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Friday, May 19, 2023

05-19-2023-1534 - Sword-and-sandal

From Wikipedia, the free encyclopedia
This poster for Goliath and the Barbarians (1959) by Carlo Campogalliani illustrates many people's expectations from films of this genre

Sword-and-sandal, also known as peplum (pepla plural), is a subgenre of largely Italian-made historical, mythological, or biblical epics mostly set in the Greco-Roman antiquity or the Middle Ages. These films attempted to emulate the big-budget Hollywood historical epics of the time, such as Samson and Delilah (1949), Quo Vadis (1951), The Robe (1953), The Ten Commandments (1956), Ben-Hur (1959), Spartacus (1960), and Cleopatra (1963).[1] These films dominated the Italian film industry from 1958 to 1965, eventually being replaced in 1965 by spaghetti Western and Eurospy films.[2][3]

The term "peplum" (a Latin word referring to the ancient Greek garment peplos), was introduced by French film critics in the 1960s.[2][3] The terms "peplum" and "sword-and-sandal" were used in a condescending way by film critics. Later, the terms were embraced by fans of the films, similar to the terms "spaghetti Western" or "shoot-'em-ups". In their English versions, peplum films can be immediately differentiated from their Hollywood counterparts by their use of "clumsy and inadequate" English language dubbing.[4] A 100-minute documentary on the history of Italy's peplum genre was produced and directed by Antonio Avati in 1977 entitled Kolossal: i magnifici Maciste (aka Kino Kolossal).[5][6][7][8][9][10][11][12]

Genre characteristics

Kirk Douglas and Silvana Mangano in a pause during the shootings of Ulysses (1954) by Mario Camerini

Sword-and-sandal films are a specific class of Italian adventure films that have subjects set in Biblical or classical antiquity, often with plots based more or less loosely on Greco-Roman history or the other contemporary cultures of the time, such as the Egyptians, Assyrians, and Etruscans, as well as medieval times. Not all of the films were fantasy-based by any means. Many of the plots featured actual historical personalities such as Julius Caesar, Cleopatra, and Hannibal, although great liberties were taken with the storylines. Gladiators and slaves rebelling against tyrannical rulers, pirates and swashbucklers were also popular subjects.

As Robert Rushing defines it, peplum, "in its most stereotypical form, [...] depicts muscle-bound heroes (professional bodybuilders, athletes, wrestlers, or brawny actors) in mythological antiquity, fighting fantastic monsters and saving scantily clad beauties. Rather than lavish epics set in the classical world, they are low-budget films that focus on the hero's extraordinary body."[13] Thus, most sword-and-sandal films featured a superhumanly strong man as the protagonist, such as Hercules, Samson, Goliath, Ursus or Italy's own popular folk hero Maciste. In addition, the plots typically involved two women vying for the affection of the bodybuilder hero: the good love interest (a damsel in distress needing rescue), and an evil femme fatale queen who sought to dominate the hero.

Also, the films typically featured an ambitious ruler who would ascend the throne by murdering those who stood in his path, and often it was only the muscular hero who could depose him. Thus the hero's often political goal: "to restore a legitimate sovereign against an evil dictator."[14]

Many of the peplum films involved a clash between two populations, one civilized and the other barbaric, which typically included a scene of a village or city being burned to the ground by invaders. For their musical content, most films contained a colorful dancing girls sequence, meant to underline pagan decadence.

Precursors of the sword-and-sandal wave (pre-1958)

Italian films of the silent era

Italian filmmakers paved the way for the peplum genre with some of the earliest silent films dealing with the subject, including the following:

The silent Maciste films (1914–1927)

The 1914 Italian silent film Cabiria was one of the first films set in antiquity to make use of a massively muscled character, Maciste (played by actor Bartolomeo Pagano), who served in this premiere film as the hero's slavishly loyal sidekick. Maciste became the public's favorite character in the film however, and Pagano was called back many times to reprise the role. The Maciste character appeared in at least two dozen Italian silent films from 1914 through 1926, all of which featured a protagonist named Maciste although the films were set in many different time periods and geographical locations.

Here is a complete list of the silent Maciste films in chronological order:

  • Cabiria (1914) introduced the Maciste character
  • Maciste (1915) a.k.a. "The Marvelous Maciste"
  • Maciste bersagliere ("Maciste the Ranger", 1916)
  • Maciste alpino ("Maciste The Warrior", 1916)
  • Maciste atleta ("Maciste the Athlete", 1917)
  • Maciste medium ("Maciste the Clairvoyant", 1917)
  • Maciste poliziotto ("Maciste the Detective", 1917)
  • Maciste turista ("Maciste the Tourist", 1917)
  • Maciste sonnambulo ("Maciste the Sleepwalker", 1918)
  • La Rivincita di Maciste ("The Revenge of Maciste", 1919)
  • Il Testamento di Maciste ("Maciste's Will", 1919)
  • Il Viaggio di Maciste ("Maciste's Journey", 1919)
  • Maciste I ("Maciste the First", 1919)
  • Maciste contro la morte ("Maciste vs Death", 1919)
  • Maciste innamorato ("Maciste in Love", 1919)
  • Maciste in vacanza ("Maciste on Vacation", 1920)
  • Maciste salvato dalle acque ("Maciste Rescued from the Waters", 1920)
  • Maciste e la figlia del re della plata ("Maciste and the Silver King's Daughter", 1922)
  • Maciste und die Japanerin ("Maciste and the Japanese", 1922)
  • Maciste contro Maciste ("Maciste vs. Maciste", 1923)
  • Maciste und die chinesische truhe ("Maciste and the Chinese Trunk", 1923)
  • Maciste e il nipote di America ("Maciste's American Nephew", 1924)
  • Maciste imperatore ("Emperor Maciste", 1924)
  • Maciste contro lo sceicco ("Maciste vs. the Sheik", 1925)
  • Maciste all'inferno ("Maciste in Hell", 1925)
  • Maciste nella gabbia dei leoni ("Maciste in the Lions' Den", 1926)
  • il Gigante delle Dolemite ("The Giant From the Dolomite", released in 1927)

Italian fascist and post-war historical epics (1937-1956)

Theodora, Slave Empress (1954) by Riccardo Freda

The Italian film industry released several historical films in the early sound era, such as the big-budget Scipione l'Africano (Scipio Africanus: The Defeat of Hannibal) in 1937. In 1949, the postwar Italian film industry remade Fabiola (which had been previously filmed twice in the silent era). The film was released in the United Kingdom and in the United States in 1951 in an edited, English-dubbed version. Fabiola was an Italian-French co-production like the following films The Last Days of Pompeii (1950) and Messalina (1951).

During the 1950s, a number of American historical epics shot in Italy were released. In 1951, MGM producer Sam Zimbalist cleverly used the lower production costs, use of frozen funds and the expertise of the Italian film industry to shoot the large-scale Technicolor epic Quo Vadis in Rome. In addition to its fictional account linking the Great Fire of Rome, the Persecution of Christians in the Roman Empire and Emperor Nero, the film - following the novel "Quo vadis" by the Polish writer Henryk Sienkiewicz - featured also a mighty protagonist named Ursus (Italian filmmakers later made several pepla in the 1960s exploiting the Ursus character). MGM also planned Ben Hur to be filmed in Italy as early as 1952.[16]

Riccardo Freda's Sins of Rome was filmed in 1953 and released by RKO in an edited, English-dubbed version the following year. Unlike Quo Vadis, there were no American actors or production crew. The Anthony Quinn film Attila (directed by Pietro Francisci in 1954), the Kirk Douglas epic Ulysses (co-directed by an uncredited Mario Bava in 1954) and Helen of Troy (directed by Robert Wise with Sergio Leone as an uncredited second unit director in 1955) were the first of the big peplum films of the 1950s. Riccardo Freda directed another peplum, Theodora, Slave Empress in 1954, starring his wife Gianna Maria Canale. Howard Hawks directed his Land of the Pharaohs (starring Joan Collins) in Italy and Egypt in 1955. Robert Rossen made his film Alexander the Great in Egypt in 1956, with a music score by famed Italian composer Mario Nascimbene.

The main sword-and-sandal period (1958-1965)

Duel of the Titans (1961) by Sergio Corbucci

To cash in on the success of the Kirk Douglas film Ulysses, Pietro Francisci planned to make a film about Hercules, but searched unsuccessfully for years for a physically convincing yet experienced actor. His daughter spotted American bodybuilder Steve Reeves in the American film Athena and he was hired to play Hercules in 1957 when the film was made. (Reeves was paid $10,000 to star in the film).[17][18]

The genre's instantaneous growth began with the U.S. theatrical release of Hercules in 1959. American producer Joseph E. Levine acquired the U.S. distribution rights for $120,000, spent $1 million promoting the film and made more than $5 million profit.[19] This spawned the 1959 Steve Reeves sequel Hercules Unchained, the 1959 re-release of Cecil B. DeMille's Samson and Delilah (1949), and dozens of imitations that followed in their wake. Italian filmmakers resurrected their 1920s Maciste character in a brand new 1960s sound film series (1960–1964), followed rapidly by Ursus, Samson, Goliath and various other mighty-muscled heroes.

Almost all peplum films of this period featured bodybuilder stars, the most popular being Steve Reeves, Reg Park and Gordon Scott.[20] Some of these stars, such as Mickey Hargitay, Reg Lewis, Mark Forest, Gordon Mitchell and Dan Vadis, had starred in Mae West's touring stage review in the United States in the 1950s.[20] Bodybuilders of Italian origin, on the other hand, would adopt English pseudonyms for the screen; thus, stuntman Sergio Ciani became Alan Steel, and ex-gondolier Adriano Bellini was called Kirk Morris.[20]

My Son, the Hero (1962) by Duccio Tessari

To be sure, many of the films enjoyed widespread popularity among general audiences, and had production values that were typical for popular films of their day. Some films included frequent re-use of the impressive film sets that had been created for Ben-Hur and Cleopatra.

