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Timeline of computer animation in film and television
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See also: History of computer animation
This is a chronological list of films and television programs that have been recognised as being pioneering in their use of computer animation.
Contents
[hide]
1 1960s
2 1970s
3 1980s
4 1990s
5 2000s
6 2010s
7 See also
8 References
9 External links
1960s[edit]
Film
Year
Notes
Rendering of a planned highway 1961 In 1960, a 49-second vector animation of a car traveling down a planned highway at 110 km/h was created at the Swedish Royal Institute of Technology on the BESK computer. The short animation was broadcast on November 9, 1961 on national television.[1][2]
Hummingbird 1967 A ten-minute computer animated film by Charles Csuri and James Shaffer. This was awarded a prize at the 4th annual International Experimental Film Competition in Brussels, Belgium and in the collection of The Museum of Modern Art, New York City. The subject was a line drawing of a hummingbird for which a sequence of movements appropriate to the bird were programmed. Over 30,000 images comprising some 25 motion sequences were generated by the computer.[3][4]
Kitty 1968 A group of Soviet mathematicians and physicists headed by N. Konstantinov created a mathematically computable model of the physics of a moving cat. The algorithms were programmed on the BESM-4 computer. The computer then printed hundreds of frames to be later converted to film.[5][6][7] An accompanying scientific paper describes the foundation of the employed physics simulation techniques that nowadays are commonly applied to animation films and computer games.[8]
1970s[edit]
Film
Year
Notes
Metadata 1971 This is an experimental 2D animated short drawn on a data tablet by Peter Foldes, who used the world's first key frame animation software, invented by Nestor Burtnyk and Marceli Wein.[9][10][11][12]
A Computer Animated Hand 1972 Produced by Ed Catmull, the short demonstrates a computer animated hand, as well as human faces. Added to the United States National Film Registry in 2011.
Westworld 1973 First use of 2D computer animation in a significant entertainment feature film. The point of view of Yul Brynner's gunslinger was achieved with raster graphics.[13][14]
The Six Million Dollar Man 1974 First television series to use CGI in the intro.
Faces (Faces & Body Parts) 1974 Fred Parke's thesis film on facial modeling at the University of Utah.[15]
Great 1975 The Oscar-winning 1975 short animated film about the life of the Victorian engineer Isambard Kingdom Brunel contains a brief sequence of a rotating wire-frame model of Brunel's final project, the iron steam ship SS Great Eastern.
Futureworld 1976 First use of 3D computer graphics for animated hand and face. Used 2D digital compositing to materialize characters over a background.[14]
Star Wars 1977 Used an animated 3D wire-frame graphic for the trench run briefing sequence. Added to the United States National Film Registry in 1989.
The Black Hole 1979 Used raster wire-frame model rendering for the open credits depicting a 3D wireframe of a black hole.[16]
Alien Used raster wire-frame model rendering for navigation monitors in the landing sequence.[14] Added to the United States National Film Registry in 2002.
1980s[edit]
Film
Year
Notes
Looker 1981 First CGI human character, Cindy. Also, first use of 3D-shaded CGI.[16][17]
The Works 1982 The New York Institute of Technology Computer Graphics Lab premiered a trailer at SIGGRAPH for their CGI project. This would have been the first feature-length CGI film, but it was never completed.
Star Trek II: The Wrath of Khan ILM computer graphics division develops "Genesis Effect", the first use of a fractal-generated landscape in a film.[18] Bill Reeves leads the Genesis Effect programming team, and creates a new graphics technique called "Particle Systems".
Tron First extensive use (15 minutes fully computer generated) of 3D CGI including the famous Light Cycle sequence.[19] Also includes very early facial animation (for the Master Control Program).
Golgo 13 1983 First animated film to incorporate CGI sequences.[20] Entirely digital models of revolvers, skeletons, helicopters, and skyscrapers (created by Toyo Links Corporation and Osaka University's CG division) are used in the film's title sequence and part of the climax; the remainder of the film is traditionally animated by Tokyo Movie Shinsha.
Dream Flight First 3D generated film telling a story, shown in Electronic Theater in SIGGRAPH ‘83
Rock & Rule First Western animated film to use computer graphics.[21]
The Last Starfighter 1984 Uses CGI for all spaceship shots, replacing traditional models. First use of "integrated CGI" where the effects are supposed to represent real world objects.[21]
Lensman: Secret of The Lens Uses CGI for spaceships and other scenes.
The Adventures of André and Wally B. Lucasfilm's computer animation division creates an all-CGI-animated short. The first CGI animation with motion blur effects and squash and stretch motion.
2010: The Year We Make Contact Jupiter's turbulent atmosphere is CGI-rendered, mostly during the black spot shots.
Tony de Peltrie 1985 First CGI-animated human character to express emotion through his face and body language.[22]
Young Sherlock Holmes Lucasfilm creates the first photorealistic CGI character, "stained glass knight" with 10 seconds of screentime.[23][24]
"Money for Nothing" The first computer-generated music video.[24] The animators would go on to found Mainframe Entertainment.
Flight of the Navigator 1986 The first use of reflection mapping in a feature film, used for the flying alien spacecraft.[24]
Labyrinth First realistic CGI animal.[24]
Star Trek IV: The Voyage Home First use of the Cyberware 3D scanner, first 3D morphing.[24]
Luxo Jr. First use of shadows in CGI, made with the specially developed software Photorealistic Renderman. First Pixar film, and first CGI film to be nominated for an Academy Award. Added to the United States National Film Registry in 2014.
The Golden Child First use of primitive photorealistic morphing.[25]
Captain Power and the Soldiers of the Future 1987 First TV series to include characters modeled entirely with computers.
Knightmare First game show with interaction between humans and computer-generated surroundings.
Rendez-vous in Montreal First 3D generated film involving virtual actors Marilyn Monroe and Humphrey Bogart
Willow 1988 First extensive photorealistic use of morphing effect in a feature film.[26]
Akira CGI is used to animate the pattern indicator, and to plot the paths of falling objects, model parallax effects on backgrounds, and tweak lighting and lens flares.[20]
Tin Toy First computer-animated short film to win an Oscar. Added to the United States National Film Registry in 2003.
The Abyss 1989 First digital 3D water effect.[27]
The Jim Henson Hour TV series with real-time and rendered CGI featuring digitally puppeteered CG character "Waldo."
