Structure.

Muscles.

The five muscles of the soft palate play important roles in swallowing and breathing. The muscles are:  These muscles are innervated by the pharyngeal plexus via the vagus nerve, with the exception of the tensor veli palatini. The tensor veli palatini is innervated by the mandibular division of the trigeminal nerve (V).

Function.

The soft palate is moveable, consisting of muscle fibers sheathed in mucous membrane. It is responsible for closing off the nasal passages during the act of swallowing, and also for closing off the airway. During sneezing, it protects the nasal passage by diverting a portion of the excreted substance to the mouth.  In humans, the uvula hangs from the end of the soft palate. Touching the uvula or the end of the soft palate evokes a strong gag reflex in most people.

Speech.

A speech sound made with the middle part of the tongue (dorsum) touching the soft palate is known as a velar consonant.  It is possible for the soft palate to retract and elevate during speech to separate the oral cavity (mouth) from the nasal cavity in order to produce the oral speech sounds. If this separation is incomplete, air escapes through the nose, causing speech to be perceived as nasal.

Modelling.

Within the microstructure of the soft palate lie a variety of variably-oriented fibers that create a nonuniform surface with a nonuniform density distribution. The tissue has been characterized as viscoelastic, nonlinear, and anisotropic in the direction of the fibers. Young modulus values range from 585 Pa at the posterior free edge of the soft palate to 1409 Pa where the soft palate attaches to the maxilla. These properties are useful when quantifying the effects of corrective orthopedic devices such as the Hotz Plate on cleft lip.  Quantitative analyses have been done on bilateral and unilateral cleft palate to better understand geometric differences in cleft palate throughout the course of its development and correction. Despite the difficulty in finding common, comparable landmarks between normal soft palates and cleft palates, analytical methods have been devised to assess differences in degree of curvature of the alveolar crest, two-dimensional and three-dimensional surface area, and slope of the alveolar crest.  Finite element analysis has demonstrated effective modeling of soft-palate extension and movement. It has also been an effective tool for evaluating the craniofacial effects of corrective orthopedic devices and cleft lip.

Clinical significance.

Disease.

Pathology of the soft palate includes mucosal lesions such as pemphigus vulgaris, herpangina and migratory stomatitis, and muscular conditions such as the congenital cleft palate and cleft uvula. Petechiae on the soft palate are mainly associated with streptococcal pharyngitis, and as such it is an uncommon but highly specific finding. 10 to 30 percent of palatal petechiae cases are estimated to be caused by suction, which can be habitual or secondary to fellatio.

The soft palate (also known as the velum, palatal velum, or muscular palate) is, in mammals, the soft tissue constituting the back of the roof of the mouth. The soft palate is part of the palate of the mouth; the other part is the hard palate. The soft palate is distinguished from the hard palate at the front of the mouth in that it does not contain bone.

Le voile du palais, aussi nommé "velum" ou "palais mou", est le prolongement postérieur du palais dur, où est attachée la luette. Il est constitué d'une membrane et de muscles dont le but est de permettre l'isolement des fosses nasales de la cavité buccale lors de la déglutition et de l'articulation de certains phones, par accolement contre le rhinopharynx.


Geography.

Tarawa has a large lagoon, in total area, and a wide reef. Although naturally abundant in fish and shellfish of all kinds, marine resources are being strained by the large and growing population. Drought is frequent, but in normal years rainfall is sufficient to maintain breadfruit, papaya and banana trees as well as coconut and pandanus.  North Tarawa consists of a string of islets from Buariki in the north to Buota in the south. The islets are separated in places by wide channels that are best crossed at low tide, and there is a ferry service between Buota and Abatao. Only Buota is connected by road to South Tarawa, via a bridge.  On South Tarawa, the construction of causeways has now created a single strip of land from Betio in the west to Tanaea in the northeast.

Climate.

Tarawa features a tropical rainforest climate under the Köppen climate classification. The climate is pleasant from April to October, with predominant northeastern winds and stable temperatures close to. From November to March, western gales bring rain and occasional cyclones.  Precipitation varies significantly between islands. For example, the annual average is 3,000 mm (120 in) in the north and 500 mm (20 in) in the south of the Gilbert Islands. Most of these islands are in the dry belt of the equatorial oceanic climatic zone and experience prolonged droughts.

Administration.

Tarawa atoll has three administrative subdivisions: Betio Town Council (or BTC), on Betio Islet; Teinainano Urban Council (or TUC), from Bairiki to Tanaea; and Eutan Tarawa Council (or ETC), for North Tarawa or "Tarawa Ieta", consisting of all the islets on the east side from Buota northwards. The meaning of Teinainano is "down of the mast", alluding to the sail-shape of the atoll.  South Tarawa hosts the capital of the Republic of Kiribati. The House of Assembly is in Ambo, and the State House is in Bairiki. The offices of the various ministries of the government range from Betio at the south-west extreme to Nawerewere (at the near-centre of North Tarawa in an easterly island in its chain). Settlements on North Tarawa include Marenanuka and Taborio.

Diplomatic missions.

Four resident diplomatic missions exist: the embassies of China and Cuba, and the high commissions of Australia and New Zealand.

History.

In Kiribati mythology, Tarawa was the earth when the land, ocean and sky had not been cleaved yet by Nareau the spider. Thus after calling the sky "karawa" and the ocean "marawa", he called the piece of rock that "Riiki" (another god that Nareau found) had stood upon when he lifted up the sky as, "Tarawa". Nareau then created the rest of the islands in Kiribati and also Samoa.  People arrived on these islands thousands of years ago, and there have been migrations to and from Kiribati since antiquity.  Evidence from a range of sources, including carbon dating and DNA analyses, confirms that the exploration of the Pacific included settlement of the Gilbert Islands by around 200 BC. The people of Kiribati are still excellent seafarers, capable of making ocean crossings in locally made vessels using traditional navigation techniques.  Thomas Gilbert, captain of the East India Company vessel, was the first European to describe Tarawa, arriving on 20 June 1788. He named it Matthew Island, after the owner of his ship "Charlott"e. He named the lagoon, Charlotte Bay. Gilbert's 1788 sketches survive.  The island was surveyed in 1841 by the US Exploring Expedition.  Tarawa Post Office opened on 1 January 1911.  Sir Arthur Grimble was a cadet administrative officer based at Tarawa (1913–1919) and became Resident Commissioner of the Gilbert and Ellice Islands colony in 1926. During World War II, Tarawa was occupied by the Japanese, and beginning on 20 November 1943 it was the scene of the bloody Battle of Tarawa. On that day United States Marines landed on Tarawa and fought Japanese soldiers occupying entrenched positions on the atoll. The Marines captured the island after 76 hours of intense fighting that killed 6,000 people on both sides.  The fierce fighting was the subject of a documentary film produced by the Combat Photographers of the Second Marine Division entitled "With the Marines at Tarawa". It was released in March 1944 at the insistence of President Roosevelt. It became the first time many Americans viewed American servicemen dead on film.  The Kiribati Government commenced a road restoration project funded in part by the World Bank in 2014 to surface the main road between Betio in the West to Bonriki in the East, upgrading the main road that transits Tarawa from a dirt road. As of 2018, all that remained to be completed of this project was the sealing of Betio Causeway, connecting Bairiki and Betio.