Although many of the bigger budget pepla were released theatrically in the US, fourteen of them were released directly to Embassy Pictures television in a syndicated TV package called The Sons of Hercules. Since few American viewers had a familiarity with Italian film heroes such as Maciste or Ursus, the characters were renamed[20] and the films molded into a series of sorts by splicing on the same opening and closing theme song and newly designed voice-over narration that attempted to link the protagonist of each film to the Hercules mythos. These films ran on Saturday afternoons in the 1960s.

Peplum films were, and still are, often ridiculed for their low budgets and bad English dubbing. The contrived plots, poorly overdubbed dialogue, novice acting skills of the bodybuilder leads, and primitive special effects that were often inadequate to depict the mythological creatures on screen all conspire to give these films a certain camp appeal now. In the 1990s, several of them have been subjects of riffing and satire in the United States comedy series Mystery Science Theater 3000.

However, in the early 1960s, a group of French critics, mostly writing for the Cahiers du cinéma, such as Luc Moullet, started to celebrate the genre and some of its directors, including Vittorio Cottafavi, Riccardo Freda, Mario Bava, Pietro Francisci, Duccio Tessari, and Sergio Leone.[21] Not only directors, but also some of the screenwriters, often put together in teams, worked past the typically formulaic plot structure to include a mixture of "bits of philosophical readings and scraps of psychoanalysis, reflections on the biggest political systems, the fate of the world and humanity, fatalistic notions of accepting the will of destiny and the gods, anthropocentric belief in the powers of the human physique, and brilliant syntheses of military treatises".[22]

With reference to the genre's free use of ancient mythology and other influences, Italian director Vittorio Cottafavi, who directed a number of peplum films, used the term "neo-mythologism".[23]

Hercules series (1958–1965)

A poster for Hercules (1958) by Pietro Francisci starring Steve Reeves

A series of 19 Hercules movies were made in Italy in the late '50s and early '60s. The films were all sequels to the successful Steve Reeves peplum Hercules (1958), but with the exception of Hercules Unchained, each film was a stand-alone story not connected to the others. The actors who played Hercules in these films were Steve Reeves followed by Gordon Scott, Kirk Morris, Mickey Hargitay, Mark Forest, Alan Steel, Dan Vadis, Brad Harris, Reg Park, Peter Lupus (billed as Rock Stevens) and Mike Lane. In a 1997 interview, Reeves said he felt his two Hercules films could not be topped by another sequel, so he declined to do any more Hercules films.[24]

The films are listed below by their American release titles, and the titles in parentheses are their original Italian titles with an approximate English translation. Dates shown are the original Italian theatrical release dates, not the U.S. release dates (which were years later in some cases).

A number of English-dubbed Italian films that featured the word "Hercules" in the title were not made as Hercules movies originally, such as:

  • Hercules Against the Moon Men, Hercules Against the Barbarians, Hercules Against the Mongols and Hercules of the Desert were all originally Maciste films. (See "Maciste" section below)
  • Hercules and the Black Pirate and Hercules and the Treasure of the Incas were both re-titled Samson movies. (See "Samson" section below)
  • Hercules, Prisoner of Evil was actually a re-titled Ursus film. (See "Ursus" section below)
  • Hercules and the Masked Rider was actually a re-titled Goliath movie. (See "Goliath" section below)

None of these films in their original Italian versions involved the Hercules character in any way. Likewise, most of the Sons of Hercules movies shown on American TV in the 1960s had nothing to do with Hercules in their original Italian versions.

(see also The Three Stooges Meet Hercules (1962), an American-made genre parody starring peplum star Samson Burke as Hercules)

Goliath series (1959–1964)

The Italians used Goliath as the superhero protagonist in a series of adventure films (pepla) in the early 1960s. He was a man possessed of amazing strength, although he seemed to be a different person in each film. After the classic Hercules (1958) became a blockbuster sensation in the film industry, a 1959 Steve Reeves film Il terrore dei barbari (Terror of the Barbarians) was re-titled Goliath and the Barbarians in the U.S. The film was so successful at the box office, it inspired Italian filmmakers to do a series of four more films featuring a generic beefcake hero named Goliath, although the films were not related to each other in any way (the 1960 Italian peplum David and Goliath starring Orson Welles was not part of this series, since that movie was just a historical retelling of the Biblical story).

The titles in the Italian Goliath adventure series were as follows: (the first title listed for each film is the film's original Italian title along with its English translation, while the U.S. release title follows in bold type in parentheses)

The name Goliath was also inserted into the English titles of three other Italian pepla that were re-titled for U.S. distribution in an attempt to cash in on the Goliath craze, but these films were not originally made as "Goliath movies" in Italy.

Both Goliath and the Vampires (1961) and Goliath and the Sins of Babylon (1963) actually featured the famed Italian folk hero Maciste in the original Italian versions, but American distributors did not feel the name "Maciste" meant anything to American audiences.

Goliath and the Dragon (1960) was originally an Italian Hercules movie called The Revenge of Hercules, but it was re-titled Goliath and the Dragon in the U.S. since at the time Goliath and the Barbarians was breaking box-office records, and the distributors may have thought the name "Hercules" was trademarked by distributor Joseph E. Levine.

Maciste series (1960–1965)

Main article: Maciste
Maciste in King Solomon's Mines (1964) by Piero Regnoli

There were a total of 25 Maciste films from the 1960s peplum craze (not counting the two dozen silent Maciste films made in Italy pre-1930). By 1960, seeing how well the two Steve Reeves Hercules films were doing at the box office, Italian producers decided to revive the 1920s silent film character Maciste in a new series of color/sound films. Unlike the other Italian peplum protagonists, Maciste found himself in a variety of time periods ranging from the Ice Age to 16th century Scotland. Maciste was never given an origin, and the source of his mighty powers was never revealed. However, in the first film of the 1960s series, he mentions to another character that the name "Maciste" means "born of the rock" (almost as if he was a god who would just appear out of the earth itself in times of need). One of the 1920s silent Maciste films was actually entitled "The Giant from the Dolomite", hinting that Maciste may be more god than man, which would explain his great strength.
The first title listed for each film is the film's original Italian title along with its English translation, while the U.S. release title follows in bold type in parentheses (note how many times Maciste's name in the Italian title is altered to an entirely different name in the American title):

In 1973, the Spanish cult film director Jesus Franco directed two low-budget "Maciste films" for French producers: Maciste contre la Reine des Amazones (Maciste vs the Queen of the Amazons) and Les exploits érotiques de Maciste dans l'Atlantide (The Erotic Exploits of Maciste in Atlantis). The films had almost identical casts, both starring Val Davis as Maciste, and appear to have been shot back-to-back. The former was distributed in Italy as a "Karzan" movie (a cheap Tarzan imitation), while the latter film was released only in France with hardcore inserts as Les Gloutonnes ("The Gobblers"). These two films were totally unrelated to the 1960s Italian Maciste series.

Ursus series (1960–1964)

Main article: Ursus (film character)

Following Buddy Baer's portrayal of Ursus in the classic 1951 film Quo Vadis, Ursus was used as a superhuman Roman-era character who became the protagonist in a series of Italian adventure films made in the early 1960s.

When the "Hercules" film craze hit in 1959, Italian filmmakers were looking for other muscleman characters similar to Hercules whom they could exploit, resulting in the nine-film Ursus series listed below. Ursus was referred to as a "Son of Hercules" in two of the films when they were dubbed in English (in an attempt to cash in on the then-popular "Hercules" craze), although in the original Italian films, Ursus had no connection to Hercules whatsoever. In the English-dubbed version of one Ursus film (retitled Hercules, Prisoner of Evil), Ursus was actually referred to throughout the entire film as "Hercules".

There were a total of nine Italian films that featured Ursus as the main character, listed below as follows: Italian title / English translation of the Italian title (American release title);

Samson series (1961–1964)

A character named Samson was featured in a series of five Italian peplum films in the 1960s, no doubt inspired by the 1959 re-release of the epic Victor Mature film Samson and Delilah. The character was similar to the Biblical Samson in the third and fifth films only; in the other three, he just appears to be a very strong man (not related at all to the Biblical figure).

The titles are listed as follows: Italian title / its English translation (U.S. release title in parentheses);

The name Samson was also inserted into the U.S. titles of six other Italian movies when they were dubbed in English for U.S. distribution, although these films actually featured the adventures of the famed Italian folk hero Maciste.

Samson Against the Sheik (1962), Son of Samson (1960), Samson and the Slave Queen (1963), Samson and the Seven Miracles of the World (1961), Samson vs. the Giant King (1964), and Samson in King Solomon's Mines (1964) were all re-titled Maciste movies, because the American distributors did not feel the name Maciste was marketable to U.S. filmgoers.

Samson and the Treasure of the Incas (a.k.a. Hercules and the Treasure of the Incas) (1965) sounds like a peplum title, but was actually a spaghetti Western.

The Sons of Hercules (TV syndication package)

Main article: The Sons of Hercules
Title card for the 1960s series The Sons of Hercules

The Sons of Hercules was a syndicated television show that aired in the United States in the 1960s. The series repackaged 14 randomly chosen Italian peplum films by unifying them with memorable title and end title theme songs and a standard voice-over intro relating the main hero in each film to Hercules any way they could. In some areas, each film was split into two one-hour episodes, so the 14 films were shown as 28 weekly episodes. None of the films were ever theatrically released in the U.S.