Indiana Jones and the Last Crusade First all-digital composite.[27]
Les Fables géométriques First broadcast series of animated CGI shorts.
1990s[edit]
Film
Year
Notes
Die Hard 2 1990 First digitally-manipulated matte painting.[27]
RoboCop 2 An early use of real-time computer graphics or "digital puppetry" to create a character in a motion picture.[28]
Backdraft 1991 First use of photorealistic CGI fire in a motion picture.[28]
Terminator 2: Judgment Day First realistic human movements on a CGI character.[27] The first partially computer-generated main character and the first blockbuster movie to feature multiple morphing effects.[29] First use of a personal computer to create major movie 3D effects.
Death Becomes Her 1992 First human skin CGI software.[27]
Jurassic Park 1993 First photorealistic CG creatures.[27]
Live & Kicking First TV program to feature a live computer-generated character as part of its cast.
VeggieTales First completely computer animated direct-to-video release.
Babylon 5 First television series to use CGI as the primary method for its visual effects. First TV use of virtual sets.
Insektors[30] First fully computer animated TV series. First use of character animation in a computer animated television series.
Radioland Murders 1994 First use of virtual CGI sets with live actors.[31]
ReBoot First half-hour computer animated TV series.[32]
The Flintstones First CGI-rendered fur.[27]
The Crow First deceased actor (Brandon Lee) to be re-created through CGI.
Waterworld 1995 First realistic CG water.[27]
Casper First CGI lead character in a feature-length live-action film, and first CGI characters to interact realistically with live actors.
Toy Story First CGI feature-length animation. Added to the United States National Film Registry in 2005.
Cassiopeia 1996 A Brazilian CGI animated feature film produced and released by NDR Filmes in Brazil on April 1, 1996. First CGI feature-length animation that did not use scanned models for heads.
The Island of Dr. Moreau First film to use motion capture CGI to portray a character.
DragonHeart First 2D all-CGI backgrounds with live actors. First film to use ILM's Caricature software (created during the film's production).
Beast Wars: Transformers First CGI Transformers animated series produced by Mainframe Entertainment. Sequel to the original Transformers.
Star Wars (Episodes IV, V and VI Special Editions) 1997 First re-release of a film to incorporate CGI characters and elements.
Marvin the Martian in 3D First CG film created for viewing with 3D glasses.
Titanic First wide-release feature film with major elements rendered under the open-source Linux operating system.[33] Also included a number of advances, specifically in the rendering of flowing water.
A Bug's Life 1998 First CG anamorphic widescreen film. First all-digital transfer to DVD. First film to be reframed for home video releases.
Invasion: Earth First major use of digital effects in a British TV series
What Dreams May Come First use of 3D CGI in combination with 3D location scanning (Lidar) and motion-analysis based 3D camera tracking in a feature film.
Fight Club 1999 First realistic close-up of detailed facial deformation on a synthetic human.
The Matrix First use of CG interpolation with bullet time effects. Added to the National Film Registry in 2012.
Star Wars: Episode I – The Phantom Menace First film to use CG extensively for thousands of shots, including backgrounds, environmental effects, vehicles, and crowds. Several CG characters stood alongside real actors in dozens of shots, making them the first CG "supporting" cast members.
2000s[edit]
Film
Year
Notes
Final Fantasy: The Spirits Within 2001 First feature-length digital film to be made based on photorealism and live action principles. The first theatrically released feature film to utilize performance capture (motion capture) for all of its characters actions.[34]
Jimmy Neutron: Boy Genius First CGI feature-length movie made using off-the-shelf hardware and software.
The Lord of the Rings: The Fellowship of the Ring First use of AI for digital actors (using the Massive software developed by Weta Digital).
The Lord of the Rings: The Two Towers 2002 First virtual actor to win an award (Critics' Choice Movie Awards by Andy Serkis playing Gollum), in the newly created category Best Digital Acting Performance
Ice Age First full-length feature animated film exclusively rendered with a ray tracer (CGI Studio).[35]
The Matrix Reloaded 2003 The Burly Brawl - the first use of "Universal Capture", the combination of dense (rather than point-based) motion capture, per-frame texture capture and optical flow of pixels over the data from 7 camera setup bought into a shared UV space by projection onto a neutral expression geometry leading to the introduction of realistic digital look-alikes, last missing piece into puzzle to make fully virtual cinematography possible.
Able Edwards 2004 First movie shot completely on a green screen using digitally scanned images as backgrounds.
Sky Captain and the World of Tomorrow First movie with all-CGI backgrounds and live actors.[36]
The Polar Express First computer-animated 3D film to be created with motion capture.
Elephants Dream 2006 First CGI short movie released as completely open source. Made with Open Source software, theatrical and DVD release under Creative Commons License.[37] Unique that all 3D models, animatics and software are included on the DVD free for any use.
Flatland 2007 First CGI feature film to be animated by one person. Made with Lightwave 3D and Adobe After Effects.[38]
Plumíferos 2009 First CGI feature-length movie made using Open Source/Free Software for all 3D models, animation, lighting and render process, under GNU/Linux operating system.
Avatar First full-length movie made using performance capture to create photorealistic 3D characters and to feature a fully CG 3D photorealistic world. The first virtual art department and complete virtual production pipeline was developed by director James Cameron and team to create the film in real-time.
Up First computer-animated feature to be nominated for Academy Award for Best Picture.
2010s[edit]
Film
Year
Notes
Toy Story 3 2010 First CGI feature-length movie to gross more than $1 billion.
See also[edit]
Animation
List of computer-animated films
References[edit]
Jump up ^ Du Rietz, Peter (20 December 2016). "Svensk datorhistoria – 1960-talet" [Swedish computer history - 1960s]. Tekniska museet (in Swedish). Tekniska museet. Retrieved 17 January 2017. In front of the oscilloscope mounted a 35 mm camera with extended magazine on a custom-made stand. The camera was controlled automatically by computer, which sent a signal to the camera when a new image has been fed on the oscilloscope. In the Nordic ADB, who counted a lot and release data stewed, they had realized that they had all the coordinates to draw perspective from the driver's seat. They took as an example of this in the future how the then nyprojekterade motorway towards Nacka, outside Stockholm, would look like. With the camera in front of the oscilloscope, they could snap a picture every twenty meters of the virtual road. The result was a 30 second fictitious trip in the virtual highway at a speed of 110 km/h. The film was transferred to 16 mm format and made in 100 copies. Technical Museum is the only known surviving copy of the film in the collections. On the film roll box says that it is the first computer-drawn film in the world. There is little other evidence that this is actually true, and that this is the world's first computer animation. The film aired on the best time in the TV November 9, 1961 in news program News.
