy. As Huxley remarked in Brave New World Revisited, the civil libertarians and rationalists who are ever on the alert to oppose tyranny "failed to take into account man's almost infinite appetite for distractions." In 1984, Huxley added, people are controlled by inflicting pain. In Brave New World, they are controlled by inflicting pleasure. In short, Orwell feared that what we hate will ruin us. Huxley feared that what we love will ruin us.

The writer Christopher Hitchens, who published several articles on Huxley and a book on Orwell, noted the difference between the two texts in the introduction to his 1999 article "Why Americans Are Not Taught History",

We dwell in a present-tense culture that somehow, significantly, decided to employ the telling expression "You're history" as a choice reprobation or insult, and thus elected to speak forgotten volumes about itself. By that standard, the forbidding dystopia of George Orwell's Nineteen Eighty-Four already belongs, both as a text and as a date, with Ur and Mycenae, while the hedonist nihilism of Huxley still beckons toward a painless, amusement-sodden, and stress-free consensus. Orwell's was a house of horrors. He seemed to strain credulity because he posited a regime that would go to any lengths to own and possess history, to rewrite and construct it, and to inculcate it by means of coercion. Whereas Huxley ... rightly foresaw that any such regime could break because it could not bend. In 1988, four years after 1984, the Soviet Union scrapped its official history curriculum and announced that a newly authorized version was somewhere in the works. This was the precise moment when the regime conceded its own extinction. For true blissed-out and vacant servitude, though, you need an otherwise sophisticated society where no serious history is taught.

Brave New World Revisited

In 1946, Huxley wrote in the foreword of the new edition of Brave New World:

If I were now to rewrite the book, I would offer the Savage a third alternative. Between the Utopian and primitive horns of his dilemma would lie the possibility of sanity... In this community economics would be decentralist and Henry-Georgian, politics Kropotkinesque and co-operative. Science and technology would be used as though, like the Sabbath, they had been made for man, not (as at present and still more so in the Brave New World) as though man were to be adapted and enslaved to them. Religion would be the conscious and intelligent pursuit of man's Final End, the unitive knowledge of immanent Tao or Logos, the transcendent Godhead or Brahman. And the prevailing philosophy of life would be a kind of Higher Utilitarianism, in which the Greatest Happiness principle would be secondary to the Final End principle—the first question to be asked and answered in every contingency of life being: "How will this thought or action contribute to, or interfere with, the achievement, by me and the greatest possible number of other individuals, of man's Final End?"

First UK edition

Brave New World Revisited (Harper & Brothers, US, 1958; Chatto & Windus, UK, 1959), written by Huxley almost thirty years after Brave New World, is a non-fiction work in which Huxley considered whether the world had moved toward or away from his vision of the future from the 1930s. He believed when he wrote the original novel that it was a reasonable guess as to where the world might go in the future. In Brave New World Revisited, he concluded that the world was becoming like Brave New World much faster than he originally thought.

Huxley analysed the causes of this, such as overpopulation, as well as all the means by which populations can be controlled. He was particularly interested in the effects of drugs and subliminal suggestion. Brave New World Revisited is different in tone because of Huxley's evolving thought, as well as his conversion to Hindu Vedanta in the interim between the two books.

The last chapter of the book aims to propose action which could be taken to prevent a democracy from turning into the totalitarian world described in Brave New World. In Huxley's last novel, Island, he again expounds similar ideas to describe a utopian nation, which is generally viewed as a counterpart to Brave New World.

Censorship

According to American Library Association, Brave New World has frequently been banned and challenged in the United States due to insensitivity, offensive language, nudity, racism, drug use, conflict with a religious viewpoint, and being sexually explicit. It landed on the list of the top ten most challenged books in 2010 (3) and 2011 (7). The book also secured a spot on the association's list of the top one hundred challenged books for 1990–1999 (54), 2000–2009 (36), and 2010–2019 (26).

The following include specific instances of when the book has been censored, banned, or challenged:

• In 1932, the book was banned in Ireland for its language, and for supposedly being anti-family and anti-religion.
• In 1965, a Maryland English teacher alleged that he was fired for assigning Brave New World to students. The teacher sued for violation of First Amendment rights but lost both his case and the appeal, with the appeals court ruling that the assignment of the book was not the reason for his firing.
• The book was banned in India in 1967, with Huxley accused of being a "pornographer".
• In 1980, it was removed from classrooms in Miller, Missouri, among other challenges.
• The version of Brave New World Revisited published in China lacks explicit mentions of China itself.

Influences and allegations of plagiarism

The English writer Rose Macaulay published What Not: A Prophetic Comedy in 1918. What Not depicts a dystopian future where people are ranked by intelligence, the government mandates mind training for all citizens, and procreation is regulated by the state. Macaulay and Huxley shared the same literary circles and he attended her weekly literary salons.

Bertrand Russell felt Brave New World borrowed from his 1931 book The Scientific Outlook, and wrote in a letter to his publisher that Huxley's novel was "merely an expansion of the two penultimate chapters of 'The Scientific Outlook.'"

H. G. Wells' novel The First Men in the Moon (1901) used concepts that Huxley added to his story. Both novels introduce a society (in Wells' case, that of the Lunar natives) consisting of a specialized caste system, in which new generations are produced in vessels, where their designated caste is decided before birth by tampering with the fetus' development, and individuals are drugged down when they are not needed.

George Orwell believed that Brave New World must have been partly derived from the 1921 novel We by Russian author Yevgeny Zamyatin. However, in a 1962 letter to Christopher Collins, Huxley says that he wrote Brave New World long before he had heard of We. According to We translator Natasha Randall, Orwell believed that Huxley was lying. Kurt Vonnegut said that in writing Player Piano (1952), he "cheerfully ripped off the plot of Brave New World, whose plot had been cheerfully ripped off from Yevgeny Zamyatin's We".