The films are not listed in chronological order, since they were not really related to each other in any way. The first title listed below for each film was its American broadcast television title, followed in parentheses by the English translation of its original Italian theatrical title:

Steve Reeves pepla (in chronological order of production)

Main article: Steve Reeves
Steve Reeves in Hercules by Pietro Francisci (1958)

Steve Reeves appeared in 14 pepla made in Italy from 1958 to 1964, and most of his films are highly regarded examples of the genre. His pepla are listed below in order of production, not in order of release. The U.S. release titles are shown below, followed by the original Italian title and its translation (in parentheses)

Other (non-series) Italian pepla

There were many 1950s and 1960s Italian pepla that did not feature a major superhero (such as Hercules, Maciste or Samson), and as such they fall into a sort of miscellaneous category. Many were of the Cappa e spada (swashbuckler) variety, though they often feature well-known characters such as Ali Baba, Julius Caesar, Ulysses, Cleopatra, the Three Musketeers, Zorro, Theseus, Perseus, Achilles, Robin Hood, and Sandokan. The first really successful Italian films of this kind were Black Eagle (1946) and Fabiola (1949).

Gladiator films

Inspired by the success of Spartacus, there were a number of Italian peplums that heavily emphasized the gladiatorial arena in their plots, with it becoming almost a peplum subgenre in itself. One group of supermen known as "The Ten Gladiators" appeared in a trilogy, all three films starring Dan Vadis in the lead role.

  • Alone Against Rome (1962) a.k.a. Vengeance of the Gladiators
  • The Arena (1974) a.k.a. Naked Warriors, co-directed by Joe D'Amato, starring Pam Grier, Paul Muller and Rosalba Neri
  • Challenge of the Gladiator (1965) starring Peter Lupus (a.k.a. Rock Stevens)
  • Fabiola (1949) a.k.a. The Fighting Gladiator
  • Gladiator of Rome (1962) a.k.a. Battle of the Gladiators, starring Gordon Scott
  • Gladiators Seven (1962) a.k.a. The Seven Gladiators, starring Richard Harrison
  • Invincible Gladiator, The (1961) Richard Harrison
  • Last Gladiator, The (1963) a.k.a. Messalina Against the Son of Hercules
  • Maciste, Gladiator of Sparta (1964) a.k.a. Terror of Rome Against the Son of Hercules
  • Revenge of Spartacus, The (1965) a.k.a. Revenge of the Gladiators, starring Roger Browne
  • Revenge of The Gladiators (1961) starring Mickey Hargitay
  • Revolt of the Seven (1964) a.k.a. The Spartan Gladiator, starring Tony Russel and Helga Line
  • Revolt of the Slaves (1961) Rhonda Fleming
  • Seven Rebel Gladiators (1965) a.k.a. Seven Against All, starring Roger Browne
  • Seven Slaves Against the World (1965) a.k.a. Seven Slaves Against Rome, a.k.a. The Strongest Slaves in the World, starring Roger Browne and Gordon Mitchell
  • Sheba and the Gladiator (1959) a.k.a. The Sign of Rome, a.k.a. Sign of the Gladiator, Anita Ekberg
  • Sins of Rome (1952) a.k.a. Spartacus, directed by Riccardo Freda
  • Slave, The (1962) a.k.a. Son of Spartacus, Steve Reeves
  • Spartacus and the Ten Gladiators (1964) a.k.a. Ten Invincible Gladiators, Dan Vadis
  • Spartan Gladiator, The (1965) Tony Russel
  • Ten Gladiators, The (1963) Dan Vadis
  • Triumph of the Ten Gladiators (1965) Dan Vadis
  • Two Gladiators, The (1964) a.k.a. Fight or Die, Richard Harrison
  • Ursus, the Rebel Gladiator (1963) a.k.a. Rebel Gladiators, Dan Vadis
  • Warrior and the Slave Girl, The (1958) a.k.a. The Revolt of the Gladiators, Gianna Maria Canale

Ancient Rome

Greek mythology

Barbarian and Viking films

Swashbucklers / pirates

  • Adventurer of Tortuga (1965) starring Guy Madison
  • Adventures of Mandrin, The (1960) a.k.a. Captain Adventure
  • Adventures of Scaramouche, The (1963) a.k.a. The Mask of Scaramouche, Gianna Maria Canale
  • Arms of the Avenger (1963) a.k.a. The Devils of Spartivento, starring John Drew Barrymore
  • At Sword's Edge (1952) dir. by Carlo Ludovico Bragaglia
  • Attack of the Moors (1959) a.k.a. The Kings of France
  • Avenger of the Seven Seas (1961) a.k.a. Executioner of the Seas, Richard Harrison
  • Avenger of Venice, The (1963) directed by Riccardo Freda, starring Brett Halsey
  • Balboa (Spanish, 1963) a.k.a. Conquistadors of the Pacific
  • Beatrice Cenci (1956) directed by Riccardo Freda
  • Beatrice Cenci (1969) directed by Lucio Fulci
  • Behind the Mask of Zorro (1966) a.k.a. The Oath of Zorro, Tony Russel
  • Black Archer, The (1959) Gerard Landry
  • Black Devil, The (1957) Gerard Landry
  • Black Duke, The (1963) Cameron Mitchell
  • Black Eagle, The (1946) a.k.a. Return of the Black Eagle, directed by Riccardo Freda
  • Black Lancers, The (1962) a.k.a. Charge of the Black Lancers, Mel Ferrer
  • Captain from Toledo, The (1966)
  • Captain of Iron, The (1962) a.k.a. Revenge of the Mercenaries, Barbara Steele
  • Captain Phantom (1953)
  • Captains of Adventure (1961) starring Paul Muller and Gerard Landry
  • Caribbean Hawk, The (1963) Yvonne Monlaur
  • Castillian, The (1963) Cesare Romero, U.S./Spanish co-production
  • Catherine of Russia (1962) directed by Umberto Lenzi
  • Cavalier in Devil's Castle (1959) a.k.a. Cavalier of Devil's Island
  • Conqueror of Maracaibo, The (1961)
  • The Count of Braggalone (1954) aka The Last Musketeer, starring Georges Marchal
  • Count of Monte Cristo, The (1962) Louis Jourdan
  • Devil Made a Woman, The (1959) a.k.a. A Girl Against Napoleon
  • Devil's Cavaliers, The (1959) a.k.a. The Devil's Riders, Gianna Maria Canale
  • Dick Turpin (1974) a Spanish production
  • El Cid (1961) Sophia Loren, Charlton Heston, U.S./ Italian film shot in Italy
  • Executioner of Venice, The (1963) Lex Barker, Guy Madison
  • Fighting Musketeers, The (1961)
  • Giant of the Evil Island (1965) a.k.a. Mystery of the Cursed Island, Peter Lupus
  • Goliath and the Masked Rider (1964) a.k.a. Hercules and the Masked Rider, Alan Steel
  • Guns of the Black Witch (1961) a.k.a. Terror of the Sea, Don Megowan
  • Hawk of the Caribbean (1963)
  • Invincible Swordsman, The (1963)
  • The Iron Swordsman (1949) a.k.a. Count Ugolino, directed by Riccardo Freda
  • Ivanhoe, the Norman Swordsman (1971) a.k.a. La spada normanna, directed by Roberto Mauri
  • Knight of a Hundred Faces, The (1960) a.k.a. The Silver Knight, starring Lex Barker
  • Knights of Terror (1963) a.k.a. Terror of the Red Capes, Tony Russel
  • Knight Without a Country (1959) a.k.a. The Faceless Rider
  • Lawless Mountain, The (1953) a.k.a. La montaña sin ley (stars Zorro)
  • Lion of St. Mark, The (1964) Gordon Scott
  • Mark of Zorro (1975) made in France, Monica Swinn
  • Mark of Zorro (1976) George Hilton
  • Masked Conqueror, The (1962)
  • Mask of the Musketeers (1963) a.k.a. Zorro and the Three Musketeers, starring Gordon Scott
  • Michael Strogoff (1956) a.k.a. Revolt of the Tartars
  • Miracle of the Wolves (1961) a.k.a. Blood on his Sword, starring Jean Marais
  • Morgan, the Pirate (1960) Steve Reeves
  • Musketeers of the Sea (1960)
  • Mysterious Rider, The (1948) directed by Riccardo Freda[26]
  • Mysterious Swordsman, The (1956) starred Gerard Landry
  • Nephews of Zorro, The (1968) Italian comedy with Franco and Ciccio
  • Night of the Great Attack (1961) a.k.a. Revenge of the Borgias
  • Night They Killed Rasputin, The (1960) a.k.a. The Last Czar
  • Nights of Lucretia Borgia, The (1959)
  • Pirate and the Slave Girl, The (1959) Lex Barker
  • Pirate of the Black Hawk, The (1958)
  • Pirate of the Half Moon (1957)
  • Pirates of the Coast (1960) Lex Barker
  • Prince with the Red Mask, The (1955) a.k.a. The Red Eagle
  • Prisoner of the Iron Mask, The (1961) a.k.a. The Revenge of the Iron Mask
  • Pugni, Pirati e Karatè (1973) a.k.a. Fists, Pirates and Karate, directed by Joe D'Amato, starring Richard Harrison (a 1970s Italian spoof of pirate movies)
  • Queen of the Pirates (1961) a.k.a. The Venus of the Pirates, Gianna Maria Canale
  • Queen of the Seas (1961) directed by Umberto Lenzi
  • Rage of the Buccaneers (1961) a.k.a. Gordon, The Black Pirate, starring Vincent Price
  • Red Cloak, The (1955) Bruce Cabot
  • Revenge of Ivanhoe, The (1965) Rik Battaglia
  • Revenge of the Black Eagle (1951) directed by Riccardo Freda
  • Revenge of the Musketeers (1963) a.k.a. Dartagnan vs. the Three Musketeers, Fernando Lamas
  • Revenge of Spartacus, The (1965) Roger Browne
  • Revolt of the Mercenaries (1961)
  • Robin Hood and the Pirates (1960) Lex Barker
  • Roland, the Mighty (1956) directed by Pietro Francisci
  • Rome 1585 (1961) a.k.a. The Mercenaries, Debra Paget, set in the 1500s
  • Rover, The (1967) a.k.a. The Adventurer, starring Anthony Quinn
  • The Sack of Rome (1953) a.k.a. The Barbarians, a.k.a. The Pagans (set in the 1500s)
  • Samson vs. the Black Pirate (1963) a.k.a. Hercules and the Black Pirate, Alan Steel
  • Samson vs. the Pirates (1963) a.k.a. Samson and the Sea Beast, Kirk Morris
  • Sandokan Fights Back (1964) a.k.a. Sandokan to the Rescue, a.k.a. The Revenge of Sandokan, Guy Madison
  • Sandokan the Great (1964) a.k.a. Sandokan, the Tiger of Mompracem, Steve Reeves
  • Sandokan, the Pirate of Malaysia (1964) a.k.a. Pirates of Malaysia, a.k.a. Pirates of the Seven Seas, Steve Reeves, directed by Umberto Lenzi
  • Sandokan vs. the Leopard of Sarawak (1964) a.k.a. Throne of Vengeance, Guy Madison
  • Saracens, The (1965) a.k.a. The Devil's Pirate, a.k.a. The Flag of Death, starring Richard Harrison
  • Sea Pirate, The (1966) a.k.a. Thunder Over the Indian Ocean, a.k.a. Surcouf, Hero of the Seven Seas
  • Secret Mark of D'artagnan, The (1962)
  • Seven Seas to Calais (1961) a.k.a. Sir Francis Drake, King of the Seven Seas, Rod Taylor
  • Seventh Sword, The (1960) Brett Halsey
  • Shadow of Zorro (1962) Frank Latimore
  • Sign of Zorro, The (1952)
  • Sign of Zorro [it] (1963) a.k.a. Duel at the Rio Grande, Sean Flynn
  • Son of Black Eagle (1968)
  • Son of Captain Blood (1962)
  • Son of d'Artagnan (1950) directed by Riccardo Freda
  • Son of El Cid, The (1965) Mark Damon
  • Son of the Red Corsair (1959) a.k.a. Son of the Red Pirate, Lex Barker
  • Son of Zorro (1973) Alberto Dell'Acqua
  • Sword in the Shadow, A (1961) starring Livio Lorenzon
  • Sword of Rebellion, The (1964) a.k.a. The Rebel of Castelmonte
  • Sword of Vengeance (1961) a.k.a. La spada della vendetta
  • Swordsman of Siena, The (1961) a.k.a. The Mercenary
  • Sword Without a Country (1960) a.k.a. Sword Without a Flag
  • Taras Bulba, The Cossack (1963) a.k.a. Plains of Battle
  • Terror of the Black Mask (1963) a.k.a. The Invincible Masked Rider
  • Terror of the Red Mask (1960) Lex Barker
  • Three Swords of Zorro, The (1963) a.k.a. The Sword of Zorro, Guy Stockwell
  • Tiger of the Seven Seas (1963)
  • Triumph of Robin Hood (1962) starring Samson Burke
  • Tyrant of Castile, The (1964) Mark Damon
  • White Slave Ship (1961) directed by Silvio Amadio
  • The White Warrior (1959) a.k.a. Hadji Murad, the White Devil, Steve Reeves
  • Women of Devil's Island (1962) starring Guy Madison
  • Zorro (1968) a.k.a. El Zorro, a.k.a. Zorro the Fox, George Ardisson
  • Zorro (1975) Alain Delon
  • Zorro and the Three Musketeers (1963) Gordon Scott
  • Zorro at the Court of England (1969) Spiros Focás as Zorro
  • Zorro at the Court of Spain (1962) a.k.a. The Masked Conqueror, Georgio Ardisson
  • Zorro of Monterrey (1971) a.k.a. El Zorro de Monterrey, Carlos Quiney
  • Zorro, Rider of Vengeance (1971) Carlos Quiney
  • Zorro's Last Adventure (1970) a.k.a. La última aventura del Zorro, Carlos Quiney
  • Zorro the Avenger (1962) a.k.a. The Revenge of Zorro, Frank Latimore
  • Zorro the Avenger (1969) a.k.a. El Zorro justiciero (1969) Fabio Testi
  • Zorro, the Navarra Marquis (1969) Nadir Moretti as Zorro
  • Zorro the Rebel (1966) Howard Ross
  • Zorro Against Maciste (1963) a.k.a. Samson and the Slave Queen (1963) starring Pierre Brice, Alan Steel

