ColourTheory The four colour theorem (or the Four Color Theorem). Mathematicians Nomore so than in the area of the four colour theorem. This http://www.adit.co.uk/html/colourtheory.html
Extractions: RGB and CYMK Colours The Four Colour Theorem (or the Four Color Theorem) Mathematicians and map makers share a lot of common ground. No more so than in the area of the four colour theorem. This theorem simply states that any map in a single plane can be coloured using four-colours in such a way that any regions sharing a common boundary (other than a single point) do not share the same colour. The theorem was first propounded by F Guthrie in 1853. Fallacious proofs have come and gone starting with Kempe in 1879 and Tait an 1880. In 1977 K. Appel and W. Haken used computer assistance to test many different combinations to effectively prove that four colours was all that was required in all instances. Since then, it may be that a mathematical proof has, at last, been arrived at. So, if we know that we can colour any map using just four colours how to we go about it. A little though would indicate that the problem is not straightforward. Simply starting with a random colour and an arbitrary polygon would soon lead to an impasse when the process met an area bounded by more than three other areas yet to be coloured. Kempe is credited with first recording that, when tackling a map of national boundaries, those with three or fewer neighbours presented no problems. His solution was to ignore (temporarily remove from the map) those countries with three or fewer neighbours. This process will immediately simplify the remainder of the map and can be repeated until only countries with three or fewer neighbours remain. The remaining areas can then be coloured. Then the missing countries can be restored in reverse order to their removal and coloured as the process proceeds. This effective procedure is know as Kempe transformations.
Maps, Colouring, Four Colour Theorem only four colours, four colour theorem Every map drawn on a plane can be colouredwith at most four colours in such a way that neighbouring contries are http://www.geog.port.ac.uk/webmap/hantsmap/hantsmap/fourcols.htm
Extractions: At a lecture given by Agustus de Morgan, professor of mathematics, University College, London, 23 October 1852, one of the students, Francis Guthrie, asked a simple question. De Morgan mentioned the suggested conjecture in a letter to Sir William Rowan Hamilton, mathematician and physicist; was it a fact that:- ... if a figure be divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted but not more. ... The conjecture was first proposed by Frederick's elder brother Francis Guthrie, and earlier student of de Morgan's at University College, graduated 1850, later a professor of mathematics in South Africa.
The Four Colour Theorem References References for The four colour theorem. DA Holton and S Purcell, The four colourtheorema short history, Austral. Math. Soc. Gaz. 6 (1) (1979), 11-14. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/~DZ929F.htm
Extractions: N L Biggs, E K Lloyd and R J Wilson, Graph Theory 1736-1936 (Oxford, 1986). N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Mathematica G D Birkhoff, Collected mathematical papers Vol. III : The four color problem, miscellaneous papers (New York 1968). L Arias C, A chronological account of the various theories developed for solving the four-color problem (Spanish), Rev. Integr. Temas Mat. D A Holton and S Purcell, The four colour theorem-a short history, Austral. Math. Soc. Gaz. P C Kainen, Is the four color theorem true?, Geombinatorics Proceedings of the Seminar on the History of Mathematics (Paris, 1982), 43-62. T L Saaty, The four-color problem : assaults and conquest (New York, 1977). J Wilson, New light on the origin of the four-color conjecture, Historia Mathematica Back to the history topic History Topics Index
Count On (A large version of the tiling is downloadable as an Acrobat PDF file as MT7.pdf)The four colour theorem A tiling is like a map and colouring maps has been a http://www.counton.org/explorer/morphing/13usedownload.shtml
Extractions: There are a number of morphing tilings on the Downloads page . The following are a few suggestions for using them and any you might create yourself to explore patterns in new ways. Morphing patterns are more complex than simply repeating patterns and so offer more possibilities for creativity. Creating tilings can be a slow process if you draw them by hand, although a computer can speed up the process, with the right drawing package. Once you have a set of tiles then you can copy them to build up the tiling. To help you create tilings easily, you can also download a set of True Type fonts which allow you to create tilings in a word-processor. Type in a character and a tile appears. Click here for instructions on how to use these fonts. These pages plus the download files are meant as starting points from which to develop your own ideas. We hope you will find new ways to create new tilings, not just copy the ones here. Analysing the patterns in the tilings Everyone sees something different in morphing tilings. As the eye wanders over them it is possible to see the changes in different ways - sometimes you can focus on a local pattern and sometimes you see a movement over a larger area which is hard to pin down. They are ideal for teaching analytical skills and being able to put discoveries into words. Seeing what is happening is only the first step in showing someone else what you can see.