In 1982, Polish author Antoni Smuszkiewicz, in his analysis of Polish science-fiction Zaczarowana gra ("The Magic Game"), presented accusations of plagiarism against Huxley. Smuszkiewicz showed similarities between Brave New World and two science fiction novels written earlier by Polish author Mieczysław Smolarski, namely Miasto światłości ("The City of Light", 1924) and Podróż poślubna pana Hamiltona ("Mr Hamilton's Honeymoon Trip", 1928). Smuszkiewicz wrote in his open letter to Huxley: "This work of a great author, both in the general depiction of the world as well as countless details, is so similar to two of my novels that in my opinion there is no possibility of accidental analogy."

Kate Lohnes, writing for Encyclopædia Britannica, notes similarities between Brave New World and other novels of the era could be seen as expressing "common fears surrounding the rapid advancement of technology and of the shared feelings of many tech-skeptics during the early 20th century". Other dystopian novels followed Huxley's work, including C.S. Lewis's That Hideous Strength (1945) and Orwell's Nineteen Eighty-Four (1949).

Legacy

In 1998–1999, the Modern Library ranked Brave New World fifth on its list of the 100 Best Novels in English of the 20th century. In 2003, Robert McCrum writing for The Observer included Brave New World chronologically at number 53 in "the top 100 greatest novels of all time", and the novel was listed at number 87 on the BBC's survey The Big Read.

On 5 November 2019, BBC News listed Brave New World on its list of the 100 Most Inspiring Novels. In 2021, Brave New World was one of six classic science fiction novels by British authors selected by Royal Mail to feature on a series of UK postage stamps.

Adaptations

Theatre

• Brave New World (opened 4 September 2015) in co-production by Royal & Derngate, Northampton and Touring Consortium Theatre Company which toured the UK. The adaptation was by Dawn King, composed by These New Puritans and directed by James Dacre.

Radio

• Brave New World (radio broadcast) CBS Radio Workshop (27 January and 3 February 1956): music composed and conducted by Bernard Herrmann. Adapted for radio by William Froug. Introduced by William Conrad and narrated by Aldous Huxley. Featuring the voices of Joseph Kearns, Bill Idelson, Gloria Henry, Charlotte Lawrence, Byron Kane, Sam Edwards, Jack Kruschen, Vic Perrin, Lurene Tuttle, Herb Butterfield, Doris Singleton.
• Brave New World (radio broadcast) BBC Radio 4 (May 2013)
• Brave New World (radio broadcast) BBC Radio 4 (22, 29 May 2016)

Film

• Brave New World (1980), a television film directed by Burt Brinckerhoff
• Brave New World (1998), a television film directed by Leslie Libman and Larry Williams
• The 1993 film Demolition Man, starring Sylvester Stallone and Sandra Bullock, is said to draw heavily from the novel.

Television

Brave New World (2010), miniseries directed by Leonard Menchiari

Brave New World (2020), series created by David Wiener

In May 2015, The Hollywood Reporter reported that Steven Spielberg's Amblin Television would bring Brave New World to Syfy network as a scripted series, adapted by Les Bohem. The adaptation was eventually written by David Wiener with Grant Morrison and Brian Taylor, with the series ordered to air on USA Network in February 2019. The series eventually moved to the Peacock streaming service and premiered on 15 July 2020. In October 2020, the series was cancelled after one season.

Golden ratio

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Golden_ratio

 

Golden ratio (φ)

Representations
Decimal	1.618033988749894 . . .
Algebraic form	${\displaystyle \frac{1+\sqrt{5}}{2}}$
Continued fraction	${\displaystyle 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\genfrac{}{}{0ex}{}{\phantom{\mathrm{}}}{\ddots }}}}}$

A golden rectangle with long side a + b and short side a can be divided into two pieces: a similar golden rectangle (shaded red, right) with long side a and short side b and a square (shaded blue, left) with sides of length a. This illustrates the relationship ⁠a + b/a⁠ = ⁠a/b⁠ = φ.

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 ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ with ⁠${\displaystyle a>b>0}$⁠, ⁠${\displaystyle a}$⁠ is in a golden ratio to ⁠${\displaystyle b}$⁠ if

$${\displaystyle \frac{a+b}{a}=\frac{a}{b}=\phi ,}$$

where the Greek letter phi (⁠${\displaystyle \phi }$⁠ or ⁠${\displaystyle \varphi }$⁠) denotes the golden ratio. The constant ⁠${\displaystyle \phi }$⁠ satisfies the quadratic equation ⁠${\displaystyle {\phi }^2=\phi +1}$⁠ and is an irrational number with a value of

${\displaystyle \phi =\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 ⁠${\displaystyle \phi }$⁠—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 quantities ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are in the golden ratio ⁠${\displaystyle \phi }$⁠ if $${\displaystyle \frac{a+b}{a}=\frac{a}{b}=\phi .}$$

Thus, if we want to find ⁠${\displaystyle \phi }$⁠, we may use that the definition above holds for arbitrary ⁠${\displaystyle b}$⁠; thus, we just set ⁠${\displaystyle b=1}$⁠, in which case ⁠${\displaystyle \phi =a}$⁠ and we get the equation ⁠${\displaystyle (\phi +1)/\phi =\phi }$⁠, which becomes a quadratic equation after multiplying by ⁠${\displaystyle \phi }$⁠: $${\displaystyle \phi +1={\phi }^2}$$ which can be rearranged to $${\displaystyle {\phi }^2-\phi -1=0.}$$

The quadratic formula yields two solutions:

${\displaystyle \frac{1+\sqrt{5}}{2}=1.618033\dots }$ and ${\displaystyle \frac{1-\sqrt{5}}{2}=-0.618033\dots .}$

Because ⁠${\displaystyle \phi }$⁠ is a ratio between positive quantities, ⁠${\displaystyle \phi }$⁠ is necessarily the positive root. The negative root is in fact the negative inverse ⁠${\displaystyle -1/\phi }$⁠, which shares many properties with the golden ratio.

