41. Count On The four colour theorem A tiling is like a map and colouring maps has been a The four colour theorem was proved in 1976 but because it was proved by http://www.mathsyear2000.co.uk/explorer/morphing/13usedownload.shtml

Using the Morphing Tilings downloads 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. A starting point for designing your own tilings 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.

42. From Karl Brace Brace@cadtls.enet.dec.com Newsgroups Rec.org information on the four color theorem, but not to so many newsgroups. for title that four color theorem or title that four colour theorem http://www.math.niu.edu/~rusin/known-math/97/4ct

From: Karl Brace , Bob McQueer

43. Four Colour Map Theorem From FOLDOC four colour theorem . four colour map theorem. mathematics, application (Or fourcolour theorem ) The theorem stating that if the plane is divided http://www.instantweb.com/foldoc/foldoc.cgi?four colour theorem

44. Citations: The Fourcolour Theorem - Robertson, Sanders, Seymour, Thomas (Researc The four colour theorem as a possible corollary of binomial. The correspondingstatement is the famous four colour theorem for which recently a shorter http://citeseer.ifi.unizh.ch/context/423868/0

38 citations found. Retrieving documents... N. Robertson, D. Sanders, P. Seymour, and R. Thomas, The four-colour theorem . J. Combin. Theory Ser. B, 70(1):2-44, 1997. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: Chromatic Numbers of Quadrangulations on Closed Surfaces - Archdeacon, Hutchinson.. (1 citation) (Correct) ....colors so that any two adjacent vertices get di erent colors. The chromatic number of G, denoted by (G) is the smallest k such that G is k colorable. We It is well known that every planar graph G has chromatic number at most 4, the Four Color Theorem However, if we restrict G to be a quadrangulation, then the chromatic number decreases to 2 since every quadrangulation on the sphere is bipartite. For general closed surfaces F , the chromatic number of a graph G which is embedded in can be bounded by: 6 6 7 7 7 This is best .... N. Robertson, D. Sanders, P. Seymour, and R. Thomas, The four colour theorem , J. Combin. Theory Ser. B , 70 (1997) 2-44. Polytope Skeletons And Paths - Kalai (1997) (1 citation) (Correct) ....all together, an even number of edges.

45. Count On The four colour theorem. A tiling is like a map and colouring maps has been a The four colour theorem was proved in 1976 but because it was proved by http://www.counton.org/explorer/morphing/using-the-downloads/

@import "/styles/general.css"; @import "/styles/home.css"; @import "/styles/events.css"; @import "/styles/numberland.css"; @import "/styles/resources.css"; @import "/styles/explorer.css"; @import "/styles/sum.css"; @import "/styles/cards.css"; @import "/styles/links.css"; @import "/styles/sitemap.css"; @import "/styles/online-stories.php?rkeyword="; @import "/styles/fractions-style.css"; Count On Explorer Explorer: Maths and Magic The Wise Wizz of Woo Patterns Maths Library ... Gridwarping Using the Morphing Tilings downloads 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

46. Verifying The Four Colour Theorem Verifying the four colour theorem. Last fall we completed a fully formal proofof the famous Four Color Theorem, which we mechanically checked using the Coq http://www.cl.cam.ac.uk/Research/HVG/ARG_Lunches/abstracts/23-Mar-2005.html

ARG Lunch Talk by Georges Gonthier Microsoft Research , on 23 March 2005: Verifying the Four Colour Theorem Last fall we completed a fully formal proof of the famous Four Color Theorem, which we mechanically checked using the Coq proof assistant. This is the first time that the technology of proof assistants is successfully used to verify a mathematical result that is beyond the reach of conventional pencil-and-paper mathematics. In this talk we will review the main obstacles of this formal proof and present the main techniques that were used to overcome them. Known as computational reflection, this technique consists in embedding working programs inside formal mathematical statements; most of the work involved in proving such computable statements is carried out by the programs embedded in them. We will also show how this computational approach reshaped and sharpened the analysis of the mathematical problem.

