text:
Monaco Grand Prix

The Monaco Grand Prix (Grand Prix de Monaco) is a Formula One motor racing event held annually on the Circuit de Monaco, in late May or early June. 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. It is the only Grand Prix that does not adhere to the FIA's mandated 305 km minimum race distance for Formula One races.

The race is held on a narrow course laid out in the streets of Monaco, with many elevation changes and tight corners as well as the tunnel, making it one of the most demanding circuits in Formula One. In spite of the relatively low average speeds, the Monaco circuit is a dangerous place to race due to how narrow the track is, and the race often involves the intervention of a safety car.

The first Monaco Grand Prix took place on 14 April 1929, and the race eventually became part of the pre-Second World War European Championship and was included in the first World Championship of Drivers in 1950. It was twice designated the European Grand Prix, in 1955 and 1963, when this title was an honorary designation given each year to one Grand Prix race in Europe. Graham Hill was known as "Mr. Monaco" due to his five Monaco wins in the 1960s. Ayrton Senna won the race more times than any other driver, with six victories, winning five races consecutively between 1989 and 1993. In the 81 editions of the Monaco Grand Prix, only two Monégasque drivers have won the race, Louis Chiron in 1931 and Charles Leclerc in 2024.

The circuit has been called "an exceptional location of glamour and prestige". The Formula One event is usually held on the last weekend of May and is known as one of the largest weekends in motor racing, as the Formula One race occurs on the same Sunday as the Indianapolis 500 (IndyCar Series) and the Coca-Cola 600 (NASCAR Cup Series).

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 was 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 who were 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 $4 1/2$ 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.

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 remained the only native of Monaco to have won the event until 2024.

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 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 lack of profits for organisers, and demand for nearly £500 (approximately £ adjusted to inflation) 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.

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 due to budgetary concerns and a lack of regulations in the sport.

1952 was 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 due to the fact that the car regulations were not finalized.

The Monaco Grand Prix returned in 1955, again as part of the Formula One World Championship, and this would begin a streak of 64 consecutive years in which the race was held. In the 1955 race, 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 participating in 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.

Era of Prost/Senna dominance
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 due to 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 René 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 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 in crashes 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. Schumacher also won the 1995 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. It returned in 2021, on 23 May, where Max Verstappen won his first Monaco Grand Prix. The 2022 event saw the Monégasque driver, Charles Leclerc of Scuderia Ferrari, achieve his first Monaco Grand Prix pole position at the Circuit de Monaco (he had taken pole the previous year but could not start due to driveshaft failure ). However, a critical strategical error meant Leclerc would drop to fourth, with Verstappen's teammate Sergio Pérez winning the race. The race was delayed due to heavy rain; two formation laps were completed before the start procedure was suspended and further delayed an hour from its 15:00 local time intended start. In addition to a red flag due to a big crash from Mick Schumacher, this dropped the laps completed from the intended 78 to 64.

In September 2022, the Grand Prix signed a new race contract to remain on the F1 calendar until the 2025 season. As part of this deal, the ACM relinquished advertising rights and television coverage to Formula One Management, with previous races having their television coverage produced by Télé Monte-Carlo.

The 2024 Grand Prix saw Charles Leclerc, driving for Ferrari, taking pole position for the third time around the principality of Monaco. The race was red-flagged after a heavy crash involving the Haas of Kevin Magnussen making contact with previous race winner Sergio Pérez and his Haas teammate Nico Hülkenberg. Leclerc won from pole position after fending off Oscar Piastri to become the second Monégasque to win after Louis Chiron won the 1931 edition of the race.