Jump up ^ "Världens äldsta datoranimation?" on YouTube
Jump up ^ Csuir, Charles. "Hummingbird, 1967". ACM SIGGRAPH. ACM SIGGRAPH. Retrieved 17 January 2017. We completed a ten minute computer animated film entitled Hummingbird. The subject was a line drawing of a hummingbird for which a sequence of movements appropriate to the bird were outlined. Over 30,000 images comprising some 25 motion sequences were generated by the computer. For these, selected sequences were used for the film. A micro-film plotter recorded the images directly to film. To facilitate control over the motion of some sequences, the programs were written to read all the controlling parameters from cards, one card for each frame. Curve fit or other date generating programs were used to punch the parameter decks. We also built a windowing option into our plot subroutine. []
Jump up ^ Charles Csuri, Fragmentation Animations, 1968 - 1970: Hummingbird (1968) on YouTube
Jump up ^ "Кошечка" [Kitty]. Mathematical Etudes (in Russian). Mathematical Etudes Foundation. Retrieved 17 January 2017. More than 40 years ago in 1968... A team led by Nikolai Nikolaevich Konstantinov created a mathematical model of the motion of the animal (a cat). The computer BESM-4 carried out the computation of an ordinary written program (in the mathematical sense) differential equations, drawing cartoon "Kitty" showing even by modern standards one of the most amazing animation movements of a cat ever created by computer.
Jump up ^ Quigley, Robert (22 March 2010). ""Kitty": One of the First-Ever Computer Animations". The Mary Sue. The Mary Sue, LLC. Retrieved 19 January 2017. “Kitty” is one of the first computer animations ever made: It was produced in Russia in 1968 under the direction of one N.Konstantinov, who led a team of physicists and mathematicians to produce something that looks a bit like an ASCII art flipbook today, but must have been remarkable at the time given the state of computing technology.
Jump up ^ "Kitty. - N.Konstantinov." on YouTube
Jump up ^ Konstantinov, V.V.; Minachin, N.N.; Ponomarenko, V.Y. (1974). "Программа, моделирующая механизм и рисующая мультфильм о нем" [The program that simulates the mechanism and draws a cartoon about it]. Проблемы кибернетики (in Russian). Moscow, USSR: Наука (28): 193–209. Retrieved 6 October 2016.
Jump up ^ [1]
Jump up ^ [2]
Jump up ^ [3]
Jump up ^ [4]
Jump up ^ Bowles, Scott (2004-09-15). "'Sky Captain' takes CGI to limit". USA Today. Retrieved 2010-05-25.
^ Jump up to: a b c Visual and Special Effects Film Milestones - Part 9
Jump up ^ Wenz, John (Jun 25, 2015). "These Early Computer Animations Show How Far We've Come". popularmechanics.com. Retrieved March 19, 2016.
^ Jump up to: a b Visual and Special Effects Film Milestones - Part 10
Jump up ^ Netzley, pg. 49.
Jump up ^ Pegoraro, Rob (June 29, 2008). "Incredibles, Inc; The story of how computer programmers transformed the art of movie animation". The Washington Post. p. W8.
Jump up ^ Tron - The 1982 movie
^ Jump up to: a b Hughes, David (2003). Comic Book Movies. Virgin Books. p. 27. ISBN 0-7535-0767-6.
^ Jump up to: a b Visual and Special Effects Film Milestones - Part 11
Jump up ^ Along the Banks of the St. Lawrence
Jump up ^ Netzley, 246.
^ Jump up to: a b c d e Visual and Special Effects Film Milestones - Part 12
Jump up ^ Dirks, Tim. "Visual and Special Effects Film Milestones". Filmsite. AMC Networks International. Retrieved 5 June 2016.
Jump up ^ Netzley, 239.
^ Jump up to: a b c d e f g h Netzley, 50.
^ Jump up to: a b Part 14
Jump up ^ "Visual and Special Effects Film Milestones". Filmsite.org. Retrieved October 3, 2012.
Jump up ^ Created in 1993. 2nd Prize for the category 3D Animation Imagina in 1993 for the episode "Some Flowers for Bakrakra" [5]
Jump up ^ Marcus Hearn (2005). The Cinema of George Lucas. New York City: Harry N. Abrams, Inc. pp. 79–80. ISBN 0-8109-4968-7.
Jump up ^ Bernstein, Sharon (1994-11-10). "'Reboot' Is First Series to Be Fully Computerized". LA Times. Retrieved 2010-08-23.
Jump up ^ Rowe, Robin (2003-01-01). "Linux and Star Trek". Linux Journal.
Jump up ^ "Cinema: A Painstaking Fantasy". Time. 2000-07-31.
Jump up ^ "Ray Tracers: Blue Sky Studios". Retrieved 2016-06-30.
Jump up ^ "Kid Robot and the World of Tomorrow". Wired.
Jump up ^ http://www.elephantsdream.org/
Jump up ^ "Flatland director Ladd Ehlinger Jr. starts column Filmmaker's Perspective for GreenCine.com". Flatland.
Netzley, Patricia D. Encyclopedia of Movie Special Effects. Checkmark Books, 2001.
External links[edit]
CG101: A Computer Graphics Industry Reference ISBN 073570046X Unique and personal histories of early computer graphics production, plus a comprehensive foundation of the industry for all reading levels.