Tarawa is an atoll and the capital of the Republic of Kiribati, in the central Pacific Ocean. It comprises North Tarawa, which has much in common with other, more remote islands of the Gilberts group; and South Tarawa, which is home to 56,284 people – half of the country's total population. The atoll was the site of the Battle of Tarawa during World War II.

Tarawa est un atoll de la république des Kiribati, où se trouve la capitale, Tarawa-Sud (TUC), mais aussi Betio (BTC) et Tarawa-Nord (ETC) (). Sa population compte près de soit plus des deux tiers des Gilbertins.


History.

Origins.

Like many European races, the Monaco Grand Prix predates the current World Championship. The principality's first Grand Prix was organised in 1929 by Antony Noghès, under the auspices of Prince Louis II, through the Automobile Club de Monaco (ACM), of which he was president. The ACM organised the Rallye Automobile Monte Carlo, and in 1928 applied to the "Association Internationale des Automobiles Clubs Reconnus" (AIACR), the international governing body of motorsport, to be upgraded from a regional French club to full national status. Their application was refused due to the lack of a major motorsport event held wholly within Monaco's boundaries. The rally could not be considered as it mostly used the roads of other European countries.  To attain full national status, Noghès proposed the creation of an automobile Grand Prix in the streets of Monte Carlo. He obtained the official sanction of Prince Louis II, and the support of Monégasque "Grand Prix" driver Louis Chiron. Chiron thought Monaco's topography well-suited to setting up a race track.  The first race, held on 14 April 1929, was won by William Grover-Williams (using the pseudonym "Williams"), driving a works Bugatti Type 35B. It was an invitation-only event, but not all of those invited decided to attend. The leading Maserati and Alfa Romeo drivers decided not to compete, but Bugatti was well represented. Mercedes sent their leading driver, Rudolf Caracciola. Starting fifteenth, Caracciola drove a fighting race, taking his SSK into the lead before wasting 41⁄2 minutes on refuelling and a tyre change to finish second. Another driver who competed using a pseudonym was "Georges Philippe", the Baron Philippe de Rothschild. Chiron was unable to compete, having a prior commitment to compete in the Indianapolis 500 on the same day.  Caracciola's SSK was refused permission to race the following year, but Chiron did compete (in the works Bugatti Type 35C), when he was beaten by privateer René Dreyfus and his Bugatti Type 35B, and finished second. Chiron took victory in the 1931 race driving a Bugatti., he remains the only native of Monaco to have won the event.

Pre-war.

The race quickly grew in importance after its inception. Because of the high number of races which were being termed 'Grands Prix', the AIACR formally recognised the most important race of each of its affiliated national automobile clubs as International Grands Prix, or "Grandes Épreuves", and in 1933 Monaco was ranked as such alongside the French, Belgian, Italian, and Spanish Grands Prix. That year's race was the first Grand Prix in which grid positions were decided, as they are now, by practice time rather than the established method of balloting. The race saw Achille Varzi and Tazio Nuvolari exchange the lead many times before the race being settled in Varzi's favour on the final lap when Nuvolari's car caught fire.  The race became a round of the new European Championship in 1936, when stormy weather and a broken oil line led to a series of crashes, eliminating the Mercedes-Benzes of Chiron, Fagioli, and von Brauchitsch, as well as Bernd Rosemeyer's "Typ C" for newcomer Auto Union; Rudolf Caracciola, proving the truth of his nickname, "Regenmeister" (Rainmaster), went on to win. In 1937, von Brauchitsch duelled Caracciola before coming out on top. It was the last prewar "Grand Prix" at Monaco, for in 1938, the demand for £500 (about US$2450) in appearance money per top entrant led AIACR to cancel the event, while looming war overtook it in 1939, and the Second World War ended organised racing in Europe until 1945.

Post-war Grand Prix.

Racing in Europe started again on 9 September 1945 at the Bois de Boulogne Park in the city of Paris, four months and one day after the end of the war in Europe. However, the Monaco Grand Prix was not run between 1945 and 1947 due to financial reasons. In 1946 a new premier racing category, Grand Prix, was defined by the Fédération Internationale de l'Automobile (FIA), the successor of the AIACR, based on the pre-war voiturette class. A Monaco Grand Prix was run to this formula in 1948, won by the future world champion Nino Farina in a Maserati 4CLT.

Formula One.

Early championship days.

The 1949 event was cancelled due to the death of Prince Louis II; it was included in the new Formula One World Drivers' Championship the following year. The race provided future five-time world champion Juan Manuel Fangio with his first win in a World Championship race, as well as third place for the 51-year-old Louis Chiron, his best result in the World Championship era. However, there was no race in 1951. In 1952, the first of the two years in which the World Drivers' Championship was run to less powerful Formula Two regulations, the race was run to sports car rules instead, and it did not form part of the World Championship.  No races were held in 1953 or 1954.  Since 1955, the Monaco Grand Prix has continuously been part of the Formula One World Championship. That year, Maurice Trintignant won in Monte Carlo for the first time and Chiron again scored points and at 56 became the oldest driver to compete in a Formula One Grand Prix. It was not until 1957, when Fangio won again, that the Grand Prix saw a double winner. Between 1954 and 1961 Fangio's former Mercedes colleague, Stirling Moss, went one better, as did Trintignant, who won the race again in 1958 driving a Cooper. The 1961 race saw Moss fend off three works Ferrari 156s in a year-old privateer Rob Walker Racing Team Lotus 18, to take his third Monaco victory.

Graham Hill's era.

Britain's Graham Hill won the race five times in the 1960s and became known as ""King of Monaco"" and ""Mr. Monaco"". He first won in 1963, and then won the next two years. In the 1965 race he took pole position and led from the start, but went up an escape road on lap 25 to avoid hitting a slow backmarker. Re-joining in fifth place, Hill set several new lap records on the way to winning. The race was also notable for Jim Clark's absence (he was doing the Indianapolis 500), and for Paul Hawkins's Lotus ending up in the harbour. Hill's teammate, Briton Jackie Stewart, won in 1966 and New Zealander Denny Hulme won in 1967, but Hill won the next two years, the 1969 event being his final Formula One championship victory, by which time he was a double Formula One world champion.

Track alterations, safety, and increasing business interests.