Circuit

The Grand Prix takes place on the Circuit de Monaco which traverses the narrow city streets of Monte Carlo and La Condamine alongside Monaco's harbour, Port Hercules. It has hosted the Grand Prix every time it has been run since 1929 – 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 requires millimetre accuracy and makes it one of the most demanding tracks in Formula One racing. Drivers will often touch the track walls to achieve a fast lap time. In 1929, La Vie Automobile magazine offered the opinion that "Any respectable traffic system would have covered the track with <> sign posts left, right and centre". , two drivers have crashed and ended up in the harbour, the most famous being Alberto Ascari in 1955. Despite undergoing minor changes throughout its history, the circuit is largely unchanged since 1929 and remains widely regarded as the ultimate test of driving skills and mental strength. If Monaco were not already an existing Grand Prix, it is unlikely that its street circuit would be permitted to host Formula One due to its short track length and concerns with safety and overtaking. However, as the 'Crown Jewel' of Formula One with a near century-old heritage in motorsport, Monaco is granted some exceptions to accommodate its limited possibilities for expansion.

Triple Formula One champion Nelson Piquet famously described racing at Monaco as "like riding a bicycle around your living room," illustrating the unique challenges posed by the circuit's narrow streets. He further emphasized the significance of victory at Monaco by stating that "a win here was worth two anywhere else".

Notably, the circuit 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 1:10.166.

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 previously ran the Junior Monaco Kart Cup.

The Monaco Grand Prix differs in several ways from other Grands Prix. The practice session for the race was traditionally held on the Thursday preceding the race instead of Friday. This allows the streets to be opened to the public again on Friday. From the 2022 event onwards the first two Formula One practice sessions will now be held on Friday, bringing the running schedule for Formula One in line with other Grands Prix. Support races will still be run on Thursday. 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. 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 start 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. Until 2017, there was no proper podium at the race. 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 2023, 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 principality 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. In the 2006 race, after retiring due to a mechanical failure while in second place, Kimi Räikkönen retired to his yacht, which was parked in the harbour.

The Grand Prix attracts big-name celebrities each year who come to experience the glamour and prestige of the event. Big parties are held in the nightclubs on the Grand Prix weekend, and the Port Hercule fills up with party-goers joining in the celebrations.

Criticism from drivers and commentators

In the 21st century, several commentators and F1 drivers have called the Grand Prix the most boring race of all circuits, both to drive and to watch as a spectator. Criticism has been directed towards how few overtake attempts are performed, as well as how frequently the driver who sets the pole position wins. Fernando Alonso has said that the race is "the most boring race ever," and Lewis Hamilton stated that the 2022 Grand Prix "wasn't really racing."

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.

* Between 1998 and 2005 built by Ilmor, funded by Mercedes

** Built by Cosworth, funded by Ford

*** Built by Porsche

By year
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.

title: Monaco Grand Prix


text:
Fission

Fission, a splitting of something into two or more parts, may refer to:

 * Fission (biology), the division of a single entity into two or more parts and the regeneration of those parts into separate entities resembling the original
 * Nuclear fission, when the nucleus of an atom splits into smaller parts
 * Fission (band), a Swedish death metal band
 * Fission (album), by Jens Johansson

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fission


text:
Fusion

Fusion, or synthesis, is the process of combining two or more distinct entities into a new whole.

Fusion may also refer to:

Physics

 * Nuclear fusion, multiple atomic nuclei combining to form one or more different atomic nuclei and subatomic particles
 * Fusion power, power generation using controlled nuclear fusion reactions
 * Cold fusion, a hypothesized type of nuclear reaction that would occur at or near room temperature
 * Heat fusion, a welding process for joining two pieces of a thermoplastic material
 * Melting, or transitioning from solid to liquid form

Biology and medicine

 * Binaural fusion, the cognitive process of combining the auditory information received by both ears
 * Binocular fusion, the cognitive process in binocular vision of combining the visual information received by both eyes
 * Cell fusion, a process in which several uninuclear cells combine to form a multinuclear cell
 * Gene fusion, a genetic event and molecular biology technique
 * Lipid bilayer fusion, a part of several cellular processes
 * Spinal fusion, a surgical technique used to combine two or more vertebrae
 * Tooth fusion, a dental abnormality in which two teeth are joined

Computing techniques

 * Image fusion, a process of combining relevant information from two or more images into a single image
 * Loop fusion, a compiler program-optimization transformation that replaces multiple loops with a single one
 * Sensor fusion, the combining of sensory data from disparate sources

