The Four Color Theorem « Gödel’s Lost Letter And P=NP Apr 24, 2009 Kenneth Appel and Wolfgang Haken made the Four Color Conjecture into the Four Color Theorem (4CT) in 1976. Their proof, as you probably know http://rjlipton.wordpress.com/2009/04/24/the-four-color-theorem/
Extractions: Kenneth Appel and Wolfgang Haken made the Four Color Conjecture into the Four Color Theorem (4CT) in 1976. Their proof, as you probably know, relies essentially on computer aided checking, which makes it not checkable by a human. Today I want to talk about their proof, the nature of computer aided proofs, and new alternative approaches to the 4CT. I also want to draw a parallel between their great achievement and some of our open problems. I was at Yale University, in the Computer Science Department, when they discovered their proof. The Yale mathematics department quickly invited Haken to give a talk on their proof. As you might expect the room was filled, and Haken give a wonderful talk: on the history of the famous problem, on previous approaches, and on their proof. The other recollection is what Haken said to me during a private conversation I had with him, before his talk. I quickly realized that Haken, who previously had been a Siemens engineer, had a very pragmatic approach to the four color problem. To him it was more like an engineering project, then a delicate mathematical proof. One of the central issues was discovering whether certain graphs were reducible, which I will define later on. The proof required that a specific set of about two thousand small planar graphs were reducible.
Four Color Theorem Applets « Division By Zero Dec 9, 2008 The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in http://divisbyzero.com/2008/12/09/four-color-theorem-applets/
Extractions: A blog about math, puzzles, teaching, and academic technology Posted by: Dave Richeson The four color theorem is a beloved result with a long and fascinating history. The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in 1852 by Francis Guthrie. After many, many failed proofs, the conjecture was finally put to rest in 1976 (using a computer) by Kenneth Appel and Wolfgang Haken. To this date there is no pencil-and-paper proof of this seemingly elementary theorem. Today I was exploring the website of the Japanese publisher and puzzle-popularizer Nikoli . The website has a number of interesting Flash-based logic games (Kakuro, Nurikabe, Sudoku, etc.). They also have five four color theorem applets such as the one below. The goal of each puzzle is to find a four-coloring of the given map. I wish they had more than five of these! Posted in Math applet four color theorem Francis Guthrie ... A new continued fraction for pi For pencil-and-paper proof(s) see my arXiv papers. For illustrations including one of the puzzle visit:
Four Color Theorem And Lie Algebras Ars Mathematica Jan 26, 2008 Thanks to Greg Muller, I m looking at this paper by Dror BarNatan that reduces the Four Color Theorem to a plausible statement about Lie http://www.arsmathematica.net/archives/2008/01/26/four-color-theorem-and-lie-alg
Serendip: The 4 Color Problem Dec 13, 2004 The four color problem is a simple and yet quite significant problem in mathematics, with implications for thinking about human http://serendip.brynmawr.edu/playground/fourcolor/
Four-colour Map Problem -- Britannica Online Encyclopedia fourcolour map problem, problem in topology, originally posed in the early 1850s and not solved until 1976, that required finding the minimum number of http://www.britannica.com/EBchecked/topic/214896/four-colour-map-problem
Extractions: document.write(''); Search Site: With all of these words With the exact phrase With any of these words Without these words Home My Britannica CREATE MY four-colour ... NEW ARTICLE ... SAVE Table of Contents: four-colour map problem Article Article Related Articles Related Articles External Web sites External Web sites Citations Primary Contributor: Robert Osserman ARTICLE from the four-colour map problem problem in topology , originally posed in the early 1850s and not solved until 1976, that required finding the minimum number of different colours required to colour a map such that no two adjacent regions (i.e., with a common boundary segment) are of the same colour. Three colours are not enough, since one can draw a map of four regions with each region contacting the three other regions. It had been proved mathematically by the English attorney Alfred Bray Kempe in 1879 that five colours will always suffice; and no map had ever been found on which four colours would not do. As is often the case in mathematics, consideration of the problem provided the impetus for the discovery of related results in topology and
Extractions: Pages : Matthew 06-September-2005, 07:26 AM So you only need four color's to color a map? What if all the localaties met at a single point? See the attached diagram. Lets a ssume the lines are perfect and borders intersect at a single point. Would that mean that more than 4 colors are needed? 06-September-2005, 08:05 AM Wikipedia: Four color theorem (http://en.wikipedia.org/wiki/Four_color_theorem) The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than 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. jfribrg 06-September-2005, 12:50 PM worzel 06-September-2005, 01:55 PM But have they proved it for all colours? jfribrg 06-September-2005, 02:13 PM But have they proved it for all colours? I'm not sure I understand what you mean. It is trivial to prove that 3 or fewer colors is not sufficient. All you need is to show that there is a map where 3 colors aren't enough. Here is a simple counterexample: Start with a thick circle with a hole in the center. Section the circle into 3 parts, each of which must be a different color. The hole in the center must also be different from the first 3 colors used on the circle, which proves that 3 colors are too few. For 5 colors, take a 4 color map and arbitrarily change one of the areas to a color that isn't anywhere else on the map. For 6 colors, take the above mentioned 5 color map and change one area to a color that isn't yet used. Do this for as many colors as you have in your crayon box.
