Claim Of Proof To Four-Color Theorem The recent announcement by two American computer scientists that they have a proof of the four colour theorem, although they certainly have not published a proof, coupled with the http://www.lawsofform.org/gsb/nature.html
Extractions: 17 December 1976 Sirs The recent announcement by two American computer scientists that they have a proof of the four colour theorem, although they certainly have not published a proof, coupled with the fact that they are widely reported as saying they believe that no simple or elegant proof of this theorem is possible, prompts me to refer to the work of me and my brother, the late D J Spencer-Brown, on this theorem as early as 1960-1964. As reported in 1969 [ ], we found during this period an extremely elegant way of expressing the four-colour conjecture (as it then was) which, if verified, would lead to a correspondingly elegant proof. As is well known, the difficulty of the foul colour problem stems from the fact that the Heawood formulae [ ], say Hmin, Hmax, giving the minimum and maximum values for the chromatic numbers of surfaces (Sg) of connectivity g, give Hmin = Hmax = [(1/2)(7 + (24g - 23)^(1/2) )] for g > 1
The Four Color Theorem The Four Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of counties of England noticed that four colors sufficed. http://people.math.gatech.edu/~thomas/FC/
Extractions: History. Why a new proof? Outline of the proof. Main features of our proof. ... References. History. The Four Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of counties of England noticed that four colors sufficed. He asked his brother Frederick if it was true that any map can be colored using four colors in such a way that adjacent regions (i.e. those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is due to Cayley in 1878. A year later the first `proof' by Kempe appeared; its incorrectness was pointed out by Heawood 11 years later. Another failed proof is due to Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe discovered what became known as Kempe chains, and Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge-coloring. The next major contribution came from Birkhoff whose work allowed Franklin in 1922 to prove that the four color conjecture is true for maps with at most 25 regions. It was also used by other mathematicians to make various forms of progress on the four color problem. We should specifically mention Heesch who developed the two main ingredients needed for the ultimate proof - reducibility and discharging. While the concept of reducibility was studied by other researchers as well, it appears that the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and that it was he who conjectured that a suitable development of this method would solve the Four Color Problem.
How To Make Leather Juggling Balls This would illustrate the FourColour theorem except that the Four-Colour theorem applies to maps on a plane rather than maps on a sphere; still, it refers to it. http://www.pjb.com.au/jug/leatherballs.html
Extractions: These are Peter Billam 's suggestions for making the most personal and the prettiest juggling balls in the universe. The dodecahedral pattern is how balls were made in ancient Greece. Modern footballs are made in a similar way, though they have more faces and are reinforced with a bladder inside. These balls as I make them are stuffed with linseed and don't bounce; I'm not sure how well they would stand up to being used in a bat-and-ball game as I've never tried - in my opinion they're far too beautiful to be hit around in the dirt. They are vulnerable to sharp objects, and to moisture. They can also get scuffed by rough surfaces. If filled tightly, they have a stage-ball feel, nothing like a bean-bag. The patterns are written in Postscript. I find Postscript a very useful language, and write a lot of it: Muscript is my music typesetting program, which produces Postscript output I use Muscript to typeset my own musical compositions and arrangements All the logo gifs and background gifs for my web site are from Postscript originals.
The Four Color Theorem N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, A new proof of the four colour theorem, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 1725 (electronic). http://people.math.gatech.edu/~thomas/FC/fourcolor.html
Extractions: History. Why a new proof? Outline of the proof. Main features of our proof. ... References. History. The Four Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of counties of England noticed that four colors sufficed. He asked his brother Frederick if it was true that any map can be colored using four colors in such a way that adjacent regions (i.e. those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is due to Cayley in 1878. A year later the first `proof' by Kempe appeared; its incorrectness was pointed out by Heawood 11 years later. Another failed proof is due to Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe discovered what became known as Kempe chains, and Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge-coloring. The next major contribution came from Birkhoff whose work allowed Franklin in 1922 to prove that the four color conjecture is true for maps with at most 25 regions. It was also used by other mathematicians to make various forms of progress on the four color problem. We should specifically mention Heesch who developed the two main ingredients needed for the ultimate proof - reducibility and discharging. While the concept of reducibility was studied by other researchers as well, it appears that the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and that it was he who conjectured that a suitable development of this method would solve the Four Color Problem.