Biblical

  • Barabbas (1961) Dino de Laurentiis, Anthony Quinn, filmed in Italy
  • Bible, The (1966) Dino de Laurentiis, John Huston, filmed in Italy
  • David and Goliath (1960) Orson Welles
  • Desert Desperadoes (1956) plot involves King Herod
  • Esther and the King (1961) Joan Collins, Richard Egan
  • Head of a Tyrant, The (1959)
  • Herod the Great (1958) Edmund Purdom
  • Jacob, the Man Who Fought with God (1964) Giorgio Cerioni
  • Mighty Crusaders, The (1957) a.k.a. Jerusalem Set Free, Gianna Maria Canale
  • Moses the Lawgiver (1973) aka Moses in Egypt, Burt Lancaster, Anthony Quayle (6-hour made-for-TV Italian/British co-production) also released theatrically
  • Old Testament, The (1962) Brad Harris
  • Pontius Pilate (1962) Jean Marais
  • The Queen of Sheba (1952), directed by Pietro Francisci
  • Samson and Gideon (1965) Fernando Rey
  • Saul and David (1963) Gianni Garko
  • Sodom and Gomorrah (1962) Rosanna Podesta, U.S./Italian film shot in Italy
  • Story of Joseph and his Brethren, The (1960)
  • Sword and the Cross, The (1958) a.k.a. Mary Magdalene, Gianna Maria Canale

Ancient Egypt

Babylon / the Middle East

The second peplum wave: the 1980s

After the peplum gave way to the spaghetti Western and Eurospy films in 1965, the genre lay dormant for close to 20 years. Then in 1982, the box-office successes of Jean-Jacques Annaud's Quest for Fire (1981), Arnold Schwarzenegger's Conan the Barbarian (1982) and Clash of the Titans (1981 film) (1981) spurred a second renaissance of sword and sorcery Italian pepla in the five years immediately following. Most of these films had low budgets, focusing more on barbarians and pirates so as to avoid the need for expensive Greco-Roman sets. The filmmakers tried to compensate for their shortcomings with the addition of some graphic gore and nudity. Many of these 1980s entries were helmed by noted Italian horror film directors (Joe D'Amato, Lucio Fulci, Luigi Cozzi, etc.) and many featured actors Lou Ferrigno, Miles O'Keeffe and Sabrina Siani. Here is a list of the 1980s pepla:

A group of so-called "porno peplum" films were devoted to Roman emperors, especially - but not only - to Caligula and Claudius' spouse Messalina:

  • Caligula (1979) directed by Tinto Brass
  • A Filha de Calígula/ The Daughter of Caligula (1981) directed by Ody Fraga; made in Brazil
  • Caligula and Messalina (1981) directed by Bruno Mattei
  • Bacanales Romanas/ My Nights with Messalina (1982) directed in Spain by Jaime J. Puig; stars Ajita Wilson
  • Nerone and Poppea (1982) directed by Bruno Mattei
  • Caligula... The Untold Story (1982) directed by Joe D'Amato, starring Laura Gemser and Gabriele Tinti
  • Orgies of Caligula (1984) a.k.a. Caligula's Slaves, a.k.a. Roma, l'antica chiave dei sensi; directed by Lorenzo Onorati

See also

References


  • Patrick Lucanio, With Fire and Sword: Italian Spectacles on American Screens, 1958–1968 (Scarecrow Press, 1994; ISBN 0810828162)

  • O'Brien, D. (2014). Classical Masculinity and the Spectacular Body on Film: The Mighty Sons of Hercules. Springer. ISBN 9781137384713. Retrieved 14 February 2019.

  • Kinnard, Roy; Crnkovich, Tony (2017). Italian Sword and Sandal Films, 1908–1990. McFarland. p. 1. ISBN 9781476662916. Retrieved 14 February 2019.

  • Bondanella, Peter; Pacchioni, Federico (2017). A History of Italian Cinema. Bloomsbury Publishing USA. p. 166. ISBN 9781501307645. Retrieved 14 February 2019.

  • Della Casa, Steve; Giusti, Marco (2013). "Il Grande Libro di Ercole". Edizione Sabinae. Page 194. ISBN 978-88-98623-051

  • Kino kolossal – Herkules, Maciste & Co. Eintrag letterboxd.com. Retrieved 19 March 2021.

  • "Cineforum" (in Italian). 29 (#1–6). Federazione italiana cineforum. 1989: 62. Retrieved 14 February 2019.

  • Pomeroy, Arthur J. (2008). 'Then it Was Destroyed by the Volcano': The Ancient World in Film and on Television. Bloomsbury Academic. p. 67. ISBN 9780715630266. Retrieved 14 February 2019.

  • Nikoloutsos, Konstantinos P. (2013). Ancient Greek Women in Film. OUP Oxford. p. 139. ISBN 9780199678921. Retrieved 14 February 2019.

  • Bayman, Louis (2011). Directory of World Cinema: Italy. Intellect Books. p. 177. ISBN 9781841504001. Retrieved 14 February 2019.