Four Colour Theorem - BlueRider.com four colour theorem listen domain availability, four colour theorem.Your search results search for four_colour_theorem on http://four_colour_theorem.bluerider.com/wordsearch/four_colour_theorem
The Four Colour Theorem The four colour theorem. Theorem 1 3 . The four colour theorem is thusequivalent to the following statement about graphs Theorem http://www.shef.ac.uk/puremath/theorems/fourcol.html
Extractions: The four colour theorem Theorem 1 Given a map drawn in the plane we can colour the countries with red, green, blue and orange in such a way that any two adjacent countries have different colours. Here countries are assumed to be connected; we disallow the United States, for example, as Alaska and Hawaii are separated from the main body of the country. Two countries are only considered to be adjacent if they have a common boundary of nonzero length, not if they just touch at a single point. The theorem is illustrated by the following colouring of the map of Africa: For a detailed history of this problem, see the St Andrews history of mathematics site. Briefly, Francis Guthrie conjectured in 1852 that the result was true, but was unable to prove it. Over the next 27 years a number of powerful mathematicians such as Cayley considered the question but were also unable to answer it. A proof was announced in 1879 by Alfred Kempe to great acclaim, but eleven years later Percy Heawood revealed a fatal error in his argument. Over the next 86 years steady progress was made, showing for example that every map with at most 95 countries can be coloured with four colours. In 1976, Appel and Haken gave a proof of the theorem, which required consideration of 1476 different special cases, with computer assistance. The four-colour theorem is thus unusual in at least two ways: one of very few major results for which an incorrect proof has remained undetected for a considerable time, and the first major result to be proved by computer. In 1996 Robertson, Sanders, Seymour and Thomas gave a
Department Report The Journey of the four colour theorem Through Time. Andreea S. Calude. Abstract Keywordsfour colour theorem. Math Review Classification Primary 05C10. http://www.math.auckland.ac.nz/deptdb/dept_report_view.php?id=466
Sci.math FAQ: The Four Colour Theorem sci.math FAQ The four colour theorem. 21, 1977, pp. 429567. N. Robertson, D. Sanders,P. Seymour, R. Thomas The four colour theorem Preprint, February 1994. http://www.uni-giessen.de/faq/archiv/sci-math-faq.fourcolour/msg00000.html
A New Proof Of The Four-Colour Theorem A new proof of the fourcolour theorem. Neil Robertson, Daniel P. Sanders,Paul Seymour, and Robin Thomas. Abstract. The four-colour http://www.mpim-bonn.mpg.de/external-documentation/era-mirror/1996-01-003/1996-0
The Map Room: Archives (November 2003) available.). Posted by Jonathan Crowe at 919 AM. Thursday, November13, 2003. The fourcolour theorem. The four-colour theorem states http://www.mcwetboy.net/maproom/2003_11_01_archive.phtml
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Four Color Theorem - InformationBlast four color theorem Information Blast. four color theorem. EnlargeExample of a four colour map. The four color theorem states that http://www.informationblast.com/Four_color_theorem.html
Extractions: Example of a four colour map 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. Significant results in that area have been done by Croatian mathematician Danilo Blanusa in the 40s.by finding original snark . 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
4-colour Theorem different computers. For the latest word see Robin Thomas, An Updateon the fourColor theorem, Notices of the Amer. Math. Soc. 45 http://mathcentral.uregina.ca/RR/database/RR.09.97/fisher1.html
Extractions: A nice discussion of map coloring can be found in "The Mathematics of Map Coloring," which Professor H.S.M. Coxeter wrote for the Journal of Recreational Mathematics, 2:1 (1969). He began by pointing out that in almost any atlas, 5 or 6 colors are used in a map of the United States to distinguish neighboring states. "Apparently the artist did not realize that four colors would have sufficed. (It is understood that two states may be colored alike if they merely have a point in common, as in the case of Arizona and Colorado.)" This leads to the mathematical question, Can every conceivable map (on a sphere or a plane) be colored with four colors, or does some particular map really need five? The question was first posed in 1852 by Francis Guthrie, a mathematics graduate student in London at the time. He had noticed the sufficiency of four colors for distinguishing the counties in a map of England. The question was passed along to several important British mathematicians (De Morgan, Hamilton), but apparently it was not seriously investigated until Cayley in 1878 challenged the members of the London Mathematical Society to solve it. From that time until its computer solution nearly 100 years later the problem stood alongside Fermat's last theorem among the great mathematical challenge of the century. Like the Fermat problem, the map-coloring question is easily stated and can easily be understood by anybody. Both problems lack any important consequences, yet have led to extraordinarily important new mathematical ideas and techniques. Both problems are alluring and elusive.
Four Colour Map Theorem From FOLDOC four colour map theorem. mathematics, application (Or four colourtheorem ) The theorem stating that if the plane is divided into http://www.swif.uniba.it/lei/foldop/foldoc.cgi?four colour map theorem
Gnist.no: Fagbokhandelen På Internett Wilson, Robert A. Graphs, colourings and the fourcolour theorem Pris 319.00. PartII - Related Topics5 Other approaches to the four-colour theorem. http://www.gnist.no/visbok.php?varenummer=85044&side=PBV
Extractions: VIEW BASKET Quick Links About OUP Career Opportunities Contacts Need help? oup.com Search the Catalogue Site Index American National Biography Booksellers' Information Service Children's Fiction and Poetry Children's Reference Dictionaries Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks Humanities International Education Unit Journals Law Medicine Music Oxford English Dictionary Reference Rights and Permissions Science School Books Social Sciences World's Classics UK and Europe Book Catalogue Help with online ordering How to order Postage Returns policy ... Table of contents Robert A. Wilson , Professor of Group Theory, The University of Birmingham
Graphs, Colourings And The Four-Colour Theorem Email a friend about this book, Graphs, colourings and the fourcolour theorem.Format, Hardcover. Subject, Mathematics / Graphic Methods. ISBN/SKU, 0198510616. http://www.booksmatter.com/b0198510616.htm
Extractions: This brief monograph considers the attempts to resolve the four-color theorem, and outlines the contributions to graph theory that grew out of those attempts. The book develops the theoretical concepts as it progresses, providing an introduction to elementary graph theory. Chapters discuss Euler's formula, Kenpe's approach, Kuratowski's theorem, reducibility, discharging, and related topics. Wilson teaches at the University of Birmingham. Annotation c. Book News, Inc., Portland, OR (booknews.com)