CG production companies and CGI in the movies - detailed historical information
Milestones in Film History: Greatest Visual and Special Effects
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Isla Fisher
From Wikipedia, the free encyclopedia
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Isla Fisher
Fisher at the 2013 Cannes Film Festival
Born
Isla Lang Fisher
(1976-02-03) 3 February 1976 (age 41)
Muscat, Oman
Nationality
Australian
Occupation
Actress
Years active
1993–present
Spouse(s)
Sacha Baron Cohen (m. 2010)
Children
3
Isla Lang Fisher (/ˈaɪlə/; born 3 February 1976)[1] is an Australian actress. Born to Scottish parents in Oman, she moved to Australia at age 6. She appeared on the children's adventure series Bay Cove and the short-lived soap opera Paradise Beach, before playing Shannon Reed on the soap opera Home and Away. She has since been known for her comedic roles in Scooby-Doo (2002), I Heart Huckabees (2004), Wedding Crashers (2005),[2] Hot Rod (2007), Definitely, Maybe (2008), Confessions of a Shopaholic (2009), Rango (2011), Bachelorette (2012), Rise of the Guardians (2012), and Arrested Development (2013).
In 2013, Fisher starred as Myrtle Wilson in The Great Gatsby and as Henley Reeves in Now You See Me.
Contents
[hide]
1 Early life
2 Career
3 Personal life
3.1 Religion
4 Filmography
4.1 Television
4.2 Film
5 Awards and nominations
6 Works and publications
7 References
8 External links
Early life[edit]
Fisher was born in Muscat, Oman, the daughter of Scottish parents Elspeth Reid and Brian Fisher.[3] Her father worked as a banker in Oman for the United Nations at that time.[4] When she was a child, Fisher and her family moved back to their hometown of Bathgate, Scotland, then to Perth, Western Australia, when she was 6 years old.[5] She has four siblings and has said that she had a "great" upbringing in Perth with a "very outdoorsy life."[2]
Fisher attended Swanbourne Primary School and Methodist Ladies' College. She appeared in lead roles in school productions such as Little Shop of Horrors. At 21, she attended L'École Internationale de Théâtre Jacques Lecoq in Paris, where she studied clown, mime and commedia dell'arte.[6]
Career[edit]
Fisher began appearing in commercials on Australian television at the age of 9, before going on to win roles in popular children's television shows Bay City and Paradise Beach. At 18, with the help of her mother, she published two teen novels, Bewitched and Seduced by Fame,[7] and in a 2005 interview, Fisher admitted that had she not been successful as an actress she would probably have been a full-time writer.[8] From 1994 to 1997 she played the role of Shannon Reed on the Australian soap opera Home and Away. In 1996, she received a Logie Award nomination for Most Popular Actress for her role. After leaving the soap, Fisher enrolled at L'École Internationale de Théâtre Jacques Lecoq, a theatre and arts training school in Paris, and went on to appear in pantomime in the United Kingdom.[9][10] She also toured with Darren Day in the musical Summer Holiday and appeared in the London theatre production Così.[11] In 2001, Fisher was cast as Kim in the German horror film Swimming Pool.[12]
In 2002, she had a role in the film version of Scooby-Doo as Mary Jane, Shaggy's love interest, who is allergic to dogs. She wore a blonde wig for portraying her role because Daphne Blake (portrayed by Sarah Michelle Gellar) was supposed to be the only redhead in the film.[13] The following year, she portrayed the character Kristy in the Australian comedy The Wannabes. Subsequently, Fisher was taken on by an American agent.[2] A larger role in Wedding Crashers (2005), alongside Vince Vaughn and Owen Wilson,[14] won her the Breakthrough Performance Award at the MTV Movie Awards. While promoting Wedding Crashers, she was officially crowned the 1000th guest on Australian talk show Rove on 2 August 2005. She entered the set ahead of Owen Wilson, winning the title by two metres.
Fisher at the 2009 Golden Globe Awards
In 2006, Fisher starred as Becca, a Manhattan party host, in the relationship drama London, co-starring Jessica Biel, Chris Evans, and Jason Statham. She played the role of Katie in the romantic comedy Wedding Daze with Jason Biggs. In 2007, she appeared in The Lookout, a thriller film co-starring Joseph Gordon-Levitt and Matthew Goode, and Hot Rod, opposite Andy Samberg. She was scheduled to appear in The Simpsons Movie, although her appearance was cut from the final version.[15] In 2008, she starred in the critically acclaimed romantic comedy film Definitely, Maybe, with Ryan Reynolds, Elizabeth Banks, Rachel Weisz, and Abigail Breslin,[2] and had a voice role in Horton Hears a Who![16] Fisher has also co-written a script entitled Groupies with Amy Poehler, as well as another project entitled The Cookie Queen.[2] She starred in the movie adaptation of the book Confessions of a Shopaholic, which opened on 13 February 2009.[17] In the film, Fisher played a college graduate who works as a financial journalist in New York City to support her shopping addiction. Confessions of a Shopaholic suffered critically but eventually opened at No.4, grossing $15,054,000 on its opening weekend. Fisher received her third Teen Choice Award nomination.[18]
Fisher has spoken out against the lack of opportunities for female comedians in Hollywood.[19] In 2010, she starred in the black comedy Burke and Hare.[20] Fisher then voiced the role of Beans in Rango, a computer-animated Western film released in March 2011. In the movie, Johnny Depp and her Definitely, Maybe co-star Abigail Breslin also provide voice-over roles.[21]
In another voice-over role, Fisher voiced the character of Tooth (the Tooth Fairy) in the animated film Rise of the Guardians, which was later released in 2012. In 2013, she starred in the Baz Luhrmann adaptation of The Great Gatsby with Leonardo DiCaprio and the magician crime thriller Now You See Me with Jesse Eisenberg and Woody Harrelson.[22] Her performance as Henley Reeves, received acclaim and popularity among viewers, who were devastated when Fisher was announced to be not returning for the sequel. Later, she also appeared in several episodes of the fourth season of Arrested Development, released in 2013.[23]