  • Diak, Nicholas (2018). The New Peplum: Essays on Sword and Sandal Films and Television Programs Since the 1990s. McFarland. p. 195. ISBN 9781476631509. Retrieved 14 February 2019.

  • Ritzer, Ivo; Schulze, Peter W. (2016). Genre Hybridisation: Global Cinematic Flow. Schüren Verlag. p. 65. ISBN 9783741000416. Retrieved 14 February 2019.

  • Klein, Amanda Ann; Palmer, R. Barton (2016). Cycles, Sequels, Spin-offs, Remakes, and Reboots: Multiplicities in Film and Television. University of Texas Press. ISBN 9781477308196. Retrieved 6 February 2019.

  • Cornelius, Michael G. (2011). Of Muscles and Men: Essays on the Sword and Sandal Film. McFarland. p. 15. ISBN 9780786489022.

  • Michelakis, Pantelis; Wyke, Maria; Pucci, Giuseppe (2013). The Ancient World in Silent Cinema. Cambridge University Press. pp. 247–261. ISBN 9781107016101. Retrieved 11 February 2019.

  • Pryor, Thomas M. "Ben-Hur to Ride for Metro Again." New York Times. December 8, 1952.

  • Jon Thurber: Steve Reeves, Mr. Universe Who Became Movie Strongman, Dies 4 May 2000 latimes.com. Retrieved 19 March 2021.

  • Frumkes, Roy, ed. (July 1994). "An Interview with Steve Reeves". The Perfect Vision Magazine. Vol. 6, no. 22.

  • p.73 Frayling, Christopher Spaghetti Westerns: Cowboys and Europeans from Karl May to Sergio Leone I.B.Tauris, 2006.

  • Hughes, Howard (2011). Cinema Italiano: The Complete Guide from Classics to Cult. Bloomsbury Academic. p. 2. ISBN 9781848856080. Retrieved 15 February 2019.

  • Brunetta, Gian Piero (2004). Cent'anni di cinema italiano (in Italian). Laterza. p. 329. ISBN 9788842073468. Retrieved 14 February 2019.

  • Brunetta, Gian Piero (2004). Cent'anni di cinema italiano (in Italian). Laterza. pp. 329–330. ISBN 9788842073468. Retrieved 14 February 2019. frammenti di letture filosofiche e briciole di psicanalisi, meditazioni sui massimi sistemi politici, sul destino del mondo e dell'umanità, concezioni fatalistiche di accetazione della volontà del destino e degli dei, fiducia antropocentrica nella potenza fisica e sintesi fulminee di trattatistica militare

  • Winkler, Martin M. (2009). Troy: From Homer's Iliad to Hollywood Epic. John Wiley & Sons. p. 14. ISBN 9781405178549. Retrieved 14 February 2019.

  • Labbe, Rod "Steve Reeves: Demi-God on Horseback" Films of the Golden Age.

  • Kinnard, Roy; Crnkovich, Tony (2017). Italian Sword and Sandal Films, 1908-1990. McFarland. ISBN 1476662916.

  • Roberto Chiti; Roberto Poppi; Enrico Lancia. Dizionario del cinema italiano: I film. Gremese 1991. ISBN 8876055487.

  • Roberto Poppi, Mario Pecorari. Dizionario del cinema italiano. I film. Gremese Editore 2007. ISBN 8884405033.

  • Hughes, Howard (30 April 2011). Cinema Italiano: The Complete Guide from Classics to Cult. ISBN 9780857730442.

    1. http://www.tcm.turner.com/tcmdb/title/78700/Hundra/.[dead link]

    Further reading

    • Diak, Nicholas, editor. The New Peplum: Essays on Sword and Sandal Films and Television Programs Since the 1990s. McFarland and Company, Inc. 2018. ISBN 978-1-4766-6762-1
    • Richard Dyer: "The White Man's Muscles" in R. Dyer: White: London: Routledge: 1997: ISBN 0-415-09537-9
    • David Chapman: Retro Studs: Muscle Movie Posters from Around the World: Portland: Collectors Press: 2002: ISBN 1-888054-69-7
    • Hervé Dumont, L'Antiquité au cinéma. Vérités, légendes et manipulations (Nouveau-Monde, 2009; ISBN 2-84736-434-X)
    • Florent Fourcart, Le Péplum italien (1946–1966) : Grandeur et décadence d'une antiquité populaire (2012, CinExploitation; ISBN 291551786X)
    • Maggie Gunsberg: "Heroic Bodies: The Culture of Masculinity in Peplums" in M. Gunsberg: Italian Cinema: Gender and Genre: Houndsmill: Palgrave Macmillan: 2005: ISBN 0-333-75115-9
    • Patrick Lucanio, With Fire and Sword: Italian Spectacles on American Screens, 1958–1968 (Scarecrow Press, 1994; ISBN 0810828162)
    • Irmbert Schenk: "The Cinematic Support to Nationalist(ic) Mythology: The Italian Peplum 1910–1930" in Natascha Gentz and Stefan Kramer (eds.) Globalization, Cultural Identities and Media Representations Albany, NY: State University of New York Press: 2006: ISBN 0-7914-6684-1
    • Stephen Flacassier: "Muscles, Myths and Movies": Rabbit's Garage: 1994 : ISBN 0-9641643-0-2

    External links

    Films
    Images and discussion

     

     

     https://en.wikipedia.org/wiki/Sword-and-sandal

    From Wikipedia, the free encyclopedia
    For other uses, see Golden ratio (disambiguation) and Golden number (disambiguation).
    Golden ratio (φ)
    Golden ratio line.svg
    Line segments in the golden ratio
    Representations
    Decimal1.618033988749894...[1]
    Algebraic form
    Continued fraction
    Binary1.10011110001101110111...
    Hexadecimal1.9E3779B97F4A7C15...
    A golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left) combine to form a similar golden rectangle with long side a + b and short side a. This illustrates the relationship

    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 and with ,

    where the Greek letter phi ( or ) denotes the golden ratio.[a] The constant satisfies the quadratic equation and is an irrational number with a value of[1]

    1.618033988749....

    The golden ratio was called the extreme and mean ratio by Euclid,[2] and the divine proportion by Luca Pacioli,[3] and also goes by several other names.[b]

    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.[7] A golden rectangle—that is, a rectangle with an aspect ratio of —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.[8] 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 and are in the golden ratio if[9]

    One method for finding 's closed form starts with the left fraction. Simplifying the fraction and substituting the reciprocal ,

    Therefore,

    Multiplying by gives

    which can be rearranged to

    The quadratic formula yields two solutions:

    and

    Because is a ratio between positive quantities, is necessarily the positive root.[10] The negative root is in fact the negative inverse , 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.[11]

    — 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;[12] the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.[13] 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.[14] Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[15][c] and contains its first known definition which proceeds as follows:[16]

    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.[17][d]

    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.[19]

    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.[20][21] Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section').[22] 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.[23] 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.[24]

    German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[25] this was rediscovered by Johannes Kepler in 1608.[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[27] 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.[28]

    18th-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".[29] Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.[30] James Sully used the equivalent English term in 1875.[31]

    By 1910, inventor Mark Barr began using the Greek letter phi () as a symbol for the golden ratio.[32][e] It has also been represented by tau (), the first letter of the ancient Greek τομή ('cut' or 'section').[35]

    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.[36] This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.[37]

    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.

    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 and the longer part then the second statement above becomes

    is to as is to

    To say that the golden ratio is rational means that is a fraction where and are integers. We may take to be in lowest terms and and to be positive. But if is in lowest terms, then the equally valued is in still lower terms. That is a contradiction that follows from the assumption that is rational.

    By irrationality 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 is rational, then is also rational, which is a contradiction if it is already known that the square root 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.

    The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

    This quadratic polynomial has two roots, and

    The golden ratio is also closely related to the polynomial

    which has roots and As the root of a quadratic polynomial, the golden ratio is a constructible number.[38]

    Golden ratio conjugate and powers

    The conjugate root to the minimal polynomial is

    The absolute value of this quantity () corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ).

    This illustrates the unique property of the golden ratio among positive numbers, that

    or its inverse:

    The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with :

    The sequence of powers of contains these values more generally, any power of is equal to the sum of the two immediately preceding powers:

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

    If then:

    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 can be expanded recursively to obtain a continued fraction for the golden ratio:[39]

    It is in fact the simplest form of a continued fraction, alongside its reciprocal form:

    The convergents of these continued fractions ( ... or ...) 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 , there are infinitely many distinct fractions such that,

    This means that the constant 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.[40]

    A continued square root form for can be obtained from , yielding:[41]

    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 golden spiral is also 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 number is equal to the sum of the preceding two, starting with the base sequence :

    (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-which each term is the sum of the previous two, however instead starts with :

    (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:[42]

    In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates .

    For example, and

    These approximations are alternately lower and higher than and converge to as the Fibonacci and Lucas numbers increase.

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

    Combining both formulas above, one obtains a formula for that involves both Fibonacci and Lucas numbers:

    Between Fibonacci and Lucas numbers one can deduce which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:

    Indeed, much stronger statements are true:

    ,
    .

    These values describe as a fundamental unit of the algebraic number field .

    Successive powers of the golden ratio obey the Fibonacci recurrence, i.e.

    The reduction to a linear expression can be accomplished in one step by using:

    This identity allows any polynomial in to be reduced to a linear expression, as in:

    Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:

    In particular, the powers of themselves round to Lucas numbers (in order, except for the first two powers, and , are in reverse order):

    and so forth.[43] The Lucas numbers also directly generate powers of the golden ratio; for :

    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 ; and, importantly, that .