History

See also: Mathematics and art and Fibonacci number § 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.

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

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.

Michael Maestlin, the first to write a decimal approximation of the ratio

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 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 ⁠${\displaystyle 0.6180340}$⁠" 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". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.

By 1910, inventor Mark Barr began using the Greek letter phi (⁠${\displaystyle \phi }$⁠) as a symbol for the golden ratio. It has also been represented by tau (⁠${\displaystyle \tau }$⁠), the first letter of the ancient Greek τομή ('cut' or 'section').

Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.

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

If φ were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so φ cannot be rational.

This is a proof by infinite descent. Recall that:

the whole is the longer part plus the shorter part;
the whole is to the longer part as the longer part is to the shorter part.

If we call the whole ⁠${\displaystyle n}$⁠ and the longer part ⁠${\displaystyle m}$⁠, then the second statement above becomes

⁠${\displaystyle n}$⁠ is to ⁠${\displaystyle m}$⁠ as ⁠${\displaystyle m}$⁠ is to ⁠${\displaystyle n-m}$⁠.

To say that the golden ratio ⁠${\displaystyle \phi }$⁠ is rational means that ⁠${\displaystyle \phi }$⁠ is a fraction ⁠${\displaystyle n/m}$⁠ where ⁠${\displaystyle n}$⁠ and ⁠${\displaystyle m}$⁠ are integers. We may take ⁠${\displaystyle n/m}$⁠ to be in lowest terms and ⁠${\displaystyle n}$⁠ and ⁠${\displaystyle m}$⁠ to be positive. But if ⁠${\displaystyle n/m}$⁠ is in lowest terms, then the equally valued ⁠${\displaystyle m/(n-m)}$⁠ is in still lower terms. That is a contradiction that follows from the assumption that ⁠${\displaystyle \phi }$⁠ 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 ⁠${\displaystyle \phi =\frac{1}{2}(1+\sqrt{5})}$⁠ is assumed to be rational, then ⁠${\displaystyle 2\phi -1=\sqrt{5}}$⁠, the square root of ⁠${\displaystyle 5}$⁠, must also be rational. This is a contradiction, as the square roots of all non-square natural numbers are irrational.

Minimal polynomial

The golden ratio φ and its negative reciprocal −φ⁻¹ are the two roots of the quadratic polynomial x² − x − 1. The golden ratio's negative −φ and reciprocal φ⁻¹ are the two roots of the quadratic polynomial x² + x − 1.

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 coefficents that has the golden ratio as a root, is $${\displaystyle x^2-x-1.}$$ This quadratic polynomial has two roots, ⁠${\displaystyle \phi }$⁠ and ⁠${\displaystyle -{\phi }^{-1}}$⁠. Because the leading coefficient of this polynomial is 1, both roots are algebraic integers. The golden ratio is also closely related to the polynomial ⁠${\displaystyle x^2+x-1}$⁠, which has roots ⁠${\displaystyle -\phi }$⁠ and ⁠${\displaystyle {\phi }^{-1}}$⁠.

The golden ratio ⁠${\displaystyle \phi }$⁠ is a fundamental unit of the quadratic field ⁠${\displaystyle \mathbb{Q}(\sqrt{5})}$⁠, sometimes called the golden field. In this field, any element can be written in the form ⁠${\displaystyle r+s\phi }$⁠, with rational coefficients ⁠${\displaystyle r}$⁠ and ⁠${\displaystyle s}$⁠; such a number has norm ⁠${\displaystyle r^2+rs-s^2}$⁠. Other units, with norm ⁠${\displaystyle \pm 1}$⁠, are the positive and negative powers of ⁠${\displaystyle \phi }$⁠. The quadratic integers in this field, which form a ring, are all numbers of the form ⁠${\displaystyle a+b\phi }$⁠ where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ 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 ⁠${\displaystyle x^2-x-1}$⁠ is

$${\displaystyle -\frac{1}{\phi }=1-\phi =\frac{1-\sqrt{5}}{2}=-0.618033\dots .}$$

The absolute value of this quantity (⁠${\displaystyle 0.618\dots }$⁠) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ⁠${\displaystyle b/a}$⁠).

This illustrates the unique property of the golden ratio among positive numbers, that $${\displaystyle \frac{1}{\phi }=\phi -1,}$$

or its inverse, $${\displaystyle \frac{1}{1/\phi }=\frac{1}{\phi }+1.}$$

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with ⁠${\displaystyle \phi }$⁠:

$${\displaystyle \begin{array}{rl}{\phi }^2& =\phi +1=2.618033\dots ,\\ \frac{1}{\phi }& =\phi -1=0.618033\dots .\end{array}}$$

The sequence of powers of ⁠${\displaystyle \phi }$⁠ contains these values ⁠${\displaystyle 0.618033\dots }$⁠, ⁠${\displaystyle 1.0}$⁠, ⁠${\displaystyle 1.618033\dots }$⁠, ⁠${\displaystyle 2.618033\dots }$⁠; more generally, any power of ⁠${\displaystyle \phi }$⁠ is equal to the sum of the two immediately preceding powers: $${\displaystyle {\phi }^n={\phi }^{n-1}+{\phi }^{n-2}=\phi \cdot {\mathrm{F}}_n+{\mathrm{F}}_{n-1}.}$$

As a result, one can easily decompose any power of ⁠${\displaystyle \phi }$⁠ into a multiple of ⁠${\displaystyle \phi }$⁠ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of ⁠${\displaystyle \phi }$⁠:

If ⁠${\displaystyle \lfloor \frac{1}{2}n-1\rfloor =m}$⁠, then: $${\displaystyle \begin{array}{rl}{\phi }^n& ={\phi }^{n-1}+{\phi }^{n-3}+\cdots +{\phi }^{n-1-2m}+{\phi }^{n-2-2m}\\ {\phi }^n-{\phi }^{n-1}& ={\phi }^{n-2}.\end{array}}$$