Application software

 * Fusion 360, a 3D CAD, CAM and CAE program by Autodesk
 * Blackmagic Fusion, a visual effects package
 * BT Fusion, a defunct UK voice-over-IP service
 * Lucidworks Fusion, a search engine platform
 * NetObjects Fusion, a web design program

Middleware and operating system components

 * Compiz Fusion, a community-maintained set of plugins for the Compiz Window Manager
 * IBRIX Fusion, a parallel file system
 * Oracle Fusion Middleware, a portfolio of standards-based software products that spans multiple services
 * VMware Fusion, a virtual machine software product

Comics

 * Fusion (Eclipse Comics)
 * Fusion (Marvel Comics), a name of two fictional supervillains
 * Fusion (Marvel/Top Cow), a crossover between Marvel and Top Cow Productions

Film and television

 * "Fusion" (Star Trek: Enterprise), a first-season episode of Star Trek: Enterprise
 * Fusion (TV channel), an American cable and satellite news channel

Gaming

 * Dancing Stage Fusion, a 2004 music video game by Konami
 * Fuzion, a merger of Interlock System and Hero System
 * Metroid Fusion, a 2002 Game Boy Advance game
 * Philadelphia Fusion, a E-sports gaming team
 * Fusion (video game), 1988

Magazines

 * Fusion (Kent State University)
 * Fusion (music magazine), an American music magazine, 1967 to 1974
 * Fusion Magazine (political magazine), founded and edited by Glenn Beck
 * Fusion Magazine (scientific magazine), predecessor to 21st Century Science and Technology magazine

Music and dance

 * Jazz fusion, genre that combines rock and jazz, starting in 1960s
 * Fusion dance, a type of partner dance that combines two or more dance styles
 * Fusion (Jimmy Giuffre 3 album), 1961
 * Fusion (Jeremy Steig album), 1972
 * Fusion (Sawthis album), 2006
 * Fusion Festival, a music festival in Lärz, Germany
 * Guardian (band), a Christian rock band formerly called Fusion

Non-profit and political organizations

 * Fusion Party, a name for multiple political parties in American history
 * Fusion Party (Australia)
 * Fusion Energy Foundation, a defunct American non-profit think tank co-founded by Lyndon LaRouche in 1974
 * Fusion International, an Australian-based Christian organisation
 * Fusion of Haitian Social Democrats, a Haitian political party
 * Fusion – Sarvodaya ICT4D Movement, Information and Communications Technology for Development, Sri Lanka

Sports teams

 * Cleveland Fusion, a women's American football team in the NWFA
 * Fort Wayne Fusion, Arena football team
 * Miami Fusion, a professional soccer club in Fort Lauderdale
 * Philadelphia Fusion, an American esports team

Other organizations

 * Fusion Academy, a private alternative school in several U.S. cities
 * Fusion Media Group, a division of Univision Communications

Law and politics

 * Fusion of powers, a feature of some parliamentary forms of government
 * Fusionism (politics), the combination of libertarianism and various types of conservatism
 * Electoral fusion, an arrangement where two or more political parties support a common candidate
 * Fusion of law and equity, combining the rules of equity and common law into one set of rules
 * Fusion of the legal profession, the elimination of the distinction between barristers and solicitors

Products

 * AMD Fusion, a combined microprocessor and GPU design, now branded as AMD Accelerated Processing Unit
 * Ford Fusion (Americas), a mid-sized car produced by Ford Motor Company
 * Ford Fusion (Europe), a mini MPV produced by Ford Motor Company
 * Schumacher Fusion, a radio-controlled car
 * Gillette Fusion, a safety razor by Gillette
 * Fusion, a brand name of computer hardware used by Arctic

Other uses

 * Fusion (phonetics), the merger of phonological features of two speech segments into one feature
 * Fusion cuisine, the combination of elements of various culinary traditions
 * Fusional language or inflected language, a type of language
 * Information fusion, the merging of information from disparate sources

title: Fusion


text:
Four color theorem

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain.