By the start of the 1970s, efforts by Jackie Stewart saw several Formula One events cancelled because of safety concerns. For the 1969 event, Armco barriers were placed at specific points for the first time in the circuit's history. Before that, the circuit's conditions were (aside from the removal of people's production cars parked on the side of the road) virtually identical to everyday road use. If a driver went off, he had a chance to crash into whatever was next to the track (buildings, trees, lamp posts, glass windows, and even a train station), and in Alberto Ascari's and Paul Hawkins's cases, the harbour water, because the concrete road the course used had no Armco to protect the drivers from going off the track and into the Mediterranean. The circuit gained more Armco in specific points for the next two races, and by 1972, the circuit was almost completely Armco-lined. For the first time in its history, the Monaco circuit was altered in 1972 as the pits were moved next to the waterfront straight between the chicane and Tabac and the chicane was moved further forward right before Tabac becoming the junction point between the pits and the course. The course was changed again for the 1973 race. The Rainier III Nautical Stadium was constructed where the straight that went behind the pits was and the circuit introduced a double chicane that went around the new swimming pool (this chicane complex is known today as "Swimming Pool"). This created space for a whole new pit facility and in 1976 the course was altered yet again; the Sainte Devote corner was made slower and a chicane was placed right before the pit straight.  By the early 1970s, as Brabham team owner Bernie Ecclestone started to marshal the collective bargaining power of the Formula One Constructors Association (FOCA), Monaco was prestigious enough to become an early bone of contention. Historically the number of cars permitted in a race was decided by the race organiser, in this case the ACM, which had always set a low number of around 16. In 1972 Ecclestone started to negotiate deals which relied on FOCA guaranteeing at least 18 entrants for every race. A stand-off over this issue left the 1972 race in jeopardy until the ACM gave in and agreed that 26 cars could participate – the same number permitted at most other circuits. Two years later, in 1974, the ACM got the numbers back down to 18.  Because of its tight confines, slow average speeds and punishing nature, Monaco has often thrown up unexpected results. In the 1982 race René Arnoux led the first 15 laps, before retiring. Alain Prost then led until four laps from the end, when he spun off on the wet track, hit the barriers and lost a wheel, giving Riccardo Patrese the lead. Patrese himself spun with only a lap and a half to go, letting Didier Pironi through to the front, followed by Andrea de Cesaris. On the last lap, Pironi ran out of fuel in the tunnel, but De Cesaris also ran out of fuel before he could overtake. In the meantime, Patrese had bump-started his car and went through to score his first Grand Prix win.  In 1983 the ACM became entangled in the disagreements between Fédération Internationale du Sport Automobile (FISA) and FOCA. The ACM, with the agreement of Bernie Ecclestone, negotiated an individual television rights deal with ABC in the United States. This broke an agreement enforced by FISA for a single central negotiation of television rights. Jean-Marie Balestre, president of FISA, announced that the Monaco Grand Prix would not form part of the Formula One world championship in 1985. The ACM fought their case in the French courts. They won the case and the race was eventually reinstated.

Prost/Senna era.

For the decade from 1984 to 1993 the race was won by only two drivers, arguably the two best drivers in Formula One at the time – Frenchman Alain Prost and Brazilian Ayrton Senna. Prost, already a winner of the support race for Formula Three cars in 1979, took his first Monaco win at the 1984 race. The race started 45 minutes late after heavy rain. Prost led briefly before Nigel Mansell overtook him on lap 11. Mansell crashed out five laps later, letting Prost back into the lead. On lap 27, Prost led from Ayrton Senna's Toleman and Stefan Bellof's Tyrrell. Senna was catching Prost and Bellof was catching both of them in the only naturally aspirated car in the race. However, on lap 31, the race was controversially stopped with conditions deemed to be undriveable. Later, FISA fined the clerk of the course, Jacky Ickx, $6,000 and suspended his licence for not consulting the stewards before stopping the race. The drivers received only half of the points that would usually be awarded, as the race had been stopped before two-thirds of the intended race distance had been completed.  Prost won 1985 after polesitter Senna retired with a blown Renault engine in his Lotus after over-revving it at the start, and Michele Alboreto in the Ferrari retook the lead twice, but he went off the track at Sainte-Devote, where Brazilian Nelson Piquet and Italian Riccardo Patrese had a huge accident only a few laps previously and oil and debris littered the track. Prost passed Alboreto, who retook the Frenchman, and then he punctured a tyre after running over bodywork debris from the Piquet/Patrese accident, which dropped him to 4th. He was able to pass his Roman countrymen Andrea De Cesaris and Elio de Angelis, but finished 2nd behind Prost. The French Prost dominated 1986 after starting from pole position, a race where the Nouvelle Chicane had been changed on the grounds of safety.  Senna holds the record for the most victories in Monaco, with six, including five consecutive wins between 1989 and 1993, as well as eight podium finishes in ten starts. His 1987 win was the first time a car with an active suspension had won a Grand Prix. He won this race after Briton Nigel Mansell in a Williams-Honda went out with a broken exhaust. His win was very popular with the people of Monaco, and when he was arrested on the Monday following the race, for riding a motorcycle without wearing a helmet, he was released by the officers after they realised who he was. Senna dominated 1988, and was able to get ahead of his teammate Prost while the Frenchman was held up for most of the race by Austrian Gerhard Berger in a Ferrari. By the time Prost got past Berger, he pushed as hard as he could and set a lap some 6 seconds faster than Senna's; Senna then set 2 fastest laps, and while pushing as hard as possible, he touched the barrier at the Portier corner and crashed into the Armco separating the road from the Mediterranean. Senna was so upset that he went back to his Monaco flat and was not heard from until the evening. Prost went on to win for the fourth time.  Senna dominated 1989 while Prost was stuck behind backmarker Rene Arnoux and others; the Brazilian also dominated 1990 and 1991. At the 1992 event Nigel Mansell, who had won all five races held to that point in the season, took pole and dominated the race in his Williams FW14B-Renault. However, with seven laps remaining, Mansell suffered a loose wheel nut and was forced into the pits, emerging behind Ayrton Senna's McLaren-Honda, who was on worn tyres. Mansell, on fresh tyres, set a lap record almost two seconds quicker than Senna's and closed from 5.2 to 1.9 seconds in only two laps. The pair duelled around Monaco for the final four laps but Mansell could find no way past, finishing just two-tenths of a second behind the Brazilian. It was Senna's fifth win at Monaco, equalling Graham Hill's record. Senna had a poor start to the 1993 event, crashing in practice and qualifying 3rd behind pole-sitter Prost and the rising German star Michael Schumacher. Both of them beat Senna to the first corner, but Prost had to serve a time penalty for jumping the start and Schumacher retired after suspension problems, so Senna took his sixth win to break Graham Hill's record for most wins at the Monaco Grand Prix. Runner-up Damon Hill commented, "If my father was around now, he would be the first to congratulate Ayrton."

Modern times.