Continued fraction and square root

See also: Lucas number § Continued fractions for powers of the golden ratio

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

The formula ⁠${\displaystyle \phi =1+1/\phi }$⁠ can be expanded recursively to obtain a simple continued fraction for the golden ratio: $${\displaystyle \phi =[1;1,1,1,\dots ]=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\genfrac{}{}{0ex}{}{\phantom{\mathrm{}}}{\ddots }}}}}$$

It is in fact the simplest form of a continued fraction, alongside its reciprocal form: $${\displaystyle {\phi }^{-1}=[0;1,1,1,\dots ]=0+\frac{1}{1+\frac{1}{1+\frac{1}{1+\genfrac{}{}{0ex}{}{\phantom{\mathrm{}}}{\ddots }}}}}$$

The convergents of these continued fractions, ⁠${\displaystyle \frac{1}{1}}$⁠, ⁠${\displaystyle \frac{2}{1}}$⁠, ⁠${\displaystyle \frac{3}{2}}$⁠, ⁠${\displaystyle \frac{5}{3}}$⁠, ⁠${\displaystyle \frac{8}{5}}$⁠, ⁠${\displaystyle \frac{13}{8}}$⁠, ... or ⁠${\displaystyle \frac{1}{1}}$⁠, ⁠${\displaystyle \frac{1}{2}}$⁠, ⁠${\displaystyle \frac{2}{3}}$⁠, ⁠${\displaystyle \frac{3}{5}}$⁠, ⁠${\displaystyle \frac{5}{8}}$⁠, ⁠${\displaystyle \frac{8}{13}}$⁠, ..., 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 ⁠${\displaystyle \xi }$⁠, there are infinitely many distinct fractions ⁠${\displaystyle p/q}$⁠ such that, $${\displaystyle \left|\xi -\frac{p}{q}\right|<\frac{1}{\sqrt{5}{q}^2}.}$$

This means that the constant ⁠${\displaystyle \sqrt{5}}$⁠ 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 ⁠${\displaystyle \phi }$⁠ can be obtained from ⁠${\displaystyle {\phi }^2=1+\phi }$⁠, yielding: $${\displaystyle \phi =\sqrt{1+\sqrt{1+\sqrt{1+\cdots \phantom{}}}}.}$$

Relationship to Fibonacci and Lucas numbers

Further information: Fibonacci number § Relation to the golden ratio

See also: Lucas number § Relationship to Fibonacci numbers

 

A Fibonacci spiral (top) which approximates the golden spiral, using Fibonacci sequence square sizes up to 21. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of Lucas numbers, here up to 76.

Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term ${\displaystyle F_n}$ is equal to the sum of the preceding two terms ${\displaystyle F_{n-1}}$ and ${\displaystyle F_{n-2}}$, starting with the base sequence ⁠${\displaystyle 0,1}$⁠ as the 0th and 1st terms ${\displaystyle F_0}$ and ${\displaystyle F_1}$:

⁠${\displaystyle 0,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 2,}$⁠ ⁠${\displaystyle 3,}$⁠ ⁠${\displaystyle 5,}$⁠ ⁠${\displaystyle 8,}$⁠ ⁠${\displaystyle 13,}$⁠ ⁠${\displaystyle 21,}$⁠ ⁠${\displaystyle 34,}$⁠ ⁠${\displaystyle 55,}$⁠ ⁠${\displaystyle 89,}$⁠ ⁠${\displaystyle \dots }$⁠ (OEIS: A000045).

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 ${\displaystyle L_n}$ is the sum of the previous two terms ${\displaystyle L_{n-1}}$ and ${\displaystyle L_{n-2}}$, however instead starts with ⁠${\displaystyle 2,1}$⁠ as the 0th and 1st terms ${\displaystyle L_0}$ and ${\displaystyle L_1}$:

⁠${\displaystyle 2,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 3,}$⁠ ⁠${\displaystyle 4,}$⁠ ⁠${\displaystyle 7,}$⁠ ⁠${\displaystyle 11,}$⁠ ⁠${\displaystyle 18,}$⁠ ⁠${\displaystyle 29,}$⁠ ⁠${\displaystyle 47,}$⁠ ⁠${\displaystyle 76,}$⁠ ⁠${\displaystyle 123,}$⁠ ⁠${\displaystyle 199,}$⁠ ⁠${\displaystyle \dots }$⁠ (OEIS: A000032).

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{F_{n+1}}{F_n}=\underset{n\to \mathrm{\infty }}{lim}\frac{L_{n+1}}{L_n}=\phi .}$$

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates ⁠${\displaystyle \phi }$⁠. For example,

${\displaystyle \frac{F_{16}}{F_{15}}=\frac{987}{610}=1.6180327\dots }$ and ${\displaystyle \frac{L_{16}}{L_{15}}=\frac{2207}{1364}=1.6180351\dots .}$

These approximations are alternately lower and higher than ⁠${\displaystyle \phi }$⁠, and converge to ⁠${\displaystyle \phi }$⁠ as the Fibonacci and Lucas numbers increase.

Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:

$${\displaystyle F\left(n\right)=\frac{{\phi }^n-(-\phi {)}^{-n}}{\sqrt{5}}=\frac{{\phi }^n-(1-\phi {)}^n}{\sqrt{5}}=\frac{1}{\sqrt{5}}\left[{\left(\frac{1+\sqrt{5}}{2}\right)}^n-{\left(\frac{1-\sqrt{5}}{2}\right)}^n\right],}$$ $${\displaystyle L\left(n\right)={\phi }^n+(-\phi {)}^{-n}={\phi }^n+(1-\phi {)}^n={\left(\frac{1+\sqrt{5}}{2}\right)}^n+{\left(\frac{1-\sqrt{5}}{2}\right)}^n.}$$

Combining both formulas above, one obtains a formula for ⁠${\displaystyle {\phi }^n}$⁠ that involves both Fibonacci and Lucas numbers: $${\displaystyle {\phi }^n=\frac{1}{2}(L_n+F_n\sqrt{5}).}$$

Between Fibonacci and Lucas numbers one can deduce ⁠${\displaystyle L_{2n}=5F_n^2+2(-1{)}^n=L_n^2-2(-1{)}^n}$⁠, which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five: $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{L_n}{F_n}=\sqrt{5}.}$$

Indeed, much stronger statements are true: $${\displaystyle \begin{array}{rl}& |L_n-\sqrt{5}{F}_n|=\frac{2}{{\phi }^n}\to 0,\\ & (\frac{1}{2}{L}_{3n}{)}^2=5(\frac{1}{2}{F}_{3n}{)}^2+(-1{)}^n.\end{array}}$$

Successive powers of the golden ratio obey the Fibonacci recurrence, ⁠${\displaystyle {\phi }^{n+1}={\phi }^n+{\phi }^{n-1}}$⁠.

The reduction to a linear expression can be accomplished in one step by using: $${\displaystyle {\phi }^n=F_n\phi +F_{n-1}.}$$

This identity allows any polynomial in ⁠${\displaystyle \phi }$⁠ to be reduced to a linear expression, as in:

$${\displaystyle \begin{array}{rl}3{\phi }^3-5{\phi }^2+4& =3({\phi }^2+\phi )-5{\phi }^2+4\\ & =3((\phi +1)+\phi )-5(\phi +1)+4\\ & =\phi +2\approx \mathrm{3.618033.}\end{array}}$$

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation: $${\displaystyle \sum _{n=1}^{\mathrm{\infty }}|F_n\phi -F_{n+1}|=\phi .}$$

In particular, the powers of ⁠${\displaystyle \phi }$⁠ themselves round to Lucas numbers (in order, except for the first two powers, ⁠${\displaystyle {\phi }^0}$⁠ and ⁠${\displaystyle \phi }$⁠, are in reverse order):

$${\displaystyle \begin{array}{rl}{\phi }^0& =1,\\ {\phi }^1& =1.618033989\dots \approx 2,\\ {\phi }^2& =2.618033989\dots \approx 3,\\ {\phi }^3& =4.236067978\dots \approx 4,\\ {\phi }^4& =6.854101967\dots \approx 7,\end{array}}$$

and so forth.^[44] The Lucas numbers also directly generate powers of the golden ratio; for ⁠${\displaystyle n\ge 2}$⁠: $${\displaystyle {\phi }^n=L_n-(-\phi {)}^{-n}.}$$

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is ⁠${\displaystyle L_n=F_{n-1}+F_{n+1}}$⁠; and, importantly, that ⁠${\displaystyle L_n{F}_n=F_{2n}}$⁠.

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.

Geometry

The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a regular icosahedron. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction

Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.

Dividing by interior division

1. Having a line segment ⁠${\displaystyle AB}$⁠, construct a perpendicular ⁠${\displaystyle BC}$⁠ at point ⁠${\displaystyle B}$⁠, with ⁠${\displaystyle BC}$⁠ half the length of ⁠${\displaystyle AB}$⁠. Draw the hypotenuse ⁠${\displaystyle AC}$⁠.
2. Draw an arc with center ⁠${\displaystyle C}$⁠ and radius ⁠${\displaystyle BC}$⁠. This arc intersects the hypotenuse ⁠${\displaystyle AC}$⁠ at point ⁠${\displaystyle D}$⁠.
3. Draw an arc with center ⁠${\displaystyle A}$⁠ and radius ⁠${\displaystyle AD}$⁠. This arc intersects the original line segment ⁠${\displaystyle AB}$⁠ at point ⁠${\displaystyle S}$⁠. Point ⁠${\displaystyle S}$⁠ divides the original line segment ⁠${\displaystyle AB}$⁠ into line segments ⁠${\displaystyle AS}$⁠ and ⁠${\displaystyle SB}$⁠ with lengths in the golden ratio.

Dividing by exterior division

1. Draw a line segment ⁠${\displaystyle AS}$⁠ and construct off the point ⁠${\displaystyle S}$⁠ a segment ⁠${\displaystyle SC}$⁠ perpendicular to ⁠${\displaystyle AS}$⁠ and with the same length as ⁠${\displaystyle AS}$⁠.
2. Do bisect the line segment ⁠${\displaystyle AS}$⁠ with ⁠${\displaystyle M}$⁠.
3. A circular arc around ⁠${\displaystyle M}$⁠ with radius ⁠${\displaystyle MC}$⁠ intersects in point ⁠${\displaystyle B}$⁠ the straight line through points ⁠${\displaystyle A}$⁠ and ⁠${\displaystyle S}$⁠ (also known as the extension of ⁠${\displaystyle AS}$⁠). The ratio of ⁠${\displaystyle AS}$⁠ to the constructed segment ⁠${\displaystyle SB}$⁠ is the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.

Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden angle

Main article: Golden angle

g ≈ 137.508°

When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure ⁠${\displaystyle g}$⁠:

$${\displaystyle \begin{array}{rl}\frac{2\pi -g}{g}& =\frac{2\pi }{2\pi -g}=\phi ,\\ 2\pi -g& =\frac{2\pi }{\phi }\approx {222.5}^{\circ },\\ g& =\frac{2\pi }{{\phi }^2}\approx {137.5}^{\circ }.\end{array}}$$

This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.