Fisher dropped out of Now You See Me sequel, Now You See Me 2, in 2014, due to her third pregnancy; she was replaced by actress Lizzy Caplan.[24] In September 2016 it was confirmed that Caplan will serve as a full-time replacement for Fisher, as she won't be returning to the films. In 2016, she starred in the comedy films Grimsby and Keeping Up with the Joneses.[25][26]
Fisher's first children's book Marge In Charge will be published in 2016.[27]
Personal life[edit]
Fisher has said that her "sensibility is Australian," she has a "laid-back attitude to life," and she feels she is "very Australian."[28] She considers herself to be a feminist.[29] Her mother and siblings live in Athens, Greece, while her father divides his time between Frankfurt, Germany and Nicaragua.[28]
Fisher first met the English comedian Sacha Baron Cohen in 2002 at a party in Sydney, Australia.[30] The couple became engaged in 2004, and then married on 15 March 2010, in a Jewish ceremony in Paris, France.[31][32] The couple have two daughters, Olive (born 2007) and Elula (born 2011), and a son, Montgomery (born 2015).[33][34][35][36]
Religion[edit]
Before marrying, Fisher converted to Judaism, saying, "I will definitely have a Jewish wedding just to be with Sacha. I would do anything—move into any religion—to be united in marriage with him. We have a future together and religion comes second to love as far as we are concerned."[37] She completed her conversion in early 2007, after three years of study.[38] She took the Hebrew name Ayala (איילה), the Hebrew word for a doe, and has described herself as keeping the Jewish Sabbath.[39][40][41]
Filmography[edit]
Television[edit]
Year
Title
Role
Notes
1993 Bay Cove Vanessa Walker Series
Paradise Beach Robyn Devereaux Barsby 2 episodes
1994 – 97 Home and Away Shannon Reed 345 episodes
1999 Oliver Twist Bet Miniseries
2000 Sunburn Woman Episode: "2.1"
Hearts and Bones Australian Barmaid Episode: "I Need a Love Song"
2001 Attila Cerca Miniseries
2002 BeastMaster Demon Manaka Episode: "The Trial"
Random Acts of Intimacy Woman in Club Short
2004 Pilot Season Butterfly Episode: "1.6"
2010 Neighbors from Hell 3 episodes; voice
2011 Bored to Death Rose 2 episodes
2013 Arrested Development Rebel Alley Season 4; 9 episodes
2015 Sofia the First Button (voice) Episode: "The Littlest Princess"
Film[edit]
Fisher at the premiere of The Dictator in London, 2012
Year
Title
Role
Notes
1997 Bum Magnet Emma Short film
1998 Furnished Room Jennie Short film
2000 Out of Depth Australian Girl #1
2001 Swimming Pool Kim
2002 Dog Days Bianca
Scooby-Doo Mary Jane
2003 The Wannabes Kirsty
Dallas 362 Redhead
Criminal Ways[42] Kirsty
2004 I Heart Huckabees Heather
2005 Wedding Crashers Gloria Cleary
London Rebecca
2006 Wedding Daze Katie
2007 The Lookout Luvlee
Hot Rod Denise
2008 Definitely, Maybe April
Horton Hears a Who! Dr. Mary Lou Larue Voice role
2009 Confessions of a Shopaholic Rebecca Bloomwood
2010 Burke and Hare Ginny Hawkins
2011 Rango Beans Voice role
2012 Bachelorette Katie
Rise of the Guardians Tooth Fairy Voice role
2013 The Great Gatsby Myrtle Wilson
Now You See Me Henley Reeves
2014 Life of Crime Melanie Ralston
2015 Visions[43] Eveleigh
Klovn Forever Herself Cameo
2016 Grimsby Jodie Figgs
Nocturnal Animals Laura Hastings
Keeping Up with the Joneses Karen Gaffney
2018 Tag Filming[44]
Awards and nominations[edit]
Year
Association
Category
Work
Result
Ref.
2006 MTV Movie Awards Best Breakthrough Performance Wedding Crashers Won [45]
Teen Choice Awards Choice Breakout (Female) Wedding Crashers Nominated
Teen Choice Awards Choice Hissy Fit Wedding Crashers Nominated
2008 Elle Women in Hollywood Awards Icon Award Won [46]
2009 Teen Choice Awards Choice Movie Actress: Comedy Confessions of a Shopaholic Nominated
2011 Alliance of Women Film Journalists Best Animated Female Rango Won [47]
2013 AACTA Awards Best Supporting Actress The Great Gatsby Nominated
2014 Screen Actors Guild Outstanding Performance by an Ensemble in a Comedy Series Arrested Development Nominated [48]
2016 AACTA Awards Trailblazer Award Won [49]
Works and publications[edit]
Fisher, Isla; Reid, Elspeth (1995). Bewitched. Ringwood, Victoria: Penguin. ISBN 978-0-14-025575-1. OCLC 38382626.
Fisher, Isla (1995). Seduced by Fame. Ringwood, Victoria: Penguin. ISBN 978-0-14-025431-0. OCLC 38376530.
Fisher, Isla; Ceulemans, Eglantine (2016). Marge in Charge. London: Piccadilly Press. ISBN 978-1-84-812540-7. OCLC 957646590.
References[edit]
Jump up ^ "Isla Fisher Biography (1976-)". FilmReference.com. Retrieved 23 October 2012.
^ Jump up to: a b c d e Roberts, Sheila (25 March 2007). "Isla Fisher Interview, The Lookout". MoviesOnline. Retrieved 26 March 2007.
Jump up ^ Marcie Muir, Kerry White (1992). Australian Children's Books: 1989-2000. Melbourne University Publishing.
Jump up ^ "Daddy's little girl! Isla Fisher joined by her father at premiere of new film Rango". Daily Mail. London. 22 February 2011.
Jump up ^ "Isla Fisher maintains she is not a shopaholic , STV Video". Video.stv.tv. Retrieved 6 May 2010.
Jump up ^ "The Vanities Girls". Vanity Fair. New York. 1 March 2009.
Jump up ^ Pringle, Gill (28 September 2007). "Hot Bod is keeping mum". The Independent.
Jump up ^ Harry, Ethan (17 July 2005). "Interrogation Isla Fisher". Sunday Mirror.
Jump up ^ "Australian TV soap opera stars in UK pantomimes". National Library of Australia. Retrieved 19 July 2017.
Jump up ^ "The Vanities". Vanity Fair. Retrieved 19 July 2017.
Jump up ^ "Isla Fisher: Why I’m promoting breastfeeding in the developing world". Metro. 19 February 2013. Retrieved 11 July 2014.
Jump up ^ "Isla Fisher - The Pool (2001)". Youtube. Retrieved 11 July 2014.
Jump up ^ "They could be sisters! Sarah Michelle Gellar links arms with lookalike Isla Fisher in cute snap for Throwback Thursday". Daily Mail. 6 June 2014. Retrieved 11 July 2014.
Jump up ^ "Isla Fisher Is A Little Afraid Of Her 'Wedding Crashers' Fans (VIDEO)". Huffington post. 24 May 2013. Retrieved 11 July 2014.