The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken. This came after many false proofs and mistaken counterexamples in the preceding decades.

The Appel–Haken proof proceeds by analyzing a very large number of reducible configurations. This was improved upon in 1997 by Robertson, Sanders, Seymour, and Thomas who have managed to decrease the number of such configurations to 633 – still an extremely long case analysis. In 2005, the theorem was verified by Georges Gonthier using a general-purpose theorem-proving software.

Precise formulation of the theorem
In graph-theoretic terms, the theorem states that for loopless planar graph $$G$$, its chromatic number is $$\chi(G) \leq 4$$.

The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.

First, regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent. (Otherwise, a map in a shape of a pie chart would make an arbitrarily large number of regions 'adjacent' to each other at a common corner, and require arbitrarily large number of colors as a result.) Second, bizarre regions, such as those with finite area but infinitely long perimeter, are not allowed; maps with such regions can require more than four colors. (To be safe, we can restrict to regions whose boundaries consist of finitely many straight line segments. It is allowed that a region has enclaves, that is it entirely surrounds one or more other regions.) Note that the notion of "contiguous region" (technically: connected open subset of the plane) is not the same as that of a "country" on regular maps, since countries need not be contiguous (they may have exclaves; e.g., the Cabinda Province as part of Angola, Nakhchivan as part of Azerbaijan, Kaliningrad as part of Russia, France with its overseas territories, and Alaska as part of the United States are not contiguous). If we required the entire territory of a country to receive the same color, then four colors are not always sufficient. For instance, consider a simplified map:

In this map, the two regions labeled A belong to the same country. If we wanted those regions to receive the same color, then five colors would be required, since the two A regions together are adjacent to four other regions, each of which is adjacent to all the others.

A simpler statement of the theorem uses graph theory. The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment. This graph is planar: it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves without crossings that lead from one region's vertex, across a shared boundary segment, to an adjacent region's vertex. Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short: every planar graph is four-colorable.

Early proof attempts
As far as is known, the conjecture was first proposed on October 23, 1852, when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, Frederick, was a student of Augustus De Morgan (the former advisor of Francis) at University College London. Francis inquired with Frederick regarding it, who then took it to De Morgan (Francis Guthrie graduated later in 1852, and later became a professor of mathematics in South Africa). According to De Morgan:

"A student of mine [Guthrie] asked me to day to give him a reason for a fact which I did not know was a fact—and do not yet. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored—four colors may be wanted but not more—the following is his case in which four colors are wanted. Query cannot a necessity for five or more be invented…"

"F.G.", perhaps one of the two Guthries, published the question in The Athenaeum in 1854, and De Morgan posed the question again in the same magazine in 1860. Another early published reference by in turn credits the conjecture to De Morgan.

There were several early failed attempts at proving the theorem. De Morgan believed that it followed from a simple fact about four regions, though he didn't believe that fact could be derived from more elementary facts. "This arises in the following way. We never need four colours in a neighborhood unless there be four counties, each of which has boundary lines in common with each of the other three. Such a thing cannot happen with four areas unless one or more of them be inclosed by the rest; and the colour used for the inclosed county is thus set free to go on with. Now this principle, that four areas cannot each have common boundary with all the other three without inclosure, is not, we fully believe, capable of demonstration upon anything more evident and more elementary; it must stand as a postulate."

One proposed proof was given by Alfred Kempe in 1879, which was widely acclaimed; another was given by Peter Guthrie Tait in 1880. It was not until 1890 that Kempe's proof was shown incorrect by Percy Heawood, and in 1891, Tait's proof was shown incorrect by Julius Petersen—each false proof stood unchallenged for 11 years.

In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem and generalized the four color conjecture to surfaces of arbitrary genus.

Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph (called a snark in modern terminology) must be non-planar.

In 1943, Hugo Hadwiger formulated the Hadwiger conjecture, a far-reaching generalization of the four-color problem that still remains unsolved.