Pentagonal symmetry system

Pentagon and pentagram

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are ⁠${\displaystyle a}$⁠, and short edges are ⁠${\displaystyle b}$⁠, then Ptolemy's theorem gives ⁠${\displaystyle a^2=b^2+ab}$⁠. Dividing both sides by ⁠${\displaystyle ab}$⁠ yields (see § Calculation above), $${\displaystyle \frac{a}{b}=\frac{a+b}{a}=\phi .}$$

The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by ⁠${\displaystyle \phi }$⁠. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is ⁠${\displaystyle \phi }$⁠, as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for ⁠${\displaystyle \phi }$⁠: $${\displaystyle \begin{array}{rl}\phi & =1+2\sin (\pi /10)=1+2\sin {18}^{\circ },\\ \phi & =\frac{1}{2}\csc (\pi /10)=\frac{1}{2}\csc {18}^{\circ },\\ \phi & =2\cos (\pi /5)=2\cos {36}^{\circ },\\ \phi & =2\sin (3\pi /10)=2\sin {54}^{\circ }.\end{array}}$$

Golden triangle and golden gnomon

Main article: Golden triangle (mathematics)

A golden triangle ABC can be subdivided by an angle bisector into a smaller golden triangle CXB and a golden gnomon XAC.

The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle ⁠${\displaystyle {36}^{\circ }}$⁠ and base angles ⁠${\displaystyle {72}^{\circ }}$⁠. Its two equal sides are in the golden ratio to its base. The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle ⁠${\displaystyle {108}^{\circ }}$⁠ and base angle ⁠${\displaystyle {36}^{\circ }}$⁠. Its base is in the golden ratio to its two equal sides. The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.

If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.

Penrose tilings

Main article: Penrose tiling

The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.

The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:

• Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.
• The kite and dart Penrose tiling uses kites with three interior angles of ⁠${\displaystyle {72}^{\circ }}$⁠ and one interior angle of ⁠${\displaystyle {144}^{\circ }}$⁠, and darts, concave quadrilaterals with two interior angles of ⁠${\displaystyle {36}^{\circ }}$⁠, one of ⁠${\displaystyle {72}^{\circ }}$⁠, and one non-convex angle of ⁠${\displaystyle {216}^{\circ }}$⁠. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.
• The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.
• The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of ⁠${\displaystyle {36}^{\circ }}$⁠ and ⁠${\displaystyle {144}^{\circ }}$⁠, and a thick rhombus with angles of ⁠${\displaystyle {72}^{\circ }}$⁠ and ⁠${\displaystyle {108}^{\circ }}$⁠. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals ⁠${\displaystyle 1:\phi }$⁠, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.

Original four-tile Penrose tiling

 

Rhombic Penrose tiling

In triangles and quadrilaterals

Odom's construction

Odom's construction: AB : BC = AC : AB = φ : 1

George Odom found a construction for ⁠${\displaystyle \phi }$⁠ involving an equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.

Kepler triangle

Main article: Kepler triangle

 

Geometric progression of areas of squares on the sides of a Kepler triangle

 

An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length

The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression: $${\displaystyle 1:\sqrt{\phi \phantom{}}:\phi .}$$ These side lengths are the three Pythagorean means of the two numbers ⁠${\displaystyle \phi \pm 1}$⁠. The three squares on its sides have areas in the golden geometric progression ⁠${\displaystyle 1:\phi :{\phi }^2}$⁠.

Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs. The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.

For a Kepler triangle with smallest side length ⁠${\displaystyle s}$⁠, the area and acute internal angles are: $${\displaystyle \begin{array}{rl}A& =\frac{1}{2}{s}^2\sqrt{\phi \phantom{}},\\ \theta & ={\sin }^{-1}\frac{1}{\phi }\approx {38.1727}^{\circ },\\ \theta & ={\cos }^{-1}\frac{1}{\phi }\approx {51.8273}^{\circ }.\end{array}}$$

Golden rectangle

Main article: Golden rectangle

To construct a golden rectangle with only a straightedge and compass in four simple steps:

Draw a square.

Draw a line from the midpoint of one side of the square to an opposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

Complete the golden rectangle.

The golden ratio proportions the adjacent side lengths of a golden rectangle in ⁠${\displaystyle 1:\phi }$⁠ ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ⁠${\displaystyle \phi }$⁠ ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).

Golden rhombus

Main article: Golden rhombus

A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly ⁠${\displaystyle 1:\phi }$⁠. For a rhombus of such proportions, its acute angle and obtuse angles are:

$${\displaystyle \begin{array}{rl}\alpha & =2\arctan \frac{1}{\phi }\approx {63.43495}^{\circ },\\ \beta & =2\arctan \phi =\pi -\arctan 2=\arctan 1+\arctan 3\approx {116.56505}^{\circ }.\end{array}}$$

The lengths of its short and long diagonals ⁠${\displaystyle d}$⁠ and ⁠${\displaystyle D}$⁠, in terms of side length ⁠${\displaystyle a}$⁠ are:

$${\displaystyle \begin{array}{rl}d& =\frac{2a}{\sqrt{2+\phi }}=2\sqrt{\frac{3-\phi }{5}}a\approx 1.05146a,\\ D& =2\sqrt{\frac{2+\phi }{5}}a\approx 1.70130a.\end{array}}$$

Its area, in terms of ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle d}$⁠:

$${\displaystyle \begin{array}{rl}A& =\sin (\arctan 2)\cdot a^2=\frac{2}{\sqrt{5}}{a}^2\approx 0.89443a^2,\\ A& =\frac{\phi }{2}{d}^2\approx 0.80902d^2.\end{array}}$$

Its inradius, in terms of side ⁠${\displaystyle a}$⁠:

$${\displaystyle r=\frac{a}{\sqrt{5}}.}$$

Golden rhombi form the faces of the rhombic triacontahedron, the two golden rhombohedra, the Bilinski dodecahedron, and the rhombic hexecontahedron.

Vesica piscis

Main article: Vesica piscis

${\displaystyle D}$ divides ${\displaystyle CX}$ in the golden ratio.