The 1994 race was an emotional and tragic affair. It came two weeks after the race at Imola in which Austrian Roland Ratzenberger and Ayrton Senna both died from massive head injuries from on-track accidents on successive days. During the Monaco event, Austrian Karl Wendlinger had an accident in his Sauber in the tunnel; he went into a coma and was to miss the rest of the season. The German Michael Schumacher won the 1994 Monaco event. The 1996 race saw Michael Schumacher take pole position before crashing out on the first lap after being overtaken by Damon Hill. Hill led the first 40 laps before his engine expired in the tunnel. Jean Alesi took the lead but suffered suspension failure 20 laps later. Olivier Panis, who started in 14th place, moved into the lead and stayed there until the end of the race, being pushed all the way by David Coulthard. It was Panis's only win, and the last for his Ligier team. Only three cars crossed the finish line, but seven were classified.  Seven-time world champion Schumacher would eventually win the race five times, matching Graham Hill's record. In his appearance at the 2006 event, he attracted criticism when, while provisionally holding pole position and with the qualifying session drawing to a close, he stopped his car at the Rascasse hairpin, blocking the track and obliging competitors to slow down. Although Schumacher claimed it was the unintentional result of a genuine car failure, the FIA disagreed and he was sent to the back of the grid.  In July 2010, Bernie Ecclestone announced that a 10-year deal had been reached with the race organisers, keeping the race on the calendar until at least 2020.  Due to the COVID-19 pandemic, the FIA announced the 2020 Monaco Grand Prix's postponement, along with the two other races scheduled for May 2020, to help prevent the spread of the virus. However, later the same day the Automobile Club de Monaco confirmed that the Grand Prix was instead cancelled, making 2020 the first time the Grand Prix was not run since 1954.

Circuit.

The Circuit de Monaco consists of the city streets of Monte Carlo and La Condamine, which includes the famous harbour. It is unique in having been held on the same circuit every time it has been run over such a long period – only the Italian Grand Prix, which has been held at Autodromo Nazionale Monza during every Formula One regulated year except 1980, has a similarly lengthy and close relationship with a single circuit.  The race circuit has many elevation changes, tight corners, and a narrow course that makes it one of the most demanding tracks in Formula One racing., two drivers have crashed and ended up in the harbour, the most famous being Alberto Ascari in 1955. Despite the fact that the course has had minor changes several times during its history, it is still considered the ultimate test of driving skills in Formula One, and if it were not already an existing Grand Prix, it would not be permitted to be added to the schedule for safety reasons. Even in 1929, 'La Vie Automobile' magazine offered the opinion that "Any respectable traffic system would have covered the track with «Danger» sign posts left, right and centre".  Triple Formula One champion Nelson Piquet was fond of saying that racing at Monaco was "like trying to cycle round your living room", but added that "a win here was worth two anywhere else".  Notably, the course includes a tunnel. The contrast of daylight and gloom when entering/exiting the tunnel presents "challenges not faced elsewhere", as the drivers have to "adjust their vision as they emerge from the tunnel at the fastest point of the track and brake for the chicane in the daylight.".  The fastest-ever qualifying lap was set by Lewis Hamilton in qualifying (Q3) for the 2019 Monaco Grand Prix, at a time of 1m 10.166s.

Viewing areas.

During the Grand Prix weekend, spectators crowd around the Monaco Circuit. There are a number of temporary grandstands built around the circuit, mostly around the harbour area. The rich and famous spectators often arrive on their boats and the yachts through the harbour. Balconies around Monaco become viewing areas for the race as well. Many hotels and residents cash in on the bird's eye views of the race.

Organization.

The Monaco Grand Prix is organised each year by the "Automobile Club de Monaco" which also runs the Monte Carlo Rally and the Junior Monaco Kart Cup.  The Monaco Grand Prix differs in several ways from other Grands Prix. The practice session for the race is held on the Thursday preceding the race instead of Friday. This allows the streets to be opened to the public again on Friday. Until the late 1990s the race started at 3:30 p.m. local time – an hour and a half later than other European Formula One races. In recent years the race has fallen in line with the other Formula One races for the convenience of television viewers. Also, earlier the event was traditionally held on the week of Ascension Day. It is now always held on the last weekend in May. For many years, the numbers of cars admitted to Grands Prix was at the discretion of the race organisers – Monaco had the smallest grids, ostensibly because of its narrow and twisting track. Only 18 cars were permitted to enter the 1975 Monaco Grand Prix, compared to 23 to 26 cars at all other rounds that year.  The erecting of the circuit takes six weeks, and the removal after the race takes three weeks. There was no podium as such at the race, until 2017. Instead, a section of the track was closed after the race to act as parc fermé, a place where the cars are held for official inspection. The first three drivers in the race left their cars there and walked directly to the royal box where the 'podium' ceremony was held, which was considered a custom for the race. The trophies were handed out before the national anthems for the winning driver and team are played, as opposed to other Grands Prix where the anthems are played first.

Fame.

The Monaco Grand Prix is widely considered to be one of the most important and prestigious automobile races in the world alongside the Indianapolis 500 and the 24 Hours of Le Mans. These three races are considered to form a "Triple Crown" of the three most famous motor races in the world. As of 2020, Graham Hill is the only driver to have won the Triple Crown, by winning all three races. The practice session for Monaco overlaps with that for the Indianapolis 500, and the races themselves sometimes clash. As the two races take place on opposite sides of the Atlantic Ocean and form part of different championships, it is difficult for one driver to compete effectively in both during his career. Juan Pablo Montoya and Fernando Alonso are the only active drivers to have won two of the three events.  In awarding its first Gold medal for motorsport to Prince Rainier III, the Fédération Internationale de l'Automobile (FIA) characterised the Monaco Grand Prix as contributing "an exceptional location of glamour and prestige" to motorsport. The Grand Prix has been run under the patronage of three generations of Monaco's royal family: Louis II, Rainier III and Albert II, all of whom have taken a close interest in the race. A large part of the principality's income comes from tourists attracted by the warm climate and the famous casino, but it is also a tax haven and is home to many millionaires, including several Formula One drivers.  Monaco has produced four native Formula One drivers - Louis Chiron, André Testut, Olivier Beretta, and Charles Leclerc - but its tax status has made it home to many drivers over the years, including Gilles Villeneuve and Ayrton Senna. Of the Formula One contenders, several have property in the principality, including Jenson Button and David Coulthard, who was part owner of a hotel there. Because of the small size of the town and the location of the circuit, drivers whose races end early can usually get back to their apartments in minutes. Ayrton Senna famously retired to his apartment after crashing out of the lead of the 1988 race.

Official names.

Unlike other venues, the Monaco Grand Prix has never been sponsored since its first race in 1950. It however, has had multiple names it goes by, namely the following:

Winners.

Repeat winners (drivers).

"Drivers in bold are competing in the Formula One championship in the current season."

Repeat winners (constructors).

"Teams in bold are competing in the Formula One championship in the current season."
 "A pink background indicates an event which was not part of the Formula One World Championship."
 "A yellow background indicates an event which was part of the pre-war European Championship."

Repeat winners (engine manufacturers).

"Manufacturers in bold are competing in the Formula One championship in the current season."
 "A pink background indicates an event which was not part of the Formula One World Championship."
 "A yellow background indicates an event which was part of the pre-war European Championship." <nowiki>*</nowiki> Between 1998-2005 built by Ilmor, funded by Mercedes  <nowiki>**</nowiki> Built by Cosworth, funded by Ford  <nowiki>***</nowiki> Built by Porsche

By year.

"A pink background indicates an event which was not part of the Formula One World Championship."<br> "A yellow background indicates an event which was part of the pre-war European Championship."