Ashay Dharwadker A New Proof of The Four Colour Theorem, http//www.dharwadker.org, 2000. A new proof of the famous Four Colour Theorem using Steiner systems, Eilenberg modules, Hall matchings http://www.dharwadker.org/profile.html
Extractions: Based on the proof of the four color theorem and the grand unification of the standard model with quantum gravity, we show how to derive the values of the famous Cabibbo angle and CKM matrix, in excellent agreement with experimental observations. We make a precise prediction for the elusive Higgs boson mass M H GeV , as a direct consequence of our theory.
Ideas, Concepts, And Definitions The Four Color Problem was famous and unsolved for many years. The proof of the 4Color-Theorem is a doorway to some interesting questions about the http://www.ccs3.lanl.gov/mega-math/gloss/math/4ct.html
Extractions: (See also: The Mathematics Behind the Maps The Most Colorful Math of All , and The Story of the Young Map Colorer The Four Color Problem was famous and unsolved for many years. Has it been solved? What do you think? Since the time that mapmakers began making maps that show distinct regions (such as countries or states), it has been known among those in that trade, that if you plan well enough, you will never need more than four colors to color the maps that you make. The basic rule for coloring a map is that no two regions that share a boundary can be the same color. (The map would look ambiguous from a distance.) It is okay for two regions that only meet at a single point to be colored the same color, however. If you look at a some maps or an atlas, you can verify that this is how all familiar maps are colored. Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors. When mathematicians picked up the thread of the conversation, they began by asking questions like: Are you sure that four colors are enough? How do you know that no one can draw a map that requires five colors? What is it about the way that regions are arranged and touch each other in a map that would make such a thing true? When the question came to the European mathematics community at the end of the 19th century, it was perceived as interesting but solvable. Prominent and experienced mathematicians who tackled the problem were surprised by their inability to solve it. Take for example, this account from
Re: Four Color Theorem Steinberg's Conjecture. It refers to a cycle surrounded by triangular faces (whose 3rd vertex points It follows from the four colour theorem. http://sci.tech-archive.net/Archive/sci.math/2006-07/msg03105.html
Four Colour Theorem Worksheet See the page on the Four Colour Theorem first! Draw any number of regions within this square. Try to colour them by four colours so that there are not two adjacent areas that are http://www.mingl.org/matematika/worksheets/fourcolourtheorem.htm
Extractions: See the page on the Four Colour Theorem first Draw any number of regions within this square. Try to colour them by four colours so that there are not two adjacent areas that are the same colour. Try again on the map provided. Do you ever get to a situation when you think you need more than four colours? Try whether you need more than four colours to colour the following map. Think of countries as single points (vertices) and their borders as lines (edges) that connect them. This kind of a graph is called planar graph. Here is one planar graph And here is another The Four Colour Theorem states that any number of points and lines reduces itself to a map which only needs four colours. This graph is said to be 'complete' if there are no more connections that can be made between the points without them crossing other connections. The Four Colour Theorem states that there is no graph which contains any set of five mutually connected vertices. You can try for yourself!
Heawood Summary Heawood made important contributions to the four colour theorem. JOC/EFR October 2003. The URL of this page is http//wwwhistory.mcs.st-andrews http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Heawood.html
The Four Color Problem - Flash Game The Four Color Problem. Color each part of the map alternately with the other player. The player that covers larger area wins the game. http://www.gamedesign.jp/flash/fourcolor/fourcolor.html
Extractions: PgNo=9; GAMEDESIGN > The Four Color Problem AC_FL_RunContent( 'codebase','http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=7,0,0,0', 'width','560', 'height','400', 'src','fourcolor', 'menu','false', 'wmode','opaque', 'quality','high', 'pluginspage','http://www.macromedia.com/go/getflashplayer', 'FlashVars','lang=1', 'movie','fourcolor' );
Question Corner -- The Four Fours Problem (If I'm wrong and you're referring to something else, such as the fourcolour theorem, please let us know). It's not really a problem that mathematicians have done in previous http://www.math.toronto.edu/mathnet/questionCorner/fourfours.html
Extractions: Question Corner and Discussion Area Asked by Hazen Simson, Grade 12, RCS-Netherwood on October 26, 1996 I would like to know if you know the answer to the four number problem that mathematicians have done in previous years and all mathematicians know of. It is something I have been trying for several months and without being able to get above 10 I am now searching for an answer. I'm not 100% sure what you're asking about, but I assume (from your remarks about not being able to get above 10) that you're referring to the problem about representing integers using four 4's and operations such as addition, multiplication, etc. (If I'm wrong and you're referring to something else, such as the four-colour theorem, please let us know). It's not really a problem "that mathematicians have done in previous years and all mathematicians know of" because it isn't really a mathematical problem. It depends on what notational convention one uses, rather than any kind of mathematical truths. In its most basic form, the puzzle is to combine four copies of the number 4, through the basic operations of negation, addition, subtraction, multiplication, division, and exponentiation, to come up with different integers.