If the two circles defining the vesica piscis are each surrounded by two concentric circles of twice the radius, then the two outer circles are tangent to the two inner circles (at the points ${\displaystyle E}$ and ${\displaystyle F}$ of the figure). The outer circles also intersect to form a lens, but one with a different angle than the vesica piscis. For these circles, the line segment ${\displaystyle \overline{XC}}$ from one of the crossing points ${\displaystyle C}$ of the inner circles to the opposite crossing point ${\displaystyle X}$ of the outer circles is subdivided in the golden ratio by the point ${\displaystyle D}$, the second crossing point of the two inner circles.

Golden spiral

Main article: Golden spiral

The golden spiral (red) and its approximation by quarter-circles (green), with overlaps shown in yellow

A logarithmic spiral whose radius grows by the golden ratio per 108° of turn, surrounding nested golden isosceles triangles. This is a different spiral from the golden spiral, which grows by the golden ratio per 90° of turn.

Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio, or their approximations generated from Fibonacci numbers, often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with ⁠${\displaystyle (r,\theta )}$⁠: $${\displaystyle r={\phi }^{2\theta /\pi }.}$$

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each ⁠${\displaystyle {108}^{\circ }}$⁠ that it turns, instead of the ⁠${\displaystyle {90}^{\circ }}$⁠ turning angle of the golden spiral. Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.

Dodecahedron and icosahedron

Cartesian coordinates of the dodecahedron :
(±1, ±1, ±1)
(0, ±φ, ±⁠1/φ⁠)
(±⁠1/φ⁠, 0, ±φ)
(±φ, ±⁠1/φ⁠, 0)
A nested cube inside the dodecahedron is represented with dotted lines.

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. A dodecahedron has ⁠${\displaystyle 12}$⁠ regular pentagonal faces, whereas an icosahedron has ⁠${\displaystyle 20}$⁠ equilateral triangles; both have ⁠${\displaystyle 30}$⁠ edges.

For a dodecahedron of side ⁠${\displaystyle a}$⁠, the radius of a circumscribed and inscribed sphere, and midradius are (⁠${\displaystyle r_u}$⁠, ⁠${\displaystyle r_i}$⁠, and ⁠${\displaystyle r_m}$⁠, respectively):

${\displaystyle r_u=a\frac{\sqrt{3}\phi }{2},}$ ${\displaystyle r_i=a\frac{{\phi }^2}{2\sqrt{3-\phi }},}$ and ${\displaystyle r_m=a\frac{{\phi }^2}{2}.}$

While for an icosahedron of side ⁠${\displaystyle a}$⁠, the radius of a circumscribed and inscribed sphere, and midradius are:

${\displaystyle r_u=a\frac{\sqrt{\phi \sqrt{5}}}{2},}$ ${\displaystyle r_i=a\frac{{\phi }^2}{2\sqrt{3}},}$ and ${\displaystyle r_m=a\frac{\phi }{2}.}$

The volume and surface area of the dodecahedron can be expressed in terms of ⁠${\displaystyle \phi }$⁠:

${\displaystyle A_d=\frac{15\phi }{\sqrt{3-\phi }}}$ and ${\displaystyle V_d=\frac{5{\phi }^3}{6-2\phi }.}$

As well as for the icosahedron:

${\displaystyle A_i=20\frac{{\phi }^2}{2}}$ and ${\displaystyle V_i=\frac{5}{6}(1+\phi ).}$

Three golden rectangles touch all of the 12 vertices of a regular icosahedron.

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving ⁠${\displaystyle \phi }$⁠. The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:

$${\displaystyle (0,\pm 1,\pm \phi ),(\pm 1,\pm \phi ,0),(\pm \phi ,0,\pm 1).}$$

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all ⁠${\displaystyle 12}$⁠ vertices of the icosahedron, or equivalently, intersect the centers of all ⁠${\displaystyle 12}$⁠ of the dodecahedron's faces.

A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is ⁠${\displaystyle 2/(2+\phi )}$⁠ times that of the dodecahedron's. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in ⁠${\displaystyle \phi :{\phi }^2}$⁠ ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's ⁠${\displaystyle 12}$⁠ vertices touch the ⁠${\displaystyle 12}$⁠ edges of an octahedron at points that divide its edges in golden ratio.

Other properties

The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation ⁠${\displaystyle x^2-x-1=0}$⁠ or on ⁠${\displaystyle x^2-5=0}$⁠ (to compute ⁠${\displaystyle \sqrt{5}}$⁠ first). The time needed to compute ⁠${\displaystyle n}$⁠ digits of the golden ratio using Newton's method is essentially ⁠${\displaystyle O(M(n))}$⁠, where ⁠${\displaystyle M(n)}$⁠ is the time complexity of multiplying two ⁠${\displaystyle n}$⁠-digit numbers. This is considerably faster than known algorithms for π and e. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers ⁠${\displaystyle F_{25001}}$⁠ and ⁠${\displaystyle F_{25000}}$⁠, each over ⁠${\displaystyle 5000}$⁠ digits, yields over ⁠${\displaystyle \mathrm{10,000}}$⁠ significant digits of the golden ratio. The decimal expansion of the golden ratio ⁠${\displaystyle \phi }$⁠ has been calculated to an accuracy of twenty trillion (⁠${\displaystyle 2\times {10}^{13}=\mathrm{20,000,000,000,000}}$⁠) digits.