47. Maps, Colouring, Four Colour Theorem The four colour theorem does not arise out of practical cartography. The proofof the four colour theorem is not simple; it involves more than a 1000 http://www.geog.port.ac.uk/webmap/hantsmap/hantsmap/fourcols.htm

Old Hampshire Mapped Four Colours? Guthrie 1850-52 Francis Guthrie's Conjecture 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. This then is the:- only four colours Four Colour Theorem Every map drawn on a plane can be coloured with at most four colours in such a way that neighbouring contries are coloured differently. Note:- Some maps can be coloured with fewer colours.

48. Four Colour Map Theorem - Computing Reference - ELook.org mathematics, application (Or four colour theorem ) The theorem stating that if Previous Terms, Terms Containing four colour map theorem, Next Terms http://www.elook.org/computing/four-colour-map-theorem.htm

eLook Home eLook Computing Reference Search Computing Reference More Resources Information US City Data Worldwide Weather Entertainment Video Game Cheats Food Nutrition Recipes Literature Literature Reference Computing Dictionary Internet Encyclopedia Programming Reference ... Thesaurus By Letter: Non-alphabet A B C ... Z Four colour map theorem The proof, due to Appel and Haken, attained notoriety by using a computer to check tens of thousands of cases and is thus not humanly checkable, even in principle. Some thought that this brought the philosophical status of the proof into doubt. There are now rumours of a simpler proof, not requiring the use of a computer. See also chromatic number Terms Containing four colour map theorem FOSIL fossil foundation FOundation for Research and Technology - Hellas ... Contact

49. Engineering Database four colour theorem. It is desired to colour a political map on a plane so thatcountries sharing a common boundary are coloured differently. http://www.diracdelta.co.uk/science/source/f/o/four colour theorem/source.html

Four Colour Theorem It is desired to colour a political map on a plane so that countries sharing a common boundary are coloured differently. Experience of map-makers has shown that four colours suffice. This assertion in its precise formulation was an unsolved problem in mathematics for over a hundred years. It has now been "proved" with the assistance of a computer. Search for Four Colour Theorem on or Web www.diracdelta.co.uk google_ad_client = "pub-9605428275898003";google_ad_width = 728;google_ad_height = 15;google_ad_format = "728x15_0ads_al_s";google_ad_channel ="9821813911";google_color_border = "CCCCCC";google_color_bg = "FFFFFF";google_color_link = "000000";google_color_url = "666666";google_color_text = "333333"; My Links: google_ad_client = "pub-9605428275898003";google_ad_width = 160;google_ad_height = 600;google_ad_format = "160x600_as";google_ad_channel ="";google_color_border = "CCCCCC";google_color_bg = "FFFFFF";google_color_link = "000000";google_color_url = "666666";google_color_text = "333333"; Dirac Delta Consultants Limited

50. Anchoring Expository Text In Formal Mathematics Verifying the four colour theorem. Georges Gonthier May 20, 2005. Abstract.The 150 year old four colour theorem is famous for being the first important http://www.cs.cornell.edu/Nuprl/PRLSeminar/PRLSeminar04_05/Gonthier/May20_05.htm

PRL Seminars Verifying the Four Colour Theorem Georges Gonthier May 20, 2005 Abstract Home Introduction Authors Topics ... PRL Project

51. La Trobe University, Bendigo Mathematics Seminars Tutte s Flow Conjectures Beyond the four colour theorem, Abstract About 30years ago the four colour theorem (for planar graphs) was proved by a http://www.latrobe.edu.au/maths/seminars/

52. Research Report Abstract This is a historical survey of the four colour theorem and a discussion of four colour theorem. Math Review Classification Primary 05C10. Last Updated http://www.math.auckland.ac.nz/Research/Reports/view.php?id=466

53. Sci.math FAQ: The Four Colour Theorem Subject sci.math FAQ The four colour theorem. This article was archived around17 Feb 2000 225204 GMT. All FAQs in Directory scimath-faq http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/fourcolour.html

Note from archiver cs.uu.nl: This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archiver. Subject: sci.math FAQ: The Four Colour Theorem This article was archived around: 17 Feb 2000 22:52:04 GMT All FAQs in Directory: sci-math-faq All FAQs posted in: sci.math Source: Usenet Version Archive-name: sci-math-faq/fourcolour Last-modified: February 20, 1998 Version: 7.5 http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