    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 5-cell. 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 construct a perpendicular at point with half the length of Draw the hypotenuse
    2. Draw an arc with center and radius This arc intersects the hypotenuse at point
    3. Draw an arc with center and radius This arc intersects the original line segment at point Point divides the original line segment into line segments and with lengths in the golden ratio.

    Dividing by exterior division

    1. Draw a line segment and construct off the point a segment perpendicular to and with the same length as
    2. Do bisect the line segment with
    3. A circular arc around with radius intersects in point the straight line through points and (also known as the extension of ). The ratio of to the constructed segment 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

    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.[44]

    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 and short edges are then Ptolemy's theorem gives Dividing both sides by yields (see § Calculation above),

    The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by . 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 as the four-color illustration shows.

    Pentagonal and pentagrammic geometry permits us to calculate the following values for :

    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 36° and base angles 72°.[45] Its two equal sides are in the golden ratio to its base.[46] 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 108° and base angle 36°. Its base is in the golden ratio to its two equal sides.[46] 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,[46] as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.[47]

    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.[46]

    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.[46]

    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.[48] 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.[49]
    • The kite and dart Penrose tiling uses kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. 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.[48]
    • 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.[48][50]
    • The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals , 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.[48]
    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 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.[51]

    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:

    .

    These side lengths are the three Pythagorean means of the two numbers . The three squares on its sides have areas in the golden geometric progression .

    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.[52] The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.[53]

    For a Kepler triangle with smallest side length , the area and acute internal angles are:

    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 ratio.[54] Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in 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).[55]

    Golden rhombus
    Main article: Golden rhombus

    A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly .[56] For a rhombus of such proportions, its acute angle and obtuse angles are:

    The lengths of its short and long diagonals and , in terms of side length are:

    Its area, in terms of ,and :

    Its inradius, in terms of side :

    Golden rhombi feature in the rhombic triacontahedron (see section below). They also are found in the golden rhombohedron, the Bilinski dodecahedron,[57] and the rhombic hexecontahedron.[56]

    Golden spiral

    Main article: Golden spiral
    A golden logarithmic spiral swirls around a golden triangle, touching its three vertices, moving inwardly inside similar fractal golden triangles.

    Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. Importantly, isosceles golden triangles can be encased by a golden logarithmic spiral, such that successive turns of a spiral generate new golden triangles inside. This special case of logarithmic spirals is called the golden spiral, and it exhibits continuous growth in golden ratio. That is, for every turn, there is a growth factor of . These spirals can be approximated by quarter-circles generated from Fibonacci and Lucas number-sized squares that are tiled together. In their exact form, they can be described by the polar equation with :

    As with any logarithmic spiral, for with at right angles:

    Its polar slope can be calculated using alongside from above,

    It has a complementary angle, :

    Golden spirals can be symmetrically placed inside pentagons and pentagrams as well, such that fractal copies of the underlying geometry are reproduced at all scales.

    In the 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. An icosahedron is made of regular pentagonal faces, whereas the icosahedron is made of equilateral triangles; both with edges.[58]

    For a dodecahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are (, and , respectively):

    , , and .

    While for an icosahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are:

    , , and .

    The volume and surface area of the dodecahedron can be expressed in terms of :

    and .

    As well as for the icosahedron:

    and
    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 . The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:

    , ,

    Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.[59][55] In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain vertices of the icosahedron, or equivalently, intersect the centers of of the dodecahedron's faces.[58]

    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 times that of the dodecahedron's.[60] 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 ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's vertices touch the edges of an octahedron at points that divide its edges in golden ratio.[61]

    Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the 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 or on (to compute first). The time needed to compute digits of the golden ratio using Newton's method is essentially , where is the time complexity of multiplying two -digit numbers.[62] This is considerably faster than known algorithms for and . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers and each over digits, yields over significant digits of the golden ratio. The decimal expansion of the golden ratio [1] has been calculated to an accuracy of ten trillion () digits.[63]

    In the complex plane, the fifth roots of unity (for an integer ) satisfying 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, is a quadratic integer, an element of Specifically,

    This also holds for the remaining tenth roots of unity satisfying

    For the gamma function , the only solutions to the equation are and .

    When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or -nary), quadratic integers in the ring – that is, numbers of the form for – 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 [64]

    The golden ratio appears in the theory of modular functions as well. For , let

    Then

    and

    where and in the continued fraction should be evaluated as . The function is invariant under , a congruence subgroup of the modular group. Also for positive real numbers and then[65]

    is a Pisot–Vijayaraghavan number.[66]

    Applications and observations

    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."[67][68]

    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.[69]

    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.[70]

    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.[71] 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.[72][73]

    Salvador Dalí, influenced by the works of Matila Ghyka,[74] 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.[71][75]

    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 with averages for individual artists ranging from (Goya) to (Bellini).[76] 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 proportions, and others with proportions like and [77]

    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."[78]

    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 and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.[79]

    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.[80]

    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.[81]

    Music

    Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[82] though other music scholars reject that analysis.[83] 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".[84]

    The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.[85] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[86]

    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.[87]

    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".[88]

    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.[89] Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".[90]

    However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[91]

    Physics

    The quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) has 8 predicted excitation states (with E8 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.[92]

    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. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[93]

    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.[94][95]
    • The shells of mollusks such as the nautilus are often claimed to be in the golden ratio.[96] 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,[97] or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.[98] However, measurements of nautilus shells do not support this claim.[99]
    • 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 [100]
    • Studies by psychologists, starting with Gustav Fechner c. 1876,[101] 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.[102][71]
    • 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.[103] 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.[104]

    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.[105]

    The Parthenon

    Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.[106]

    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.[107] 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."[108] Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."[109]

    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.[110] 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.[111] 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.[112] (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.)[113] The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception".[114] 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,[115] and Marcel Duchamp said as much in an interview.[116] 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.[116][117] 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.[118]

    Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,[119] though other experts (including critic Yve-Alain Bois) have discredited these claims.[71][120]

    See also

    References

    Explanatory footnotes


  • If the constraint on and each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. is defined as the positive solution. The negative solution is The sum of the two solutions is , and the product of the two solutions is .

  • Other names include the golden mean, golden section,[4] golden cut,[5] golden proportion, golden number,[6] medial section, and divine section.

  • Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.

  • "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."[18]

    1. After Classical Greek sculptor Phidias (c. 490–430 BC);[33] Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.[34]

    Citations


  • Sloane, N. J. A. (ed.). "Sequence A001622 (Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

  • Euclid. "Book 6, Definition 3". Elements.

  • Pacioli, Luca (1509). De divina proportione. Venice: Luca Paganinem de Paganinus de Brescia (Antonio Capella).

  • Livio 2002, pp. 3, 81.

  • Summerson, John (1963). Heavenly Mansions and Other Essays on Architecture. New York: W.W. Norton. p. 37. And the same applies in architecture, to the rectangles representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.

  • Herz-Fischler 1998.

  • Herz-Fischler 1998, pp. 20–25.

  • Strogatz, Steven (2012-09-24). "Me, Myself, and Math: Proportion Control". The New York Times.

  • Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.

  • Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio". The Mathematical Gazette. 62 (421): 197–198. doi:10.2307/3616690.

  • Livio 2002, p. 6.

  • Livio 2002, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..."

  • Livio 2002, pp. 7–8.

  • Livio 2002, pp. 4–5.

  • Livio 2002, p. 78.

  • Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21.

  • Livio 2002, p. 3.

  • Euclid (2007). Euclid's Elements of Geometry. Translated by Fitzpatrick, Richard. p. 156. ISBN 978-0615179841.

  • Livio 2002, pp. 88–96.

  • Mackinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette. 77 (479): 130–219. doi:10.2307/3619717.

  • Livio 2002, pp. 131–132.

  • Baravalle, H. V. (1948). "The geometry of the pentagon and the golden section". Mathematics Teacher. 41: 22–31. doi:10.5951/MT.41.1.0022.

  • Livio 2002, pp. 134–135.

  • Livio 2002, p. 141.

  • Schreiber, Peter (1995). "A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'". Historia Mathematica. 22 (4): 422–424. doi:10.1006/hmat.1995.1033.

  • Livio 2002, pp. 151–152.

  • O'Connor, John J.; Robertson, Edmund F. (2001). "The Golden Ratio". MacTutor History of Mathematics archive. Retrieved 2007-09-18.

  • Fink, Karl (1903). A Brief History of Mathematics. Translated by Beman, Wooster Woodruff; Smith, David Eugene (2nd ed.). Chicago: Open Court. p. 223. (Originally published as Geschichte der Elementar-Mathematik.)

  • Beutelspacher, Albrecht; Petri, Bernhard (1996). "Fibonacci-Zahlen". Der Goldene Schnitt (in German). Vieweg+Teubner Verlag. pp. 87–98. doi:10.1007/978-3-322-85165-9_6.

  • Herz-Fischler 1998, pp. 167–170.

  • Posamentier & Lehmann 2011, p. 8.

  • Posamentier & Lehmann 2011, p. 285.

  • Cook, Theodore Andrea (1914). The Curves of Life. London: Constable. p. 420.

  • Barr, Mark (1929). "Parameters of beauty". Architecture (NY). Vol. 60. p. 325. Reprinted: "Parameters of beauty". Think. Vol. 10–11. IBM. 1944.

  • Livio 2002, p. 5.

  • Gardner, Martin (2001). "7. Penrose Tiles". The Colossal Book of Mathematics. Norton. pp. 73–93.

  • Livio 2002, pp. 203–209
    Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. doi:10.1016/j.crhy.2019.05.009.
    Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
    Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. JSTOR 29775669.

  • Martin, George E. (1998). Geometric Constructions. Springer. pp. 13–14. doi:10.1007/978-1-4612-0629-3.

  • Hailperin, Max; Kaiser, Barbara K.; Knight, Karl W. (1999). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole. p. 63.

