81. Four Color Theorem - Encyclopedia: Article And Reference Information You are here Encyclopedia four color theorem four color theorem.The four color theorem states that every possible geographical http://encyclopedia.ekist.de/f/fo/four_color_theorem.html

Anthropology Architecture Archaeology Biology ... See live article You are here: Encyclopedia > Four color theorem Four color theorem The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie. It is obvious that three colors are completely inadequate, and it is not difficult to prove that five colors are sufficient to color a map. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken. They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See experimental mathematics.

82. The Four Color Theorem Daniel P. Sanders, Paul Seymour, and Robin Thomas created this page to providea short description of their new proof of the four color theorem and a four http://anduin.eldar.org/~ben/scout/html/567.html

Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created this page to provide a short description of their new proof of the four color theorem and a four-coloring algorithm. The four color theorem began as the concept that a map can be shaded so that no connected areas have the same color. On this page the creators of this new proof provide a history of the proof, the need for a new proof, and a brief overview of the proof itself. The full text paper is available as a downloadable postscript file or online at the Electronic Research Announcements of the American Mathematical Society (link provided in reference 8 at the site). Programs and data used in the proof are also available under the section Pointers. [KH] http://www.math.gatech.edu/~thomas/FC/fourcolor.html ====== Brought to You by the Internet Scout Project the Internet Scout Project, located in the University of Wisconsin-Madison's Department of Computer Sciences. copies. The InterNIC provides information about the Internet to the US research and education community under a cooperative agreement with the National Science Foundation: NCR-9218742. The Government has certain

83. Untitled Shortly after proofs of the fourcolours theorem were published by Arthur BrayKempe, a professional barrister (1879), and Peter Guthrie Tait (1880). http://www.mimuw.edu.pl/delta/delta7/mapy/mapy.htm

Map colouring In 1852 a student of the University College in London, Francis Guthrie, informed his brother Frederick that, to his knowledge, any map can be painted with only four colours in such a way that no two neighbouring countries (i.e. which have a common frontier segment) share the same colour. Indeed, as Francis had noticed, four colours are sufficient to paint the counties on the map of England, so why wouldn't they be sufficient in all other cases, for arbitrary maps drawn on paper (i.e. on the plane) or on a globe (i.e. on a sphere)? To make things precise, let's assume the frontier of each country consists of one closed curve (which is not always the case with modern states). Frederick Guthrie passed the problem on to one of his teachers, Augustus de Morgan. De Morgan proved that there is no (plane) map with five pairwise neighbouring countries. Unfortunately, this is not enough to prove the non-existence of a map requiring at least five colours (see Fig. 1) and it is not clear whether de Morgan was conscious of this gap, when he mentioned the question to Hamilton in a letter of October 23 rd Fig. 1. There are no four pairwise neighbouring countries on this map, nevertheless four colours are indispensable for the colouring.

84. Encyclopedia4U - Four Color Theorem - Encyclopedia Article four color theorem. This article is licensed under the GNU Free DocumentationLicense. It uses material from the Wikipedia article four color theorem . http://www.encyclopedia4u.com/f/four-color-theorem.html

ENCYCLOPEDIA U com Lists of articles by category ... Encyclopedia Home Page SEARCH : Four color theorem The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie. It is obvious that three colors are completely inadequate, and it is not difficult to prove that five colors are sufficient to color a map. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

85. Mbox: Qed Applied To Four Color Theorem qed applied to four color theorem. John McCarthy (jmc@sail.Stanford.EDU)Tue, 1 Nov 1994 114224 0800 http://www-unix.mcs.anl.gov/qed/mail-archive/volume-2/0051.html

qed applied to four color theorem John McCarthy jmc@sail.Stanford.EDU Tue, 1 Nov 1994 11:42:24 -0800 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: David McAllester: "Semantics" Previous message: John McCarthy: "Re: Who are the customers for QED." The 1979 proof of the four color theorem involved at least two very large computer programs. One of them showed that each of 1936 graph configurations was reducible and therefore could not occur in a minimal counterexample to the theorem. The other program showed that every (standardized) map graph contains at last one of the 1936 configurations. Mathematicians doubted the correctness of these programs. If I recall correctly, the referees wrote some of their own programs. These programs, or new programs that perform the same tasks, would be very suitable for qed verification and very likely would impress mathematicians. Next message: David McAllester: "Semantics" Previous message: John McCarthy: "Re: Who are the customers for QED."