Jump up ^ Wolf, Jeanne (29 March 2007). "What Creeps Out Borat's Future Wife?". Retrieved 6 April 2007.
Jump up ^ Finn, Natalie (16 March 2007). "Family Guy Has Burnett Feeling Copyrighteous". E! Online. Retrieved 26 March 2007.
Jump up ^ "Isla Fisher signs on for Shopaholic role". ABC News (Australia). 21 September 2007. Retrieved 6 December 2007.
Jump up ^ "Shocker! Twilight Leads Teen Choice Noms". E! Online. Retrieved 22 April 2014.
Jump up ^ "Fisher plans Hollywood sex revolution". The Sydney Morning Herald. 8 October 2007. Retrieved 8 October 2007.
Jump up ^ "Two More Cast Members for Burke and Hare". Dreadcentral.com. 22 January 2010. Retrieved 6 May 2010.
Jump up ^ "'Rango' Photo - Isla Fisher". About.com. Retrieved 22 April 2014.
Jump up ^ "Now You See me Actress is living the Dream". hitfix.com. 28 May 2013. Retrieved 22 April 2014.
Jump up ^ "'Arrested Development': Isla Fisher, Terry Crews Join Season 4". Huffingtonpost. 2012-09-14. Retrieved 22 April 2014.
Jump up ^ "Lizzy Caplan in talks to replace Isla Fisher in Now You See Me 2". digitalspy.com. 15 October 2014. Retrieved 3 October 2015.
Jump up ^ "Isla Fisher and Rebel Wilson join cast of Sacha Baron Cohen comedy film Grimsby". The Telegraph. Retrieved 3 October 2015.
Jump up ^ "Isla Fisher Joins Jon Hamm, Zach Galifianakis in ‘Keeping Up With the Joneses’". thewrap.com. Retrieved 3 October 2015.
Jump up ^ Eyre, Charlotte (29 March 2016). "Isla Fisher pens children’s fiction series". The Bookseller. Retrieved 2016-05-31.
^ Jump up to: a b "Isla an Aussie at heart". Sunday Mail. 25 March 2007. Retrieved 26 March 2007.
Jump up ^ "Isla Fisher: Q&A". 20 February 2009. Retrieved 1 April 2017.
Jump up ^ "Sacha Baron Cohen and Isla Fisher Wed". People. 22 March 2010. Retrieved 6 May 2010.
Jump up ^ "Isla Fisher and Sacha Baron Cohen Wed". Showbiz.sky.com. Retrieved 6 May 2010.
Jump up ^ "Isla Fisher and Sacha Baron Cohen's Secret, Six-Guest Wedding". People. Retrieved 25 May 2010.
Jump up ^ "Isla Fisher, Sacha Baron Cohen Welcome a Baby". People. 19 October 2007. Retrieved 3 September 2012.
Jump up ^ "Baby Borat's Beverly Hills Outing". People. 22 November 2007. Retrieved 3 September 2012.
Jump up ^ "Revealed! Find Out The Name of Isla Fisher's Baby". Us Weekly. 10 March 2011. Retrieved 3 September 2012.
Jump up ^ "Sacha Baron Cohen & Isla Fisher Holy Moses It's A Boy!!". TMZ. 16 April 2015. Retrieved 16 April 2015.
Jump up ^ "Isla Fisher says marriage with Sacha Baron Cohen is like 'winning the lottery'". hellomagazine.com. 30 October 2012.
Jump up ^ Simpson, Richard; Weaver, Clair (4 March 2004). "I is marrying me Julie". Evening Standard. London. Retrieved 21 July 2011.
Jump up ^ Hellard, Peta (3 September 2007). "Isla Fisher's pregnant pause". NEWS.com.au. Retrieved 18 February 2008.
Jump up ^ Lipworth, Elaine (19 February 2009). "Fisher: I'm addicted to breastfeeding". Metro. Retrieved 19 February 2009.
Jump up ^ "Bruno star set to wed Isla Fisher". Stuff.com. Retrieved 10 May 2013.
Jump up ^ https://www.rottentomatoes.com/m/criminal_ways
Jump up ^ "Kevin Greutert: Visions". Bloody Disgusting. 2012-09-14. Retrieved 22 April 2014.
Jump up ^ http://www.hollywoodreporter.com/news/isla-fisher-joins-ed-helms-jeremy-renner-new-line-comedy-tag-1013149
Jump up ^ "Isla Fisher Awards". imdb. Retrieved 19 September 2014.
Jump up ^ "Elle magazine women in Hollywood custom awards". www.bennettawards.com. Retrieved 8 August 2015.
Jump up ^ "2011 EDA Awards Winners". /awfj.org. Retrieved 8 August 2015.
Jump up ^ "The 20th Annual Screen Actors Guild Awards". sagawards.org. Retrieved 8 August 2015.
Jump up ^ https://au.be.yahoo.com/entertainment/celebrity/a/33444649/isla-fisher-thanks-donald-trump-for-her-2016-aacta-award-gingers-winyahoo7-be/#page1
External links[edit]
Wikimedia Commons has media related to Isla Fisher.
Isla Fisher on IMDb
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Anna Kendrick (2010)
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WorldCat Identities
VIAF: 37242317
LCCN: nb98026226
ISNI: 0000 0001 1948 6635
GND: 1012033937
SUDOC: 155598414
BNF: cb156311888 (data)
NLA: 35454834
NKC: xx0077125
BNE: XX4797705
Retrieved from "https://en.wikipedia.org/w/index.php?title=Isla_Fisher&oldid=804564147"
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Four color theorem
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Example of a four-colored map
A four-coloring of a map of the states of the United States (ignoring lakes).
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.[1]
Unlike the five color theorem, a theorem that states that five colors are enough to color a map, which was proved in the 1800's, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken, but only after many false proofs and counterexamples. It was the first major theorem to be proved using a computer. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubts remain (Wilson 2014, 216–222).
To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally, in 2005, the theorem was proved by Georges Gonthier with general-purpose theorem-proving software.