  • Hardy, G. H.; Wright, E. M. (1960) [1938]. "§11.8. The measure of the closest approximations to an arbitrary irrational". An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 163–164.

  • Sizer, Walter S. (1986). "Continued roots". Mathematics Magazine. 59 (1): 23–27. doi:10.1080/0025570X.1986.11977215. JSTOR 2690013. MR 0828417.

  • Tattersall, James Joseph (1999). Elementary number theory in nine chapters. Cambridge University Press. p. 28.

  • Parker, Matt (2014). Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284.

  • King, S.; Beck, F.; Lüttge, U. (2004). "On the mystery of the golden angle in phyllotaxis". Plant, Cell and Environment. 27 (6): 685–695. doi:10.1111/j.1365-3040.2004.01185.x.

  • Fletcher, Rachel (2006). "The golden section". Nexus Network Journal. 8 (1): 67–89. doi:10.1007/s00004-006-0004-z.

  • Loeb, Arthur (1992). "The Golden Triangle". Concepts & Images: Visual Mathematics. Birkhäuser. pp. 179–192. doi:10.1007/978-1-4612-0343-8_20.

  • Miller, William (1996). "Pentagons and Golden Triangles". Mathematics in School. 25 (4): 2–4. JSTOR 30216571.

  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. pp. 537–547.

  • Penrose, Roger (1978). "Pentaplexity". Eureka. Vol. 39. p. 32. (original PDF)

  • Frettlöh, D.; Harriss, E.; Gähler, F. "Robinson Triangle". Tilings Encyclopedia.

    Clason, Robert G (1994). "A family of golden triangle tile patterns". The Mathematical Gazette. 78 (482): 130–148. doi:10.2307/3618569.


  • Odom, George; van de Craats, Jan (1986). "E3007: The golden ratio from an equilateral triangle and its circumcircle". Problems and solutions. The American Mathematical Monthly. 93 (7): 572. doi:10.2307/2323047. JSTOR 2323047.

  • Busard, Hubert L. L. (1968). "L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr". Journal des Savants (in French and Latin). 1968 (2): 65–124. doi:10.3406/jds.1968.1175. See problem 51, reproduced on p. 98

  • Bruce, Ian (1994). "Another instance of the golden right triangle" (PDF). Fibonacci Quarterly. 32 (3): 232–233.

  • Posamentier & Lehmann 2011, p. 11.

  • Burger, Edward B.; Starbird, Michael P. (2005) [2000]. The Heart of Mathematics: An Invitation to Effective Thinking (2nd ed.). Springer. p. 382.

  • Grünbaum, Branko (1996). "A new rhombic hexecontahedron" (PDF). Geombinatorics. 6 (1): 15–18.

  • Senechal, Marjorie (2006). "Donald and the golden rhombohedra". In Davis, Chandler; Ellers, Erich W. (eds.). The Coxeter Legacy. American Mathematical Society. pp. 159–177. ISBN 0-8218-3722-2.

  • Livio (2002, pp. 70–72)

  • Gunn, Charles; Sullivan, John M. (2008). "The Borromean Rings: A video about the New IMU logo". In Sarhangi, Reza; Séquin, Carlo H. (eds.). Proceedings of Bridges 2008. Leeuwarden, the Netherlands. Tarquin Publications. pp. 63–70.; Video at "The Borromean Rings: A new logo for the IMU". International Mathematical Union.

  • Hume, Alfred (1900). "Some propositions on the regular dodecahedron". The American Mathematical Monthly. 7 (12): 293–295. doi:10.2307/2969130.

  • Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies. p. 4. Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.

  • Muller, J. M. (2006). Elementary functions : algorithms and implementation (2nd ed.). Boston: Birkhäuser. p. 93. ISBN 978-0817643720.

  • Yee, Alexander J. (2021-03-13). "Records Set by y-cruncher". numberword.org. Two independent computations done by Clifford Spielman.

  • Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.

  • Berndt, Bruce C.; Chan, Heng Huat; Huang, Sen-Shan; Kang, Soon-Yi; Sohn, Jaebum; Son, Seung Hwan (1999). "The Rogers–Ramanujan Continued Fraction" (PDF). Journal of Computational and Applied Mathematics. 105 (1–2): 9–24. doi:10.1016/S0377-0427(99)00033-3.

  • Duffin, Richard J. (1978). "Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers". Pacific Journal of Mathematics. 74 (1): 47–56. doi:10.2140/pjm.1978.74.47.

  • Le Corbusier, The Modulor, p. 25, as cited in Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 316. doi:10.4324/9780203477465.

  • Frings, Marcus (2002). "The Golden Section in Architectural Theory". Nexus Network Journal. 4 (1): 9–32. doi:10.1007/s00004-001-0002-0.

  • Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 320. doi:10.4324/9780203477465. Both the paintings and the architectural designs make use of the golden section

  • Urwin, Simon (2003). Analysing Architecture (2nd ed.). Routledge. pp. 154–155.

  • Livio, Mario (2002). "The golden ratio and aesthetics". Plus Magazine. Retrieved November 26, 2018.

  • Devlin, Keith (2007). "The Myth That Will Not Go Away". Retrieved September 26, 2013. Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.

  • Simanek, Donald E. "Fibonacci Flim-Flam". Archived from the original on January 9, 2010. Retrieved April 9, 2013.

  • Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.

  • Hunt, Carla Herndon; Gilkey, Susan Nicodemus (1998). Teaching Mathematics in the Block. pp. 44, 47. ISBN 1-883001-51-X.

  • Olariu, Agata (1999). "Golden Section and the Art of Painting". arXiv:physics/9908036.

  • Tosto, Pablo (1969). La composición áurea en las artes plásticas [The golden composition in the plastic arts] (in Spanish). Hachette. pp. 134–144.

  • Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. p. 43 Fig 4. ISBN 0-88179-116-4. Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.

  • Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. pp. 27–28. ISBN 0-88179-116-4.

  • Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist. 11: 44–52. Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?

    Johnson, Art (1999). Famous problems and their mathematicians. Teacher Ideas Press. p. 45. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.

    Stakhov, Alexey P.; Olsen, Scott (2009). "§1.4.1 A Golden Rectangle with a Side Ratio of τ". The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. World Scientific. p. 20–21. A credit card has a form of the golden rectangle

    Cox, Simon (2004). Cracking the Da Vinci Code. Barnes & Noble. p. 62. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.


  • Posamentier & Lehmann 2011, chapter 4, footnote 12: "The Togo flag was designed by the artist Paul Ahyi (1930–2010), who claims to have attempted to have the flag constructed in the shape of a golden rectangle".

  • Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.

  • Livio 2002, p. 190.

  • Smith, Peter F. (2003). The Dynamics of Delight: Architecture and Aesthetics. Routledge. p. 83.

  • Howat, Roy (1983). "1. Proportional structure and the Golden Section". Debussy in Proportion: A Musical Analysis. Cambridge University Press. pp. 1–10.

  • Trezise, Simon (1994). Debussy: La Mer. Cambridge University Press. p. 53.

  • Mongoven, Casey (2010). "A style of music characterized by Fibonacci and the golden ratio" (PDF). Congressus Numerantium. 201: 127–138.

    Hasegawa, Robert (2011). "Gegenstrebige Harmonik in the Music of Hans Zender". Perspectives of New Music. Project Muse. 49 (1): 207–234. doi:10.1353/pnm.2011.0000. JSTOR 10.7757/persnewmusi.49.1.0207.

    Smethurst, Reilly (2016). "Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal". In Torrence, Eve; et al. (eds.). Proceedings of Bridges 2016. Jyväskylä, Finland. Tessellations Publishing. pp. 519–522.


  • Livio 2002, p. 154.

  • Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. pp. 305–306. doi:10.4324/9780203477465.

    Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.


  • Zeising, Adolf (1854). "Einleitung [preface]". Neue Lehre von den Proportionen des menschlichen Körpers [New doctrine of the proportions of the human body] (in German). Weigel. pp. 1–10.

  • Pommersheim, James E.; Marks, Tim K.; Flapan, Erica L., eds. (2010). Number Theory: A Lively Introduction with Proofs, Applications, and Stories. Wiley. p. 82.

  • Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. doi:10.1126/science.1180085.

  • "A Disco Ball in Space". NASA. 2001-10-09. Retrieved 2007-04-16.

  • Pheasant, Stephen (1986). Bodyspace. Taylor & Francis.

  • van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach.

  • Dunlap, Richard A. (1997). The Golden Ratio and Fibonacci Numbers. World Scientific. p. 130.

  • Falbo, Clement (March 2005). "The golden ratio—a contrary viewpoint". The College Mathematics Journal. 36 (2): 123–134. doi:10.1080/07468342.2005.11922119.

  • Moscovich, Ivan (2004). The Hinged Square & Other Puzzles. New York: Sterling. p. 122.

  • Peterson, Ivars (1 April 2005). "Sea shell spirals". Science News. Archived from the original on 3 October 2012. Retrieved 10 November 2008.

  • Man, John (2002). Gutenberg: How One Man Remade the World with Word. Wiley. pp. 166–167. The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.

  • Fechner, Gustav (1876). Vorschule der Ästhetik [Preschool of Aesthetics] (in German). Leipzig: Breitkopf & Härtel. pp. 190–202.

  • Livio 2002, p. 7.

  • Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68. Archived (PDF) from the original on 2007-05-12. 38.2 percent and 61.8 percent retracements of recent rises or declines are common,

  • Batchelor, Roy; Ramyar, Richard (2005). Magic numbers in the Dow (Report). Cass Business School. pp. 13, 31. Popular press summaries can be found in: Stevenson, Tom (2006-04-10). "Not since the 'big is beautiful' days have giants looked better". The Daily Telegraph. "Technical failure". The Economist. 2006-09-23.

  • Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.

    Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to , and itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56

    Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334.

    Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. Mathematical Association of America. 23 (1): 2–19. doi:10.2307/2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of much less incorporated it in their buildings


  • Livio 2002, pp. 74–75.

  • Van Mersbergen, Audrey M. (1998). "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic". Communication Quarterly. 46 (2): 194–213. doi:10.1080/01463379809370095.

  • Devlin, Keith J. (2005). The Math Instinct. New York: Thunder's Mouth Press. p. 108.

  • Gazalé, Midhat J. (1999). Gnomon: From Pharaohs to Fractals. Princeton. p. 125.

  • Foutakis, Patrice (2014). "Did the Greeks Build According to the Golden Ratio?". Cambridge Archaeological Journal. 24 (1): 71–86. doi:10.1017/S0959774314000201.

  • Le Salon de la Section d'Or, October 1912, Mediation Centre Pompidou

  • Jeunes Peintres ne vous frappez pas !, La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, n° 1, 9 octobre 1912, pp. 1–7 Archived 2020-10-30 at the Wayback Machine, Bibliothèque Kandinsky

  • Herz-Fischler, Roger (1983). "An Examination of Claims Concerning Seurat and the Golden Number" (PDF). Gazette des Beaux-Arts. 101: 109–112.

  • Herbert, Robert (1968). Neo-Impressionism. Guggenheim Foundation.[page needed]

  • Livio 2002, p. 169.

  • Camfield, William A. (March 1965). "Juan Gris and the golden section". The Art Bulletin. 47 (1): 128–134. doi:10.1080/00043079.1965.10788819.

  • Green, Christopher (1992). Juan Gris. Yale. pp. 37–38.

    Cottington, David (2004). Cubism and Its Histories. Manchester University Press. p. 112, 142.


  • Allard, Roger (June 1911). "Sur quelques peintres". Les Marches du Sud-Ouest: 57–64. Reprinted in Antliff, Mark; Leighten, Patricia, eds. (2008). A Cubism Reader, Documents and Criticism, 1906–1914. The University of Chicago Press. pp. 178–191.

  • Bouleau, Charles (1963). The Painter's Secret Geometry: A Study of Composition in Art. Harcourt, Brace & World. pp. 247–248.

    1. Livio 2002, pp. 177–178.

    Works cited

    Further reading

    External links

    Wikimedia Commons has media related to Golden ratio.


     

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

    A genetic operator is an operator used in genetic algorithms to guide the algorithm towards a solution to a given problem. There are three main types of operators (mutation, crossover and selection), which must work in conjunction with one another in order for the algorithm to be successful. Genetic operators are used to create and maintain genetic diversity (mutation operator), combine existing solutions (also known as chromosomes) into new solutions (crossover) and select between solutions (selection).[1] In his book discussing the use of genetic programming for the optimization of complex problems, computer scientist John Koza has also identified an 'inversion' or 'permutation' operator; however, the effectiveness of this operator has never been conclusively demonstrated and this operator is rarely discussed.[2][3]

    Mutation (or mutation-like) operators are said to be unary operators, as they only operate on one chromosome at a time. In contrast, crossover operators are said to be binary operators, as they operate on two chromosomes at a time, combining two existing chromosomes into one new chromosome.[4] 

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

     

    From Wikipedia, the free encyclopedia
    Not to be confused with genetic algorithm, generic programming, genetic engineering, or DNA computing.
    Part of a series on the
    Evolutionary algorithm
    Genetic algorithm
    Genetic programming

    In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to the population of programs.

    The operations are: selection of the fittest programs for reproduction (crossover) and mutation according to a predefined fitness measure, usually proficiency at the desired task. The crossover operation involves swapping random parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Mutation involves substitution of some random part of a program with some other random part of a program. Some programs not selected for reproduction are copied from the current generation to the new generation. Then the selection and other operations are recursively applied to the new generation of programs.

    Typically, members of each new generation are on average more fit than the members of the previous generation, and the best-of-generation program is often better than the best-of-generation programs from previous generations. Termination of the evolution usually occurs when some individual program reaches a predefined proficiency or fitness level.

    It may and often does happen that a particular run of the algorithm results in premature convergence to some local maximum which is not a globally optimal or even good solution. Multiple runs (dozens to hundreds) are usually necessary to produce a very good result. It may also be necessary to have a large starting population size and variability of the individuals to avoid pathologies. 

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

    Part of a series on the
    Evolutionary algorithm
    Genetic algorithm
    Genetic programming

    Artificial development, also known as artificial embryogeny or machine intelligence or computational development, is an area of computer science and engineering concerned with computational models motivated by genotype–phenotype mappings in biological systems. Artificial development is often considered a sub-field of evolutionary computation, although the principles of artificial development have also been used within stand-alone computational models.

    Within evolutionary computation, the need for artificial development techniques was motivated by the perceived lack of scalability and evolvability of direct solution encodings (Tufte, 2008). Artificial development entails indirect solution encoding. Rather than describing a solution directly, an indirect encoding describes (either explicitly or implicitly) the process by which a solution is constructed. Often, but not always, these indirect encodings are based upon biological principles of development such as morphogen gradients, cell division and cellular differentiation (e.g. Doursat 2008), gene regulatory networks (e.g. Guo et al., 2009), degeneracy (Whitacre et al., 2010), grammatical evolution (de Salabert et al., 2006), or analogous computational processes such as re-writing, iteration, and time. The influences of interaction with the environment, spatiality and physical constraints on differentiated multi-cellular development have been investigated more recently (e.g. Knabe et al. 2008).

    Artificial development approaches have been applied to a number of computational and design problems, including electronic circuit design (Miller and Banzhaf 2003), robotic controllers (e.g. Taylor 2004), and the design of physical structures (e.g. Hornby 2004). 

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

    From Wikipedia, the free encyclopedia
    Part of a series on the
    Evolutionary algorithm
    Genetic algorithm
    Genetic programming
    "Linear genetic programming" is unrelated to "linear programming".

    Linear genetic programming (LGP)[1] is a particular method of genetic programming wherein computer programs in a population are represented as a sequence of instructions from an imperative programming language or machine language. The adjective "linear" stems from the fact that the sequence of instructions is normally executed in a linear fashion. Like in other programs, the data flow in LGP can be modeled as a graph that will visualize the potential multiple usage of register contents and the existence of structurally noneffective code (introns) which are two main differences of this genetic representation from the more common tree-based genetic programming (TGP) variant.[2][3] [4]

    Like other Genetic Programming methods, Linear genetic programming requires the input of data to run the program population on. Then, the output of the program (its behaviour) is judged against some target behaviour, using a fitness function. However, LGP is generally more efficient than tree genetic programming due to its two main differences mentioned above: Intermediate results (stored in registers) can be reused and a simple intron removal algorithm exists[1] that can be executed to remove all non-effective code prior to programs being run on the intended data. These two differences often result in compact solutions and substantial computational savings compared to the highly constrained data flow in trees and the common method of executing all tree nodes in TGP.

    Linear genetic programming has been applied in many domains, including system modeling and system control with considerable success.[5][6][7][8]

    Linear genetic programming should not be confused with linear tree programs in tree genetic programming, program composed of a variable number of unary functions and a single terminal. Note that linear tree GP differs from bit string genetic algorithms since a population may contain programs of different lengths and there may be more than two types of functions or more than two types of terminals.[9] 

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

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

     

     

     

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

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

     

    In economics, a monopsony is a market structure in which a single buyer substantially controls the market as the major purchaser of goods and services offered by many would-be sellers. The microeconomic theory of monopsony assumes a single entity to have market power over all sellers as the only purchaser of a good or service. This is a similar power to that of a monopolist, which can influence the price for its buyers in a monopoly, where multiple buyers have only one seller of a good or service available to purchase from.


    		one few
    sellers monopoly oligopoly
    buyers monopsony oligopsony

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


    From Wikipedia, the free encyclopedia
    David and Goliath
    David and Goliath 1960.jpg
    Directed by
    Written by
    Cinematography
    Edited byFranco Fraticelli
    Music byCarlo Innocenzi
    Distributed byAllied Artists
    Release date
    • 22 January 1960
    Running time
    		113 minutes
    CountryItaly
    Languages
    • English
    • Italian

    David and Goliath (Italian: David e Golia) is a 1960 Italian film directed by Ferdinando Baldi and Richard Pottier with sequences filmed in Israel and Yugoslavia.

    Plot

    The Prophet Samuel foretells a new king will rule Israel to the dismay of King Saul and his cousin and commander in chief Abner. King Saul has been having a streak of bad luck since the Philistine captivity of the Ark and fears the newcomer but doesn't know who the new king will be.

    The unsuspecting shepherd David visits Jerusalem where he is identified as the king. Abner decides to test his wisdom by asking how the Israelites can get around the Philistines' imposed edict that the only ones who may lawfully bear arms in defeated Israel are the officers of Saul's court and his palace guard. David replies that the Philistines have set no limit on the number of officers or palace guards.

    Meanwhile, King Asrod of the Philistines plots another attack on the riches of Israel, this time accompanied by the fearsome giant Goliath.

    Cast

    Production

    A part of the production took place in Jerusalem, another in Yugoslavia.[2]

    Comic book adaptation

    References


  • Film.zam.it

  • Hughes, p.70

  • "Dell Four Color #1205". Grand Comics Database.

    1. Dell Four Color #1205 at the Comic Book DB (archived from the original)

    Bibliography

    • Hughes, Howard (2011). Cinema Italiano - The Complete Guide From Classics To Cult. London - New York: I.B.Tauris. ISBN 978-1-84885-608-0.

    External links


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