54. Paul's Web Space Edgecolouring pairs of 3-regular trees and the four colour theorem of Planar Maps.14th Postgraduate Combinatorics Conference, Department of Mathematics, http://www.csc.liv.ac.uk/~pauls/index.php?section=research&subsection=graph_theo

55. Stanford Computer Forum - Events Calendar - Detail & Abstract The 150 year old four colour theorem is famous for being the first importantmathematical result whose proof, completed in 1976 by Appel and Haken, http://forum.stanford.edu/events/calendar/abstract.php?eventId=1132

56. University Of Wuppertal - Dep. Of Mathematics - Poster Consequently, proving the four colour theorem correct also means proving thatthe program is implemented correctly and that the computer works correctly. http://www.math.uni-wuppertal.de/org/Poster/TextE.htm

The four colour problem Primes Crash Test Computerized Tomography The four colour problem In 1852, while colouring a map representing the english counties, the british mathematician Francis Guthrie realized that only four colours where necessary to satisfy the criterion that neighbouring counties should have different colours. It turns out that this is true for any (real or imaginary) map. The poster shows such a colouring for the countries of Europe and, in small, for three other imaginary maps. It has taken more than 100 years before a correct proof for the four colour theorem has been found. The proof by Appel and Haken in 1976 has solicited much of a controversial discussion since it heavily relies on computer calculations. These are so extensive that humans cannot verify them "by hand". Consequently, proving the four colour theorem correct also means proving that the program is implemented correctly and that the computer works correctly. Instead of taking Europe and its countries, we could illustrate the four colour theorem with Germany and its regions. This can be found at MathePrisma (in German). There you can also colour some other maps on your own and develop strategies to get the right colouring fast.

57. 'Four Colours Suffice' The four colour theorem the statement that four colours suffice to fill in any map In the case of the four colour theorem, this includes the important http://plus.maths.org/issue25/reviews/book2/

about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 25 May 2003 Contents Features A whirlpool of numbers How maths can make you rich and famous: Part II Model behaviour The crystal ball Career interview Career interview: Primary teacher Regulars Plus puzzle Pluschat Reviews 'Four Colours Suffice' 'Mindbenders and Brainteasers' 'Kepler's Conjecture' 'Dissections: Plane and Fancy' ... posters! May 2003 Reviews 'Four Colours Suffice' reviewed by Helen Joyce Four Colours Suffice The Four Colour Theorem - the statement that four colours suffice to fill in any map so that neighbouring countries are always coloured differently - has had a long and controversial history. It was first conjectured 150 years ago, and finally (and infamously) proved in 1976 with much of the work done by a computer. The published proof relied on checking 1432 special cases, which took more than 1,000 hours of computer time. Reactions to the news that the theorem had succumbed to this brute force attack varied from excitement to scepticism, but with mostly disappointment on the way. There was a widespread feeling that this "proof" was no proof at all. Robin Wilson's book is a clear and well-written description of the history of the theorem; the people who worked on it, the mathematics they invented in their attempts to solve it, and the controversy that followed when it was finally proved. It is also that rarity - a book where mathematics is actually done in front of the reader, and which is also widely intelligible to a lay audience.

58. Mathematics | Proof And Beauty | Economist.com The first is the “four colour theorem”, which is perhaps the mathematical The idea behind both proofs of the four colour theorem is to suppose that the http://www.economist.com/science/displayStory.cfm?story_id=3809661

59. Powell's Books - Graphs, Colourings And The Four-Colour Theorem (Oxford Science Powell s Books is the largest independent used and new bookstore in the world.We carry an extensive collection of out of print rare, and technical titles http://www.powells.com/biblio?isbn=0198510624

60. Powell's Books - Graphs, Colourings And The Four-Colour Theorem By Robert A Wils This text discusses various attempts to solve the famous fourcolour theorem andsome of the mathematics which developed out of these attempts. http://www.powells.com/cgi-bin/biblio?inkey=17-0198510616-0

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