Proof by computer
During the 1960s and 1970s, German mathematician Heinrich Heesch developed methods of using computers to search for a proof. Notably he was the first to use discharging for proving the theorem, which turned out to be important in the unavoidability portion of the subsequent Appel–Haken proof. He also expanded on the concept of reducibility and, along with Ken Durre, developed a computer test for it. Unfortunately, at this critical juncture, he was unable to procure the necessary supercomputer time to continue his work.

Others took up his methods, including his computer-assisted approach. While other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University of Illinois announced, on June 21, 1976, that they had proved the theorem. They were assisted in some algorithmic work by John A. Koch.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts:

 * 1) An unavoidable set is a set of configurations such that every map that satisfies some necessary conditions for being a minimal non-4-colorable triangulation (such as having minimum degree 5) must have at least one configuration from this set.
 * 2) A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, the map can be reduced to a smaller map. This smaller map has the condition that if it can be colored with four colors, this also applies to the original map. This implies that if the original map cannot be colored with four colors the smaller map cannot either and so the original map is not minimal.

Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. Their proof reduced the infinitude of possible maps to 1,834 reducible configurations (later reduced to 1,482) which had to be checked one by one by computer and took over a thousand hours. This reducibility part of the work was independently double checked with different programs and computers. However, the unavoidability part of the proof was verified in over 400 pages of microfiche, which had to be checked by hand with the assistance of Haken's daughter Dorothea Blostein.

Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the University of Illinois used a postmark stating "Four colors suffice." At the same time the unusual nature of the proof—it was the first major theorem to be proved with extensive computer assistance—and the complexity of the human-verifiable portion aroused considerable controversy.

In the early 1980s, rumors spread of a flaw in the Appel–Haken proof. Ulrich Schmidt at RWTH Aachen had examined Appel and Haken's proof for his master's thesis that was published in 1981. He had checked about 40% of the unavoidability portion and found a significant error in the discharging procedure. In 1986, Appel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof. They replied that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article. Their magnum opus, Every Planar Map is Four-Colorable, a book claiming a complete and detailed proof (with a microfiche supplement of over 400 pages), appeared in 1989; it explained and corrected the error discovered by Schmidt as well as several further errors found by others.

Simplification and verification
Since the proving of the theorem, a new approach has led to both a shorter proof and a more efficient algorithm for 4-coloring maps. In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic-time algorithm (requiring only O(n2) time, where n is the number of vertices), improving on a quartic-time algorithm based on Appel and Haken's proof. The new proof, based on the same ideas, is similar to Appel and Haken's but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations. Both the unavoidability and reducibility parts of this new proof must be executed by a computer and are impractical to check by hand. In 2001, the same authors announced an alternative proof, by proving the snark conjecture. This proof remains unpublished, however.

In 2005, Benjamin Werner and Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant. This removed the need to trust the various computer programs used to verify particular cases; it is only necessary to trust the Coq kernel.

Summary of proof ideas
The following discussion is a summary based on the introduction to Every Planar Map is Four Colorable. Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. The explanation here is reworded in terms of the modern graph theory formulation above.

Kempe's argument goes as follows. First, if planar regions separated by the graph are not triangulated (i.e., do not have exactly three edges in their boundaries), we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region. If this triangulated graph is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed. So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

Suppose v, e, and f are the number of vertices, edges, and regions (faces). Since each region is triangular and each edge is shared by two regions, we have that 2e = 3f. This together with Euler's formula, v − e + f = 2, can be used to show that 6v − 2e = 12. Now, the degree of a vertex is the number of edges abutting it. If vn is the number of vertices of degree n and D is the maximum degree of any vertex,
 * $$6v - 2e = 6\sum_{i=1}^D v_i - \sum_{i=1}^D iv_i = \sum_{i=1}^D (6 - i)v_i = 12.$$

But since 12 > 0 and 6 − i ≤ 0 for all i ≥ 6, this demonstrates that there is at least one vertex of degree 5 or less.