86. M Link- Four-Color Theorem M link four-Color theorem. M link- four-Color theorem. Happy exploring have a good time kx21 (Report this post to the moderator). http://superstringtheory.com/forum/extraboard/messages9/165.html

String Theory Discussion Forum String Theory Home Forum Index M link- Four-Color Theorem Follow Ups Post Followup Extra Dimensions IX FAQ Posted by on December 24, 2002 at 21:50:18: In Reply to: Re: DickT, this could be a rehash of posted by DickT on December 23, 2002 at 07:43:38: M link- Four-Color Theorem (Report this post to the moderator) Follow Ups: (Reload page to see most recent) Post a Followup Follow Ups Post Followup Extra Dimensions IX FAQ

87. Rudolf Fritsch, The Four-Color Theorem Earth Sciences & Geography Books Reviews Rudolf Fritsch, The fourColor theorem in Earth Sciences Geography Books / BookReviews reviews at Review Centre. RUDOLF FRITSCH, THE four-COLOR theorem. http://www.reviewcentre.com/reviews30805.html

Rudolf Fritsch, The Four-Color Theorem Login/Register List Your Site Tell a Friend Bookmark ... Cookery Books Search All Items Activity Holidays America Attractions Beauty Book Reviews Buying a car Car Accessories Car Reviews Caravans Careers Caribbean Catalogues City Breaks Computer Game Reviews Computer Hardware Computer Software Cruises Digital Camera Reviews DIY Drink Reviews DVD Players and Televisions Education Electrical Appliances Entertainment Reviews Far East and Asia Fashion Film Cameras Film Reviews Finance Reviews Food Reviews Gadgets Gardening Golf HiFi Reviews High Street Shop Reviews Holiday Reviews Insurance Internet Magazine Reviews Mobile Phone Reviews Motorcycles Motoring Music Reviews Musical Instrument Reviews Office Equipment Online Shop Reviews Parenting Pets Recreation Reference Restaurant Reviews Shooting Skiing Holidays Sports Equipment Reviews Toy Reviews Travel Services Utility Companies Water Sports Book Reviews Magazine Reviews You are here : Welcome Books Book Reviews Rudolf Fritsch, The Four-Color Theorem RUDOLF FRITSCH, THE FOUR-COLOR THEOREM Amazon Reviews Review No : I had high hopes for this book, but alas it was not to be. Poor writing, even worse editing (shame on the publisher), terrible production (you need industrial tools to keep the book open). If you get ...

88. Four Color Theorem Article on four color theorem from WorldHistory.com, licensed fromWikipedia, the free encyclopedia. Return Index four color theorem. http://www.worldhistory.com/wiki/F/Four-color-theorem.htm

World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History Four color theorem Four color theorem in the news The four color theorem states that every possible geographical map can be colored with at most four color s in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie. It is obvious that three colors are completely inadequate, and it is not difficult to prove that five colors are sufficient to color a map. However, it was not until 1977 that the four-color conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

89. Joho The Blog: Four Color Maps In 3D The W Resume Back to Blog A Note to Spammers ». May 15, 2003. four Color Maps in 3D Boston Globe recently about four Colors Suffice, a book by that you only need four colors to ensure that neighboring every country on the four color map, so you need http://www.hyperorg.com/blogger/mtarchive/001505.html

Joho the Blog An Entry from the Archives The W Resume Back to Blog ... A Note to Spammers May 15, 2003 Four Color Maps in 3D There was an interesting the Boston Globe recently about Four Colors Suffice , a book by Robin Wilson on the history of the famous 4-color problem: How do you prove that you only need four colors to ensure that neighboring countries are colored differently. (More important: Why is Greenland pink?) The proof (according to the article about the book that I didn't read) was the first generated by a computer that couldn't be checked by humans: in 1976, a Cray ground through every conceivable variation and found none that required more than four colors. I have a question for the mathematically inclined (i.e., people unlike me): How many colors would you need for a 3-D map? Or, if you prefer, how many colors would you need to ensure that blocks (of any shape) stacked in any arbitrary way have differently colored neighbors? I am so bad at 3D stuff that you could tell me the answer is 2 and I would believe you, just so long as you looked at me with those doe-eyes of yours. Posted by D. Weinberger at May 15, 2003 09:35 AM

90. Math G Mission College Santa Clara One of the concepts used in the development of proof for the four ColourTheorem was the use of the ìgreedyî or ìsingle mindedî algorithm. http://www.missioncollege.org/depts/math/beard2.htm

Math Department, Mission College, Santa Clara, California Go to Math Dept Main Page Mission College Main Page This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it. To explore other such papers go to the Math G Projects Page. The Four Color Theorem Michelle Beard Math G, Spring 1999 Mission College, Santa Clara How many colors are required to color any map so that no countries with common borders are the same color? It is generally held that four colors, for any flat map, will suffice. But a belief that is commonly held and easily observed, is not a mathematical certainty. Nor does the simplicity of a question reflect the ease with which the answer can be proven. The mathematical evidence to create a valid proof that four colors are all that is required had evaded mathematicians for nearly 140 years. What became known as the Four Color Conjecture has been the cause of great fascination and frustration. It has also been the stimulus for new ideas in topology, knot theory, and the concept of mathematical proof. Historical Overview: The question was originally posed by Francis Guthrie, a former student of the famous mathematician Augustus De Morgan, in 1852. Although Francis moved on to study law, his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie presented his work on the idea to his brother asking that he pass it along to De Morgan.

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