In the complex plane, the fifth roots of unity ⁠${\displaystyle z=e^{2\pi ki/5}}$⁠ (for an integer ⁠${\displaystyle k}$⁠) satisfying ⁠${\displaystyle z^5=1}$⁠ are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, ⁠${\displaystyle z+\overline{z}}$⁠, is a quadratic integer, an element of ⁠${\displaystyle \mathbb{Z}[\phi ]}$⁠. Specifically,

$${\displaystyle \begin{array}{rl}{e}^0+e^{-0}& =2,\\ e^{2\pi i/5}+e^{-2\pi i/5}& ={\phi }^{-1}=-1+\phi ,\\ e^{4\pi i/5}+e^{-4\pi i/5}& =-\phi .\end{array}}$$

This also holds for the remaining tenth roots of unity satisfying ⁠${\displaystyle z^{10}=1}$⁠,

$${\displaystyle \begin{array}{rl}{e}^{\pi i}+e^{-\pi i}& =-2,\\ e^{\pi i/5}+e^{-\pi i/5}& =\phi ,\\ e^{3\pi i/5}+e^{-3\pi i/5}& =-{\phi }^{-1}=1-\phi .\end{array}}$$

For the gamma function ⁠${\displaystyle \mathrm{\Gamma }}$⁠, the only solutions to the equation ⁠${\displaystyle \mathrm{\Gamma }(z-1)=\mathrm{\Gamma }(z+1)}$⁠ are ⁠${\displaystyle z=\phi }$⁠ and ⁠${\displaystyle z=-{\phi }^{-1}}$⁠.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or ⁠${\displaystyle \phi }$⁠-nary), quadratic integers in the ring ⁠${\displaystyle \mathbb{Z}[\phi ]}$⁠ – that is, numbers of the form ⁠${\displaystyle a+b\phi }$⁠ for ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ in ⁠${\displaystyle \mathbb{Z}}$⁠ – have terminating representations, but rational fractions have non-terminating representations.

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is ⁠${\displaystyle 4\log (\phi )}$⁠.

The golden ratio appears in the theory of modular functions as well. For ${\displaystyle |q|<1,}$ let $${\displaystyle R(q)=\frac{q^{1/5}}{1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\genfrac{}{}{0ex}{}{\phantom{\mathrm{}}}{\ddots }}}}}.}$$ Then $${\displaystyle R(e^{-2\pi })=\sqrt{\phi \sqrt{5}}-\phi ,R(-e^{-\pi })={\phi }^{-1}-\sqrt{2-{\phi }^{-1}}}$$ and $${\displaystyle R(e^{-2\pi i/\tau })=\frac{1-\phi R(e^{2\pi i\tau })}{\phi +R(e^{2\pi i\tau })}}$$ where ⁠${\displaystyle \mathrm{Im}\tau >0}$⁠ and ⁠${\displaystyle (e^z{)}^{1/5}}$⁠ in the continued fraction should be evaluated as ⁠${\displaystyle e^{z/5}}$⁠. The function ⁠${\displaystyle \tau \mapsto R(e^{2\pi i\tau })}$⁠ is invariant under ⁠${\displaystyle \mathrm{\Gamma }(5)}$⁠, a congruence subgroup of the modular group. Also for positive real numbers ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ such that ⁠${\displaystyle ab={\pi }^2,}$⁠

$${\displaystyle \begin{array}{rl}(\phi +R(e^{-2a}))(\phi +R(e^{-2b}))& =\phi \sqrt{5},\\ ({\phi }^{-1}-R(-e^{-a}))({\phi }^{-1}-R(-e^{-b}))& ={\phi }^{-1}\sqrt{5}.\end{array}}$$

⁠${\displaystyle \phi }$⁠ is a Pisot–Vijayaraghavan number.

Applications and observations

Rhythms apparent to the eye: rectangles in aspect ratios φ (left, middle) and φ² (right side) tile the square.

Architecture

Further information: Mathematics and architecture

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

Art

Further information: Mathematics and art and History of aesthetics

Da Vinci's illustration of a dodecahedron from Pacioli's Divina proportione (1509)

Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.

Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is ⁠${\displaystyle 1.34}$⁠, with averages for individual artists ranging from ⁠${\displaystyle 1.04}$⁠ (Goya) to ⁠${\displaystyle 1.46}$⁠ (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and ⁠${\displaystyle \sqrt{5}}$⁠ proportions, and others with proportions like ⁠${\displaystyle \sqrt{2}}$⁠, ⁠${\displaystyle 3}$⁠, ⁠${\displaystyle 4}$⁠, and ⁠${\displaystyle 6}$⁠.

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."

Books and design

Main article: Canons of page construction

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions ⁠${\displaystyle 2:3}$⁠, ⁠${\displaystyle 1:\sqrt{3}}$⁠, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.

Flags

The flag of Togo, whose aspect ratio uses the golden ratio

The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position".

The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.

Nature

Detail of the saucer plant, Aeonium tabuliforme, showing the multiple spiral arrangement (parastichy)

Main article: Patterns in nature

See also: Fibonacci number § Nature

Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".

The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.

Physics

The quasi-one-dimensional Ising ferromagnet ${\text{CoNb}}_2^{}{\text{O}}_6^{}$ (cobalt niobate) has ⁠${\displaystyle 8}$⁠ predicted excitation states (with ⁠${\displaystyle E_8}$⁠ symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean.

Optimization

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. ⁠${\displaystyle {360}^{\circ }/\phi \approx {222.5}^{\circ }}$⁠. This method was used to arrange the ⁠${\displaystyle 1500}$⁠ mirrors of the student-participatory satellite Starshine-3.

The golden ratio is a critical element to golden-section search as well.

Disputed observations

Examples of disputed observations of the golden ratio include the following:

Nautilus shells are often erroneously claimed to be golden-proportioned.

• Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.
• The shells of mollusks such as the nautilus are often claimed to be in the golden ratio. The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio, or sometimes claimed that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
• Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is ⁠${\displaystyle 1.45}$⁠.
• Studies by psychologists, starting with Gustav Fechner c. 1876, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.

Egyptian pyramids

The Great Pyramid of Giza

The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.

The Parthenon

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.

The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

Modern art

Albert Gleizes, Les Baigneuses (1912)

The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat. (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.) The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception". However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition. Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.