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable. Call this graph G. Then G cannot have a vertex of degree 3 or less, because if d(v) ≤ 3, we can remove v from G, four-color the smaller graph, then add back v and extend the four-coloring to it by choosing a color different from its neighbors.

Kempe also showed correctly that G can have no vertex of degree 4. As before we remove the vertex v and four-color the remaining vertices. If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors. Such a path is called a Kempe chain. There may be a Kempe chain joining the red and blue neighbors, and there may be a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored. Suppose it is the red and blue neighbors that are not chained together. Explore all vertices attached to the red neighbor by red-blue alternating paths, and then reverse the colors red and blue on all these vertices. The result is still a valid four-coloring, and v can now be added back and colored red.

This leaves only the case where G has a vertex of degree 5; but Kempe's argument was flawed for this case. Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument (changing only that the minimal counterexample requires 6 colors) and use Kempe chains in the degree 5 situation to prove the five color theorem.

In any case, to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex. Rather the form of the argument is generalized to considering configurations, which are connected subgraphs of G with the degree of each vertex (in G) specified. For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well. A configuration for which this is possible is called a reducible configuration. If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable. The argument above began by giving an unavoidable set of five configurations (a single vertex with degree 1, a single vertex with degree 2, ..., a single vertex with degree 5) and then proceeded to show that the first 4 are reducible; to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem.

Because G is triangular, the degree of each vertex in a configuration is known, and all edges internal to the configuration are known, the number of vertices in G adjacent to a given configuration is fixed, and they are joined in a cycle. These vertices form the ring of the configuration; a configuration with k vertices in its ring is a k-ring configuration, and the configuration together with its ring is called the ringed configuration. As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can be extended without modification to a coloring of the configuration is called initially good. For example, the single-vertex configuration above with 3 or fewer neighbors were initially good. In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors; for a general configuration with a larger ring, this requires more complex techniques. Because of the large number of distinct four-colorings of the ring, this is the primary step requiring computer assistance.

Finally, it remains to identify an unavoidable set of configurations amenable to reduction by this procedure. The primary method used to discover such a set is the method of discharging. The intuitive idea underlying discharging is to consider the planar graph as an electrical network. Initially positive and negative "electrical charge" is distributed amongst the vertices so that the total is positive.

Recall the formula above:

 * $$\sum_{i=1}^D (6 - i)v_i = 12.$$

Each vertex is assigned an initial charge of 6-deg(v). Then one "flows" the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure. Since charge is preserved, some vertices still have positive charge. The rules restrict the possibilities for configurations of positively charged vertices, so enumerating all such possible configurations gives an unavoidable set.

As long as some member of the unavoidable set is not reducible, the discharging procedure is modified to eliminate it (while introducing other configurations). Appel and Haken's final discharging procedure was extremely complex and, together with a description of the resulting unavoidable configuration set, filled a 400-page volume, but the configurations it generated could be checked mechanically to be reducible. Verifying the volume describing the unavoidable configuration set itself was done by peer review over a period of several years.

A technical detail not discussed here but required to complete the proof is immersion reducibility.

False disproofs
The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history. At first, The New York Times refused, as a matter of policy, to report on the Appel–Haken proof, fearing that the proof would be shown false like the ones before it. Some alleged proofs, like Kempe's and Tait's mentioned above, stood under public scrutiny for over a decade before they were refuted. But many more, authored by amateurs, were never published at all.

Generally, the simplest, though invalid, counterexamples attempt to create one region which touches all other regions. This forces the remaining regions to be colored with only three colors. Because the four color theorem is true, this is always possible; however, because the person drawing the map is focused on the one large region, they fail to notice that the remaining regions can in fact be colored with three colors.

This trick can be generalized: there are many maps where if the colors of some regions are selected beforehand, it becomes impossible to color the remaining regions without exceeding four colors. A casual verifier of the counterexample may not think to change the colors of these regions, so that the counterexample will appear as though it is valid.

Perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches. If this were the restriction, planar graphs would require arbitrarily large numbers of colors.