The Monaco Grand Prix () is a Formula One motor race held annually on the Circuit de Monaco on the last weekend in May. Run since 1929, it is widely considered to be one of the most important and prestigious automobile races in the world, and is one of the races—along with the Indianapolis 500 and the 24 Hours of Le Mans—that form the Triple Crown of Motorsport. The circuit has been called "an exceptional location of glamour and prestige".

Le Grand Prix de Monaco est une des plus anciennes et l'une des trois courses les plus prestigieuses au monde, disputée en principauté de Monaco, sur un circuit urbain conçu en 1929, par Antony Noghès fils du président de l'Automobile Club de Monaco, sous les auspices du prince Louis II de Monaco. Cette création répondait au défi d'organiser une compétition sur le territoire exigu de Monaco (environ à l'époque), condition requise par la Commission Sportive Internationale pour que l'Automobile Club de Monaco soit reconnu internationalement. En effet, l'autre grande compétition monégasque, le rallye automobile Monte-Carlo, créé dès 1911, était disputé sur les routes de nombreux pays d'Europe.


Nomenclature.

Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see below).  In an organizational context, the following terms are often used related to hierarchies:  In a mathematical context (in graph theory), the general terminology used is different.  Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structures, objects are known as nodes, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss and a peer is a colleague.

Degree of branching.

Degree of branching refers to the number of direct subordinates or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: "linear" and "branching".  In a linear hierarchy, the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. Note that this is referring to the "objects" and not the "levels"; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects. An example of a linear hierarchy is the hierarchy of life.  In a branching hierarchy, one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher). For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including organizations and classification schemes. The broad category of branching hierarchies can be further subdivided based on the degree.  A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example, diamonds and graphite are flat hierarchies of numerous carbon atoms that can be further decomposed into subatomic particles.  An overlapping hierarchy is a branching hierarchy in which at least one object has two parent objects. For example, a graduate student can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the university hierarchy (i.e., they have the same position or tenure status).

History of the term.

Possibly the first use of the English word "hierarchy" cited by the "Oxford English Dictionary" was in 1881, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite (5th–6th centuries). Pseudo-Dionysius used the related Greek word (ἱεραρχία "hierarchia") both in reference to the celestial hierarchy and the ecclesiastical hierarchy. The Greek term ἱεραρχία means "rule of a high priest" (from ἱεράρχης "hierarches", meaning "president of sacred rites, high-priest" and that from ἱερεύς "hiereus", "priest" and ἀρχή "arche", amongst others "first place or power, rule"), and Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings.

Visually representing hierarchies.

A hierarchy is typically depicted as a pyramid, where the height of a level represents that level's status and width of a level represents the quantity of items at that level relative to the whole. For example, the few Directors of a company could be at the apex, and the base could be thousands of people who have no subordinates.  These pyramids are typically diagrammed with a tree or triangle diagram (but note that not all triangle/pyramid diagrams are hierarchical; for example, the 1992 USDA food guide pyramid), both of which serve to emphasize the size differences between the levels. An example of a triangle diagram appears to the right. An organizational chart is the diagram of a hierarchy within an organization, and is depicted in tree form in, below.  More recently, as computers have allowed the storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer's screen. Examples include fractal maps, TreeMaps and Radial Trees.

Visual hierarchy.

In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on the rules of visual hierarchy. Visual hierarchy is also important for proper organization of files on computers.  An example of visually representing hierarchy is through the Nested clusters. The Nested clusters represents hierarchical relationships by using layers of information. The child element is within the parent element, such as in a Venn diagram. This structure of representing hierarchy is most effective in representing simple relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located.  For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at the left and descending on the right. The child elements are towards the bottom of the stairs and the parent elements are at the top. This structure is effective when representing more complicated hierarchies where steps are not placed in obvious sequences. Further steps are concealed unless all of the steps are revealed in sequence. In the computer desktop example, a file that is being sought after can only be found once another file is opened. The link for the desired file is within another document. All the steps must be completed until the final destination is reached.

Informal representation.

In plain English, a hierarchy can be thought of as a set in which: The first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always transitive. The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.

Mathematical representation.

Mathematically, in its most general form, a hierarchy is a partially ordered set or "poset". The system in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resulting levels is referred to as a class.  "Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parenthesis is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example: (2 + 5) × (7 - 4). In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.

Subtypes.

Nested hierarchy.

A nested hierarchy or "inclusion hierarchy" is a hierarchical ordering of nested sets. The concept of nesting is exemplified in Russian matryoshka dolls. Each doll is encompassed by another doll, all the way to the outer doll. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll; a generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. The general concept is both demonstrated and mathematically formulated in the following example:  A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can "only" be a quadrilateral; it can never be a triangle, hexagon, etc.  Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy (the version he laid out in the 10th edition of "Systema Naturae"), a human can be formulated as:  Taxonomies may change frequently (as seen in biological taxonomy), but the underlying concept of nested hierarchies is always the same.  In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of self-similarity and recursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.

Containment hierarchy.

A containment hierarchy is a direct extrapolation of the nested hierarchy concept. All of the ordered sets are still nested, but every set must be "strict"—no two sets can be identical. The shapes example above can be modified to demonstrate this:  The notation formula_4 means "x" is a subset of "y" but is not equal to "y".  A general example of a containment hierarchy is demonstrated in class inheritance in object-oriented programming.  Two types of containment hierarchies are the "subsumptive" containment hierarchy and the "compositional" containment hierarchy. A subsumptive hierarchy "subsumes" its children, and a compositional hierarchy is "composed" of its children. A hierarchy can also be both subsumptive "and" compositional.

Subsumptive containment hierarchy.

A "subsumptive" containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "IS-A hierarchy". The last term describes the relationship between each level—a lower-level object "is a" member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that is part of the level "Mammalia" "is a" member of the level "Animalia"; more specifically, a human "is a" primate, a primate "is a" mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of "concepts". For example, with the Linnaean hierarchy outlined above, an entity name like "Animalia" is a way to group all the species that fit the conceptualization of an animal.

Compositional containment hierarchy.

A "compositional" containment hierarchy is an ordering of the parts that make up a system—the system is "composed" of these parts. Most engineered structures, whether natural or artificial, can be broken down in this manner.  The compositional hierarchy that every person encounters at every moment is the hierarchy of life. Every person can be reduced to organ systems, which are composed of organs, which are composed of tissues, which are composed of cells, which are composed of molecules, which are composed of atoms. In fact, the last two levels apply to all matter, at least at the macroscopic scale. Moreover, each of these levels inherit all the properties of their children.  In this particular example, there are also "emergent properties"—functions that are not seen at the lower level (e.g., cognition is not a property of neurons but is of the brain)—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not a required general property. These "level hierarchies" are characterized by bi-directional causation. "Upward causation" involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children. "Downward causation" refers to the effect that the incorporation of entity "x" into a higher-level entity can have on "x"'s properties and interactions. Furthermore, the entities found at each level are "autonomous".

Contexts and applications.