Contents
[hide]
1 Precise formulation of the theorem
2 History
2.1 Early proof attempts
2.2 Proof by computer
2.3 Simplification and verification
3 Summary of proof ideas
4 False disproofs
5 Three-coloring
6 Generalizations
7 Relation to other areas of mathematics
8 Use Outside of Mathematics
9 See also
10 Notes
11 References
12 External links
Precise formulation of the theorem[edit]
A map of the world using just four colors for countries. The map also uses a fifth unique color (white) for oceans and lakes. This could be eliminated by recoloring, but then some landlocked countries would share color with the ocean, and some lakes and the ocean would differ.
The intuitive statement of the four color theorem, i.e. 'that given any separation of a plane into contiguous regions, called a map, 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, all corners, points that belong to (technically, are in the closure of) three or more countries, must be ignored. In addition, bizarre maps (using regions of finite area but infinite perimeter) can require more than four colors.[2] Second, for the purpose of the theorem, every "country" has to be a connected region, or contiguous. In the real world, this is not true (e.g. the Upper and Lower Peninsula of Michigan, Nakhchivan as part of Azerbaijan, and Kaliningrad as part of Russia are not contiguous). Because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map:
In this map, the two regions labeled A belong to the same country, and must be the same color. This map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps. For maps in which more than one country may have multiple disconnected regions, six or more colors might be required.
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 is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only): 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 that lead without crossing within each region from the vertex location to each shared boundary point of the region. 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" (Thomas 1998, p. 849; Wilson 2014).
History[edit]
Early proof attempts[edit]
Letter of De Morgan to Hamilton, 23 Oct. 1852
As far as is known, the conjecture was first proposed on October 23, 1852[3] 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…" (Wilson 2014, p. 18)
"F.G.", perhaps one of the two Guthries, published the question in The Athenaeum in 1854,[4][5] and De Morgan posed the question again in the same magazine in 1860.[6] Another early published reference by Arthur Cayley (1879) 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 colors 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 color 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.[6]
One alleged proof was given by Alfred Kempe in 1879, which was widely acclaimed;[7] 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 (Thomas 1998, p. 848).
In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem (Heawood 1890) and generalized the four color conjecture to surfaces of arbitrary genus—see below.
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.[8]
In 1943, Hugo Hadwiger formulated the Hadwiger conjecture (Hadwiger 1943), a far-reaching generalization of the four-color problem that still remains unsolved.
Proof by computer[edit]
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 (Wilson 2014).
Others took up his methods and 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,[9] that they had proved the theorem. They were assisted in some algorithmic work by John A. Koch (Wilson 2014).
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 (Wilson 2014; Appel & Haken 1989; Thomas 1998, pp. 852–853):
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.
A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, then the map can be reduced to a smaller map. This smaller map has the condition that if it can be colored with four colors, then the original map can also. This implies that if the original map cannot be colored with four colors the smaller map can't 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 1,936 reducible configurations (later reduced to 1,476) 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 & Haken 1989).
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 (Wilson 2014).
In the early 1980s, rumors spread of a flaw in the Appel-Haken proof. Ulrich Schmidt at RWTH Aachen examined Appel and Haken's proof for his master's thesis (Wilson 2014, 225). He had checked about 40% of the unavoidability portion and found a significant error in the discharging procedure (Appel & Haken 1989). 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 responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article (Wilson 2014, 225–226). 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 and explained Schmidt's discovery and several further errors found by others (Appel & Haken 1989).
Simplification and verification[edit]
Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O(n2) time, where n is the number of vertices. In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic-time algorithm, improving on a quartic-time algorithm based on Appel and Haken’s proof (Thomas 1995; Robertson et al. 1996). This new proof 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 computer and are impractical to check by hand (Thomas 1998, pp. 852–853). In 2001, the same authors announced an alternative proof, by proving the snark theorem (Thomas; Pegg et al. 2002).
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.[10]
Summary of proof ideas[edit]
The following discussion is a summary based on the introduction to Appel and Haken's book Every Planar Map is Four Colorable (Appel & Haken 1989). 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,
6 v − 2 e = 6 ∑ i = 1 D v i − ∑ i = 1 D i v i = ∑ i = 1 D ( 6 − i ) v i = 12. {\displaystyle 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 less 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:
∑ i = 1 D ( 6 − i ) v i = 12. {\displaystyle \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[edit]
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 (Wilson 2014). 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.
The map (left) has been colored with five colors, and it is necessary to change at least four of the ten regions to obtain a coloring with only four colors (right).
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 in unexpected ways, 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[edit]
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.[11]
Generalizations[edit]
By joining the single arrows together and the double arrows together, one obtains a torus with seven mutually touching regions; therefore seven colors are necessary
This construction shows the torus divided into the maximum of seven regions, each one of which touches every other.
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 Nash-Williams (1967). 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.
One can also consider the coloring problem on surfaces other than the plane (Weisstein). 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 = ⌊ 7 + 49 − 24 χ 2 ⌋ , {\displaystyle 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 = ⌊ 7 + 1 + 48 g 2 ⌋ {\displaystyle p=\left\lfloor {\frac {7+{\sqrt {1+48g}}}{2}}\right\rfloor } (Weisstein).
Interactive Szilassi polyhedron model with each face a different color. In the SVG image, move the mouse to rotate it.[12]
This formula, the Heawood conjecture, was conjectured by P. J. Heawood in 1890 and 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) and requires only 6 colors, as shown by P. Franklin in 1934 (Weisstein).
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.
Tietze's subdivision of a Möbius strip into six mutually adjacent regions, requiring six colors. The vertices and edges of the subdivision form an embedding of Tietze's graph onto the strip.
A Möbius strip requires six colors (Tietze 1910) as do 1-planar graphs (graphs drawn with at most one simple crossing per edge) (Borodin 1984). 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 (Borodin 1984).
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 if you consider 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 (Wilson 2014). 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 (Reed & Allwright 2008; Magnant & Martin (2011)).
Relation to other areas of mathematics[edit]
Dror Bar-Nathan gave a statement concerning Lie algebras and Vassiliev invariants which is equivalent to the four color theorem.[13]
Use Outside of Mathematics[edit]
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" (Wilson 2014, 2). The theorem also does not guarantee the usual cartographic requirement that non-contiguous regions of the same country (such as Great Britain and Northern Ireland) be colored identically.
Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proved in the late 19th century (Heawood 1890).
See also[edit]
Discrete mathematics portal
Mathematics portal
Apollonian network
Graph coloring
Grötzsch's theorem: triangle-free planar graphs are 3-colorable.