Sci.math FAQ: The Four Colour Theorem Feb 20, 1998 An equivalent combinatorial interpretation is Theorem 3 Four Colour Theorem Every loopless planar graph admits a vertexcolouring with at http://www.faqs.org/faqs/sci-math-faq/fourcolour/
4 Colour Theorem « Bodmas Blog The four colour theorem was the first major theorem to be solved by relying on a computer algorithm (with the provisio that the theorem depends on the http://bodmas.org/blog/maths/4-colour-theorem/
Joseph Malkevitch: Four-Color Problem Tidbit Sep 13, 2003 Robertson, N., and D. Sanders, P. Seymour, R. Thomas, The four color theorem, J. Combinatorial Theory Ser. B. 70 (1997) 244. http://www.york.cuny.edu/~malk/tidbits/tidbit-four-color.html
Extractions: A common way to assist people get a feel for world geography is to display the various countries on a (spherical) globe. To help with understanding which countries share a border it is useful to have countries on the globe which share a common boundary (an edge, in graph theory terminology) have different colors. What is the minimum number of colors necessary to color the countries on a sphere so that countries that meet along an edge get different colors? To answer this mathematical question for spheres is a bit of a nuisance because as one looks at a globe one can only see the countries on one hemisphere, because the other countries are hidden from view. Is there some way of displaying the information on a sphere in a more convenient way? In fact there is! The problem of coloring maps on a sphere can be reduced to the problem of coloring maps in the plane by a simple geometrical transformation. This approach to problem solving is typical of the way that mathematicians work. Taking a problem in one setting and trying to recast into an easier related setting. The transformation this time is know as stereographic projection. Take a sphere and let N represent its North pole and S represent its South pole. Now let P be a plane tangent (i.e. touching at exactly one point) to the sphere at S. If R is any point of the sphere except N, the image of R in P, which will be denoted s(R) will be the point where the line NR meets the plane P. The two dimensional analog of stereographic projection, which should make clear what happens in 3-dimensions, is illustrated in the diagram below:. Notice that points which are close to the North pole are transformed to points which are very far away from the South pole.
490F08:Four Color Theorem - Classwiki The Four Color Theorem is a problem whose proof avoided mathematicians for quite some time. It has been proven, disproven and reproven multiple times, http://192.160.243.156/~cstaecker/classwiki/index.php/490F08:Four_Color_Theorem
Extractions: Jump to: navigation search This page was somebody's semester paper, and its topic was probably not covered in our class or textbook. The Four Color Theorem is a problem whose proof avoided mathematicians for quite some time. It has been proven, disproven and reproven multiple times, showing that a true understanding of the math behind it still eludes the mathematical community. The theorem itself states that any two-dimensional planar map can be colored with at most four colors so that no edge is colored by the same color on both sides. The history of the problem begins in 1852, when the “Four Color Conjecture” was proposed to Frederick Guthrie, by his brother Francis. Francis had noticed that he needed 4 colors for a map of Europe, and felt that four would be sufficient for all maps. Both Francis and Frederick were unable to arrive at any conclusion on the topic, so help was sought after. Augustus DeMorgan, a professor at the University of London, was recruited, but he too was unable to find a proof. In his efforts, he reached out to Sir William Rowan Hamilton (the originator of ideas such as Hamiltonian Circuits). Hamilton, like DeMorgan and both Guthries, found the problem to be quite perplexing and was unable to find any kind of proof. However, none of them had been able to find a counter-example, meaning that no conclusion could be made (Math Pages). From this point, the problem entered the mathematical landscape but would not see a valid proof for over 100 years.