According to Kulish, V. V. (2002), almost every system of organization applied to the world is arranged hierarchically. By their common definitions, every nation has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification of societies), and all systematic classification schemes (taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God. Many Christian denominations have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree) and inheritance (succession and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs. Learning must often follow a hierarchical scheme—to learn differential equations one must first learn calculus; to learn calculus one must first learn elementary algebra; and so on. Even nature itself has its own hierarchies, as numerous schemes such as Linnaean taxonomy, the organization of life, and biomass pyramids attempt to document. Hierarchies are so infused into daily life that they are viewed as trivial.  While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal code systems are hierarchical. Using the Canadian postal code system as an example, the top level's binding concept is the "postal district", and consists of 18 objects (letters). The next level down is the "zone", where the objects are the digits 0–9. This is an example of an overlapping hierarchy, because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format "A0A 0A0" (the second and third letter position allow 20 objects each). Most library classification systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.

Organizations.

Organizations can be structured as a dominance hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including governments, companies, militia and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart.  In a reverse hierarchy, the conceptual pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings.

Life.

Empirically, we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity. System hierarchies analysis performed in the 1950s, laid the empirical foundations for a field that would be, from the 1980s, hierarchical ecology.  The theoretical foundations are summarized by Thermodynamics. When biological systems are modeled as physical systems, in its most general abstraction, they are thermodynamic open systems that exhibit self-organised behavior, and the set/subset relations between dissipative structures can be characterized in a hierarchy.

Computer graphic imaging.

CGI and computer animation programs mostly use hierarchies for models. On a 3D model of a human for example, the chest is a parent of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand. This is used in modeling and animation for almost everything built as a 3D digital model.

Linguistics.

Many grammatical theories, such as phrase-structure grammar, involve hierarchy.  Direct–inverse languages such as Cree and Mapudungun distinguish subject and object on verbs not by different subject and object markers, but via a hierarchy of persons.  In this system, the three (or four with Algonquian languages) persons are placed in a hierarchy of salience. To distinguish which is subject and which object, "inverse markers" are used if the object outranks the subject.  On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun phrase to which it refers, commonly crosses grammatical boundaries in non-hierarchical ways.

Music.

The structure of a musical composition is often understood hierarchically (for example by Heinrich Schenker (1768–1835, see Schenkerian analysis), and in the (1985) Generative Theory of Tonal Music, by composer Fred Lerdahl and linguist Ray Jackendoff). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are the foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background or Ursatz, which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature.  The pitches and form of tonal music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic key, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her book "Feminine Endings", even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."

Criticisms of views, concerning distinctions of type and categories well as distinguishability.

In the work of diverse theorists such as William James (1842–1910), Michel Foucault (1926–1984) and Hayden White, important critiques of hierarchical epistemology are advanced. James famously asserts in his work "Radical Empiricism" that clear distinctions of type and category are a constant but unwritten goal of scientific reasoning, so that when they are discovered, success is declared. But if aspects of the world are organized differently, involving inherent and intractable ambiguities, then scientific questions are often considered unresolved.  Hierarchy in ethics emerged in Western Europe, West Asia and North Africa around the 1600s. In this aspect, the term hierarchy refers to how distinguishable they are from real to unreal. Feminists, Marxists, anarchists, communists, critical theorists and others, all of whom have multiple interpretations, criticize the hierarchies commonly found within human society, especially in social relationships. Hierarchies are present in all parts of society: in businesses, schools, families, etc. These relationships are often viewed as necessary. Entities that stand in hierarchical arrangements are animals, humans, plants, etc.

Ethics, behavioral psychology, philosophies of identity.

In ethics, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory.  In some of these random examples, there is an asymmetry of 'compositional' significance between levels of structure, so that small parts of the whole hierarchical array depend, for their meaning, on their membership in larger parts. There is a hierarchy of activities in human life: productive activity serves or is guided by the moral life; the moral life is guided by practical reason; practical reason (used in moral and political life) serves contemplative reason (whereby we contemplate God). Practical reason sets aside time and resources for contemplative reason.

See also.

Strucure-related concepts.

"(For example, in )"

A hierarchy (from the Greek ἱεραρχία "hierarkhia", "rule of a high priest", from "hierarkhes", "president of sacred rites") is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as philosophy, mathematics, computer science, organizational theory, systems theory, and the social sciences (especially political philosophy).

Le concept de hiérarchie tiré des vocables grec "hieros" (« sacré ») et "archos" (« commencement », ou « ce qui est premier ») ou plus certainement "arkhê" (« pouvoir », ou « commandement ») s'applique à plusieurs domaines, physiques ou moraux.


Life and career.

After his high class studies at the Neuilly Lycée Pasteur—though asserted here and there, he never studied at "Institut d'Études Politiques de Paris (Sciences Po") —he started his actor career with the comedic theater "troupe" Splendid, which had hits with films like "Les Bronzés font du ski" and "Le Père Noël est une ordure".  His most notable success without the Splendid group, and by far his biggest hit to date, was in the 1993 film "les Visiteurs", where he played a character known as Jacquouille la Fripouille; the character's cry of "Okkkayyy!!" became a popular exclamation after the movie's success.  After "les Visiteurs" he was a star, participating in big-budget films like "Astérix et Obélix contre César", "", and the sequel and remake of "Visiteurs". He also played several dramatic roles on television, including M. Thénardier in "Les Misérables" (2000 television version) and Napoléon in a biographical television film.  Clavier has played in notable duos with:  He also runs a production company, Ouille Production.  He was made "Chevalier" (Knight) of the Ordre national du Mérite on 13 June 1998, and promoted "Officier" (Officer) in 2005. He was made "Chevalier" (Knight) of the Légion d'honneur in 2008.  He is a friend of former French président Nicolas Sarkozy, and in 2012 emigrated to The United Kingdom. He denies this was as a result of Francois Hollande's punitive tax policy.

Filmography.

Box-office.

Movies starring Christian Clavier with more than a million of admissions in France.

Christian Jean-Marie Clavier (; born 6 May 1952) is a French actor, screenwriter, film producer and director. He became widely popular after starring in two hit comedy series: Patrice Leconte's "Les Bronzés" and "Les Visiteurs" directed by Jean-Marie Poiré. He furthered his popularity by taking a role of Asterix in the screen adaptations of the comic books by Albert Uderzo and René Goscinny.

Christian Clavier est un acteur, scénariste et producteur français, né le à Paris. Il débute avec la troupe du Splendid à la fin des années 1970 et s'impose avec ses compagnons dans des films restés célèbres, tels que "Les Bronzés" (1978), "Les Bronzés font du ski" (1979), "Le père Noël est une ordure" (1982) ou "Papy fait de la résistance" (1983).


Definition.

The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by. In physics, sometimes the notation is used, though this is avoided in mathematics to avoid confusion with the exterior product.  The cross product is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.  The cross product is defined by the formula  where "θ" is the angle between a and b in the plane containing them (hence, it is between 0° and 180°), ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule (illustrated). If the vectors a and b are parallel (i.e., the

Names.