Formal Proof—The Four- Color Theorem File Format PDF/Adobe Acrobat Quick View http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.141.714&rep=rep1&am
Four Colour Map Theorem From FOLDOC four colour theorem four colour map theorem mathematics, application (Or four colour theorem ) The theorem stating that if the plane is divided into connected regions which http://foldoc.org/four colour theorem
Four Color Theorem Intro Of course, the Four Color Theorem (previously called the Four Color Conjecture) was recently proven (by Wolfgang Haken and Kenneth Appel using a super http://www.jimloy.com/geometry/4color.htm
Extractions: Go to my home page We have all seen maps in which adjacent countries (or areas) are colored with different colors, so we can easily see the boundaries between them. Mathematicians asked, "Just how many colors are necessary?" They weren't trying to help out the map makers who occasionally bungle the job (I have seen several maps with mistakes in the coloring). The mathematicians found this an interesting, and diabolically difficult puzzle. Of course, the Four Color Theorem (previously called the Four Color Conjecture) was recently proven (by Wolfgang Haken and Kenneth Appel using a super computer at the University of Illinois, in 1976), showing that four colors is all you ever need, on a plane map. That proof is very long, and I will not show it. Instead, let's prove a "three color theorem:" Three color theorem - More than three colors are required for some map or maps. Proof: Look at the diagram, above left. Can't color it with just three colors, can you? That was a little informal. But, that is essentially the proof. We wanted to show that three colors were not enough for some map. All we have to do is show a map that requires four colors, and we have proved our conjecture. I could have given some reasoning why you can't color this map with three colors. But it should be fairly obvious.
The Four Color Theorem File Format PDF/Adobe Acrobat Quick View http://www.cs.washington.edu/homes/brun/pubs/pubs/Brun02four-color.pdf
Four Color Theorem Around 1998 Paul Kainen and I worked on an approach to the Four Color Theorem. He is a coauthor of a book on this topic reprinted by Dover Publications. http://cis.csuohio.edu/~somos/4ct.html
Extractions: Around 1998 Paul Kainen and I worked on an approach to the Four Color Theorem. He is a co-author of a book on this topic reprinted by Dover Publications. AUTHOR Saaty, Thomas L. TITLE The four-color problem : assaults and conquest / Thomas L. Saaty and Paul C. Kainen. PUBLISH INFO New York : Dover Publications, 1986. DESCRIPT'N vi, 217 p. : ill. ; 21 cm. NOTE Includes bibliographical references (p. 197-211) and index. SUBJECTS Four-color problem. LC NO QA612.19 .S2 1986. DEWEY NO 511/.5 19. OCLC # 12975758. ISBN 0486650928 (pbk.) : $6.00. AUTHOR Saaty, Thomas L. TITLE The four-color problem : assaults and conquest / Thomas L. Saaty and Paul C. Kainen. PUBLISHER New York : McGraw-Hill International Book Co., c1977. DESCRIPTION ix, 217 p. : ill. ; 25 cm. NOTES Bibliography: p. 197-211. Includes index. OCLC NO. 3186236. ISBN 0070543828 : $23.00. We take a pair of triangulations of a polygon and four color the vertices such that no two of the same color are connected by an edge of the triangulations. Polygon triangulations are easy to represent using data structures and the topological considerations of planarity are avoided. This turns the problem into a combinatorial one. The planarity reduces to circular order along the polygon and the non-crossing of diagonals. The history of this approach going back to Hassler Whitney and other references to this approach are in the book.
Sci.math FAQ: The Four Colour Theorem Note from archiver at 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 http://faqs.cs.uu.nl/na-dir/sci-math-faq/fourcolour.html
Extractions: 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. This article was archived around: 17 Feb 2000 22:52:04 GMT All FAQs in Directory: sci-math-faq