Hadwiger–Nelson problem: how many colors are needed to color the plane so that no two points at unit distance apart have the same color?
Notes[edit]
Jump up ^ Georges Gonthier (December 2008). "Formal Proof—The Four-Color Theorem". Notices of the AMS. 55 (11): 1382–1393. From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors.
Jump up ^ Hud Hudson (May 2003). "Four Colors Do Not Suffice". The American Mathematical Monthly. 110 (5): 417–423. doi:10.2307/3647828. JSTOR 3647828.
Jump up ^ Donald MacKenzie, Mechanizing Proof: Computing, Risk, and Trust (MIT Press, 2004) p103
Jump up ^ F. G. (June 10, 1854), "Tinting Maps", The Athenaeum: 726 .
Jump up ^ Brendan D. McKay (2012). "A note on the history of the four-colour conjecture". arXiv:1201.2852  .
^ Jump up to: a b De Morgan (anonymous), Augustus (April 14, 1860), "The Philosophy of Discovery, Chapters Historical and Critical. By W. Whewell.", The Athenaeum: 501–503
Jump up ^ W. W. Rouse Ball (1960) The Four Color Theorem, in Mathematical Recreations and Essays, Macmillan, New York, pp 222-232.
Jump up ^ Tait, P. G. (1880), "Remarks on the colourings of maps", Proc. R. Soc. Edinburgh, 10: 729
Jump up ^ Gary Chartrand and Linda Lesniak, Graphs & Digraphs (CRC Press, 2005) p.221
Jump up ^ (Gonthier 2008)
Jump up ^ Dailey, D. P. (1980), "Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete", Discrete Mathematics, 30 (3): 289–293, doi:10.1016/0012-365X(80)90236-8
Jump up ^ Branko Grünbaum, Lajos Szilassi, Geometric Realizations of Special Toroidal Complexes, Contributions to Discrete Mathematics, Volume 4, Number 1, Pages 21-39, ISSN 1715-0868
Jump up ^ Dror Bar-Natan, Lie algebras and the four color theorem. Combinatorica 17 (1997), no. 1, 43–52.
References[edit]
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Appel, Kenneth; Haken, Wolfgang (1977), "Every Planar Map is Four Colorable. I. Discharging", Illinois Journal of Mathematics, 21 (3): 429–490, MR 0543792
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Appel, Kenneth; Haken, Wolfgang (October 1977), "Solution of the Four Color Map Problem", Scientific American, 237 (4), pp. 108–121, doi:10.1038/scientificamerican1077-108
Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four-Colorable, Contemporary Mathematics, 98, With the collaboration of J. Koch., Providence, RI: American Mathematical Society, doi:10.1090/conm/098, ISBN 0-8218-5103-9, MR 1025335
Bernhart, Frank R. (1977), "A digest of the four color theorem", Journal of Graph Theory, 1 (3), pp. 207–225, doi:10.1002/jgt.3190010305, MR 0465921
Borodin, O. V. (1984), "Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs", Metody Diskretnogo Analiza (41): 12–26, 108, MR 0832128 .
Cayley, Arthur (1879), "On the colourings of maps", Proc. Royal Geographical Society, Blackwell Publishing, 1 (4), pp. 259–261, doi:10.2307/1799998, JSTOR 1799998
Fritsch, Rudolf; Fritsch, Gerda (1998), The Four Color Theorem: History, Topological Foundations and Idea of Proof, Translated from the 1994 German original by Julie Peschke., New York: Springer, doi:10.1007/978-1-4612-1720-6, ISBN 978-0-387-98497-1, MR 1633950
Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, 55 (11), pp. 1382–1393, MR 2463991
Gonthier, Georges (2005), A computer-checked proof of the four colour theorem (PDF), unpublished
Hadwiger, Hugo (1943), "Über eine Klassifikation der Streckenkomplexe", Vierteljschr. Naturforsch. Ges. Zürich, 88: 133–143
Heawood, P. J. (1890), "Map-Colour Theorem", Quarterly Journal of Mathematics, Oxford, 24, pp. 332–338
Kempe, A. B. (1879), "On the Geographical Problem of the Four Colours", American Journal of Mathematics, The Johns Hopkins University Press, 2 (3): 193–220
Magnant, C.; Martin, D. M. (2011), "Coloring rectangular blocks in 3-space", Discussiones Mathematicae Graph Theory, 31 (1): 161–170, doi:10.7151/dmgt.1535
Nash-Williams, C. St. J. A. (1967), "Infinite graphs—a survey", Journal of Combinatorial Theory, 3: 286–301, doi:10.1016/s0021-9800(67)80077-2, MR 0214501 .
O'Connor; Robertson (1996), The Four Colour Theorem, MacTutor archive
Pegg, Ed, Jr.; Melendez, J.; Berenguer, R.; Sendra, J. R.; Hernandez, A.; Del Pino, J. (2002), "Book Review: The Colossal Book of Mathematics" (PDF), Notices of the American Mathematical Society, 49 (9): 1084–1086, Bibcode:2002ITED...49.1084A, doi:10.1109/TED.2002.1003756
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Ringel, G.; Youngs, J.W.T. (1968), "Solution of the Heawood Map-Coloring Problem", Proc. Natl. Acad. Sci. USA, 60 (2), pp. 438–445, Bibcode:1968PNAS...60..438R, doi:10.1073/pnas.60.2.438, PMC 225066  , PMID 16591648
Robertson, Neil; Sanders, Daniel P.; Seymour, Paul; Thomas, Robin (1996), "Efficiently four-coloring planar graphs", Proceedings of the 28th ACM Symposium on Theory of Computing (STOC 1996), pp. 571–575, doi:10.1145/237814.238005, MR 1427555
Robertson, Neil; Sanders, Daniel P.; Seymour, Paul; Thomas, Robin (1997), "The Four-Colour Theorem", J. Combin. Theory Ser. B, 70 (1), pp. 2–44, doi:10.1006/jctb.1997.1750, MR 1441258
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External links[edit]
Wikimedia Commons has media related to Four-color theorem.
Hazewinkel, Michiel, ed. (2001) [1994], "Four-colour problem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Blanuša snarks". MathWorld.
Weisstein, Eric W. "Map coloring". MathWorld.
List of generalizations of the four color theorem on MathOverflow
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