In 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period () and an "x" (), respectively, to denote them.  In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names

Computing the cross product.

Coordinate notation.

The standard basis vectors i, j, and k satisfy the following equalities in a right hand coordinate system:  which imply, by the anticommutativity of the cross product, that  The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that  These equalities, together with the distributivity and linearity of the cross product (but both do not follow easily from the definition given above), are sufficient to determine the cross product of any two vectors a and

Matrix notation.

The cross product can also be expressed as the formal determinant:  This determinant can be computed using Sarrus's rule

Properties.

Geometric meaning.

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides (see Figure 1):  Indeed, one can also compute the volume "V" of a parallelepiped having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2):  Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value. For instance,  Because the magnitude of the

Algebraic properties.

If the cross product of two vectors is the zero vector (i.e. ), then either one or both of the inputs is the zero vector, ( or ) or else they are parallel or antiparallel () so that the sine of the angle between them is zero ( or and ).  The self cross product of a vector is the zero vector: The cross product is anticommutative, distributive over addition, and compatible with scalar multiplication so that  It is not associative, but satisfies the Jacobi identity: Distributivity, linearity and Jacobi identity show that the R vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3). The cross product does not obey the cancellation

Differentiation.

The product rule of differential calculus applies to any bilinear operation, and therefore

Triple product expansion.

The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as  It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that's an even permutation of the above ordering. The following therefore are equal:  The vector

Alternative formulation.

The cross product and the dot product are related by:  The right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle

Lagrange's identity.

The relation:  can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as:  where a and b may be "n"-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case where, combining these two equations results

Infinitesimal generators of rotations.

The cross product conveniently describes the infinitesimal generators of rotations in R. Specifically, if n is a unit vector in R and "R"("φ", n) denotes a rotation about the axis through the origin specified by n, with angle φ (measured in radians,

Alternative ways to compute the cross product.

Conversion to matrix multiplication.

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:  where superscript refers to the transpose operation, and [a] is defined by:  The columns [a] of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors, i.e.:  or  where formula_67 is the outer product operator.  Also, if a is itself expressed as a cross product:  then  This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is

Index notation for tensors.

The cross product can alternatively be defined in terms of the Levi-Civita symbol "ε" and a dot product "η" (= δ for an orthonormal basis), which are useful in converting vector notation for tensor applications:  where the indices formula_74 correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as

Mnemonic.

The word "xyzzy" can be used to remember the definition of the cross product.  If  where:  then:  The second and third equations can be obtained from the first by simply vertically rotating the subscripts,. The problem, of course,

Cross visualization.

Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula.  If  then:  If we want to obtain the formula for formula_84 we simply drop the formula_85 and formula_86 from the formula, and take the next two components down:  When doing this for formula_88 the next two elements down should "wrap around" the matrix so that after the z component comes the x

Applications.

The cross product has applications in various contexts: e.g. it is used in computational geometry, physics and engineering. A non-exhaustive list of examples follows.

Computational geometry.

The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space.  The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.  In computational geometry of the

Angular momentum and torque.

The angular momentum formula_108 of a particle about a given origin is defined as:  where formula_110 is the position vector of the particle relative to the origin, formula_111 is the linear momentum of the particle.  In the same way, the

Rigid body.

The cross product frequently appears in the description of rigid motions. Two points "P" and "Q" on a rigid body can be related by:

Lorentz force.

The cross product is used to describe the Lorentz force experienced by a moving electric

Other.

In vector calculus, the cross product is used to define the formula for the vector operator curl.

Cross product as an external product.

The cross product can be defined in terms of the exterior product. In this context, it is an external product. This view allows for a natural geometric interpretation of the cross product. In exterior algebra the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors "a" and "b", one can view the bivector as the oriented parallelogram spanned by "a" and "b".

Cross product and handedness.

When measurable quantities involve cross products, the "handedness" of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is "not" fixed a priori, the result is not a (true) vector but a pseudovector. Therefore,

Generalizations.

There are several ways to generalize the cross product to the higher dimensions.

Lie algebra.

The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms

Quaternions.

The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on R. The unit vectors i, j, k correspond to "binary" (180 deg) rotations about their respective axes (Altmann, S. L., 1986, Ch. 12), said rotations being represented by "pure" quaternions (zero real part) with unit norms.  For instance,

Octonions.

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence

Exterior product.

In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As

External product.

As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. In any finite "n" dimensions, the Hodge dual

Commutator product.

Interpreting the three-dimensional vector space of the algebra as the 2-vector (not the 1-vector) subalgebra of the three-dimensional geometric algebra, where formula_138, formula_139, and formula_140, the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol formula_2. The commutator product is defined for 2-vectors formula_142 and formula_143 in geometric algebra as:  where formula_145 is the geometric product.  The commutator product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/true vectors) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator

Multilinear algebra.

In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index.  In detail, the 3-dimensional volume form defines a product formula_146 by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function formula_147 (fixing any two inputs gives a function formula_148 by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism formula_149 and thus this yields a map formula_150 which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".  Translating the above algebra into geometry, the function "volume of the parallelepiped defined by formula_151" (where the first two vectors are fixed and the last is an input), which defines a function formula_148, can be "represented" uniquely as the dot product with a vector: this vector is the cross product formula_153 From this perspective, the

Skew-symmetric matrix.

If the cross product is defined as a binary operation, it takes as "input" exactly two vectors. If its "output" is not required to be a vector or a pseudovector but instead a "matrix", then it can be generalized in an arbitrary number of dimensions.  In mechanics, for example, the angular velocity can be interpreted either as a pseudovector formula_175 or as an anti-symmetric matrix or skew-symmetric tensor

History.

In 1773, Joseph-Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Given two quaternions and, where u and v are vectors in R, their quaternion product can be summarized as. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.  In 1878 William Kingdon Clifford published his Elements of Dynamic which was an advanced text for its time. He defined the product of two vectors to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.  Oliver Heaviside and Josiah Willard Gibbs also felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20 to the four commonly seen today.  Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. In 1853 Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product. Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.  The cross notation and the name "cross product" began with Gibbs. Originally they appeared in privately published notes for his students in 1881 as "Elements of Vector Analysis". The utility for mechanics was noted by Aleksandr Kotelnikov. Gibbs's notation and the name "cross product" later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson, a former student. Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. He divided vector analysis into three parts: Two main kinds of vector multiplications were defined, and they were called as follows: Several kinds of triple products and products of more than three vectors were also examined. The above-mentioned triple product expansion was also included.

In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space formula_1 and is denoted by the symbol formula_2. Given two linearly independent vectors formula_3 and formula_4, the cross product, formula_5 (read "a cross b"), is a vector that is perpendicular to both formula_3 and formula_4 and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).

En mathématiques, et plus précisément en géométrie, le produit vectoriel est une opération vectorielle effectuée dans les espaces euclidiens orientés de dimension 3. Le formalisme utilisé actuellement est apparu en 1881 dans un manuel d'analyse vectorielle écrit par Josiah Willard Gibbs pour ses étudiants en physique. Les travaux de Hermann Günther Grassmann et William Rowan Hamilton sont à l'origine du produit vectoriel défini par Gibbs.