Other false disproofs violate the assumptions of the theorem, such as using a region that consists of multiple disconnected parts, or disallowing regions of the same color from touching at a point.

Three-coloring
While every planar map can be colored with four colors, it is NP-complete in complexity to decide whether an arbitrary planar map can be colored with just three colors.

A cubic map can be colored with only three colors if and only if each interior region has an even number of neighboring regions. In the US states map example, landlocked Missouri (MO) has eight neighbors (an even number): it must be differently colored from all of them, but the neighbors can alternate colors, thus this part of the map needs only three colors. However, landlocked Nevada (NV) has five neighbors (an odd number): one of the neighbors must be differently colored from it and all the others, thus four colors are needed here.

Infinite graphs
The four color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn–Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable. This can also be seen as an immediate consequence of Kurt Gödel's compactness theorem for first-order logic, simply by expressing the colorability of an infinite graph with a set of logical formulae.

Higher surfaces
One can also consider the coloring problem on surfaces other than the plane. The problem on the sphere or cylinder is equivalent to that on the plane. For closed (orientable or non-orientable) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula
 * $$p=\left\lfloor\frac{7 + \sqrt{49 - 24 \chi}}{2}\right\rfloor,$$

where the outermost brackets denote the floor function.

Alternatively, for an orientable surface the formula can be given in terms of the genus of a surface, g:
 * $$p=\left\lfloor\frac{7 + \sqrt{1 + 48g }}{2}\right\rfloor.$$

This formula, the Heawood conjecture, was proposed by P. J. Heawood in 1890 and, after contributions by several people, proved by Gerhard Ringel and J. W. T. Youngs in 1968. The only exception to the formula is the Klein bottle, which has Euler characteristic 0 (hence the formula gives p = 7) but requires only 6 colors, as shown by Philip Franklin in 1934.

For example, the torus has Euler characteristic χ = 0 (and genus g = 1) and thus p = 7, so no more than 7 colors are required to color any map on a torus. This upper bound of 7 is sharp: certain toroidal polyhedra such as the Szilassi polyhedron require seven colors.

A Möbius strip requires six colors as do 1-planar graphs (graphs drawn with at most one simple crossing per edge). If both the vertices and the faces of a planar graph are colored, in such a way that no two adjacent vertices, faces, or vertex-face pair have the same color, then again at most six colors are needed.

For graphs whose vertices are represented as pairs of points on two distinct surfaces, with edges drawn as non-crossing curves on one of the two surfaces, the chromatic number can be at least 9 and is at most 12, but more precise bounds are not known; this is Gerhard Ringel's Earth–Moon problem.

Solid regions
There is no obvious extension of the coloring result to three-dimensional solid regions. By using a set of n flexible rods, one can arrange that every rod touches every other rod. The set would then require n colors, or n+1 including the empty space that also touches every rod. The number n can be taken to be any integer, as large as desired. Such examples were known to Fredrick Guthrie in 1880. Even for axis-parallel cuboids (considered to be adjacent when two cuboids share a two-dimensional boundary area), an unbounded number of colors may be necessary.

Relation to other areas of mathematics
Dror Bar-Natan gave a statement concerning Lie algebras and Vassiliev invariants which is equivalent to the four color theorem.

Use outside of mathematics
Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to cartographers. According to an article by the math historian Kenneth May, "Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property". The theorem also does not guarantee the usual cartographic requirement that non-contiguous regions of the same country (such as the exclave Alaska and the rest of the United States) be colored identically.

title: Four color theorem


text:
User:FallenAngel~enwiki

Welcome, and thanks a lot for your work on the analysis pages! --AxelBoldt

Nice job on Basic Set Theory, except it should be basic set theory --User:Buz Cory

''It's well worth doing a search before righting a wholly new article. You've duplicated a lot of effort with set theory -- User:GWO'' I believe that the author, and let "her" correct me, was attempting to introduce the subject on a more intuitive level than that the existing article. I applaud that. RoseParks

title: User:FallenAngel~enwiki