Plot.

A man (Davos Hanich) is a prisoner in the aftermath of World War III in post-apocalyptic Paris, where survivors live underground in the "Palais de Chaillot" galleries. Scientists research time travel, hoping to send test subjects to different time periods "to call past and future to the rescue of the present". They have difficulty finding subjects who can mentally withstand the shock of time travel. The scientists eventually settle upon the prisoner; his key to the past is a vague but obsessive memory from his pre-war childhood of a woman (Hélène Châtelain) he had seen on the observation platform ("the jetty") at Orly Airport shortly before witnessing a startling incident there. He did not understand exactly what happened, but knew he had seen a man die.  After several attempts, he reaches the pre-war period. He meets the woman from his memory, and they develop a romantic relationship. After his successful passages to the past, the experimenters attempt to send him into the far future. In a brief meeting with the technologically advanced people of the future, he is given a power unit sufficient to regenerate his own destroyed society.  Upon his return, with his mission accomplished, he discerns that he is to be executed by his jailers. He is contacted by the people of the future, who offer to help him escape to their time permanently; but he asks instead to be returned to the pre-war time of his childhood, hoping to find the woman again. He is returned to the past, placed on the jetty at the airport, and it occurs to him that the child version of himself is probably also there at the same time. He is more concerned with locating the woman, and quickly spots her. However, as he rushes to her, he notices an agent of his jailers who has followed him and realizes the agent is about to kill him. In his final moments, he comes to understand that the incident he witnessed as a child, which has haunted him ever since, was his own death.

Production.

"La Jetée" is constructed almost entirely from optically printed photographs playing out as a photomontage of varying rhythm. It contains only one brief shot (of the woman mentioned above sleeping and suddenly waking up) originating on a motion-picture camera, this due to the fact that Marker could only afford to hire one for an afternoon. The stills were taken with a Pentax Spotmatic and the motion-picture segment was shot with a 35 mm Arriflex. The film has no dialogue aside from small sections of muttering in German and people talking in an airport terminal. The story is told by a voice-over narrator. The scene in which the hero and the woman look at a cut-away trunk of a tree is a reference to Alfred Hitchcock's 1958 film "Vertigo" which Marker also references in his 1983 film "Sans soleil".  The editing of "La Jetée," adds to the intensity of the film. With the use of cut-ins and fade-outs, it produces the eerie and unsettling nature adding to the theme of the apocalyptic destruction of World World III. Director of "12 Monkeys", Terry Gilliam, describes the editing as "simply poetic" in the combination of editing and soundtrack that is used in the short film.  As the film plays out as a photomontage, the only continuous variable is the sound. The sound used in this production is minimal, showing up in the form of narration, Orchestral score and sound effect. The rhythmic patterns of the soundtrack act as a framework to add to the intensity of the film. This is seen as "The dissolve is synchronized with the sound. As the story moves from the past to the present, La Jetee creates mental continuity." The soundtrack adds to the Illusion of movement within the film and the change of time.

Interpretation.

In "Black and Blue", her study of postwar French fiction, Carol Mavor describes "La Jetée" as taking "place in a no-place (u-topia) in no-time (u-chronia)" which she connects to the time and place of the fairy tale. She goes on to say "even the sound of the title resonates with the fairy-tale surprise of finding oneself in another world: La Jetée evokes 'là j'étais' (there I was)". By "u-topia", Mavor does not refer to "utopia" as the word is commonly used; she also describes an ambiguity of dystopia/utopia in the film: "It is dystopia with the hope of utopia, or is it utopia cut by the threat of dystopia."  Tor Books blogger Jake Hinkson summed up his interpretation in the title of an essay about the film, "There's No Escape Out of Time". He elaborated:  Hinkson also addresses the symbolic use of imagery: "The Man is blindfolded with some kind of padded device and he sees images. The Man is chosen for this assignment because... he has maintained a sharp mind because of his attachment to certain images. Thus a film told through the use of still photos becomes about looking at images." He further observes that Marker himself did not refer to "La Jetée" as a film, but as photo novel.  Yannis Karpouzis makes a structuralistic analysis on "La Jetée", examining it as an intermedial artwork: Chris Marker creates an "archive" of objects and conditions that have a photographic quality of their own and they are followed by the same predicates as pictures. The dialogue between the media (photography and cinematography) and the filmic signifier (film stills, storyline and narration) is constantly in the backdrop.

Reception.

In 2010, "Time" ranked "La Jetée" first in its list of "Top 10 time-travel movies". In 2012, in correspondence with the "Sight & Sound" Poll, the British Film Institute deemed "La Jetée" as the 50th greatest film of all time.  In 1963, Prix Jean Vigo awarded "La Jetée" for "Best Short Film."  In 1963, "La Jetée" was part of the Locarno International Film Festival.  In 2009, the films was featured in "Buenos Aires Festival Internacional de Cine Independiente.  "La Jetée" was featured in the "Cine//B Film Festival" in 2011.  The International Documentary Film Festival Amsterdam had "La Jetée" as a featured film in 2019.  Science fiction writer William Gibson considers the film one of his main influences.

Legacy.

The video for Sigue Sigue Sputnik's 1989 single "Dancerama" is also an homage to "La Jetée".  The film is one of the influences in the video for David Bowie's "Jump They Say" (1993).  Terry Gilliam's "12 Monkeys" (1995) was inspired by and takes several concepts directly from "La Jetée" (acknowledging this debt in the opening credits).  The 2003 short film "La puppé" is both an homage to and a parody of "La Jetée".  The 2007 Mexican film "Year of the Nail", which is told entirely through still photographs, was inspired by "La Jetée".  Kode9 in collaboration with Ms. Haptic, Marcel Weber (aka MFO), and Lucy Benson created an homage to "La Jetée" in 2011, for the Unsound Festival.  Northern Irish rock band Two Door Cinema Club screened the film at the launch party for their 2016 album "Gameshow". The final track on the album, "Je viens de la", is inspired by "La Jetée" and describes the journey of the film's protagonist.  The film was included in the "1001 Movies You Must See Before You Die<nowiki>"</nowiki> by producer, Steven Schneider.

Related media.

In 1996, Zone Books released a book which reproduced the film's original images along with the script in both English and French.

Home media release.

In Region 2, the film is available with English subtitles in the "La Jetée/Sans soleil" digipack released by Arte Video. In Region 1, the Criterion Collection has released a "La Jetée/Sans soleil" combination DVD / Blu-ray, which features the option of hearing the English or French narration.

La Jetée () is a 1962 French science fiction featurette directed by Chris Marker and associated with the Left Bank artistic movement. Constructed almost entirely from still photos, it tells the story of a post-nuclear war experiment in time travel. It is 28 minutes long and shot in black and white.

La Jetée est un film français de science-fiction de Chris Marker, sorti en 1962 et d'une durée de 28 minutes.