Untitled Document But in 1890, percy heawood, an Oxford eccentric known for his immense mustacheand flowing cape, produced a map that showed that one of Kempe s chains http://www.jaschahoffman.com/articles/globescience/ittakesfour.html
Extractions: IT TAKES FOUR The Strange Career of a Cartographic Conjecture math why not five? by Jascha Hoffman Boston Globe SCIENCE / May 13, 2003 Here's a problem Lewis Carroll enjoyed posing to kids like Alice: how many colors do you need to fill in any map so that neighboring countries are always colored differently? It sounds simple enough. But when a Victorian law student first posed the question, guessing that it could be done with a mere four colors, logician Augustus De Morgan was stumped. While no one could devise a map that required more, a proof that every map requires only four colors proved remarkably elusive. Mapmakers didn't care, but problem-solvers were obsessed for decades, including the Bishop of London, a Kentucky colonel, and a California traffic cop. The question's very intractability has inspired innovations in computing and network theory, but some say it still has no satisfying solution. Oxford professor Robin Wilson's Four Colors Suffice: How the Map Problem was Solved (Princeton: 2003) presents the colorful history of this conjecture, with an unassuming lucidity that will appeal to the mathematical novice. It's thrilling to see great mathematicians fall for seductively simple proofs, then stumble on equally simple counter-examples. Or swallow their pride: after telling his class that the problem had been wasted on third-rate minds, the great number-theorist Herman Minkowski took weeks at the blackboard trying to solve it, finally admitting, "Heaven is angered by my arrogance; my proof is also defective."
Math Forum - Ask Dr. Math In 1879 a British mathematician, Alfred Kempe, published a proof that wasaccepted by the mathematics establishment until in 1890 percy heawood of Durham http://mathforum.org/library/drmath/view/52466.html
Extractions: Associated Topics Dr. Math Home Search Dr. Math Date: 11/11/97 at 23:25:27 From: Lisa Subject: The Four Color Map problem I just need help on what it is and where I can find information on it. I am having trouble finding information on it over the Internet. Thanks a lot. Date: 11/12/97 at 10:00:34 From: Doctor Anthony Subject: Re: The Four Color Map problem You can try Eric Weisstein's World of Mathematics, which will then refer you to a great deal of literature on the subject. http://mathworld.wolfram.com/Four-ColorTheorem.html http://mathforum.org/dr.math/ Associated Topics
Math Forum Discussions flaw in the proof was pointed out by percy John heawood in a paper in theQuarterly Journal of Mathematics. That s a good example. http://mathforum.org/kb/message.jspa?messageID=3848742&tstart=0
The Four Color Problem And Its Connection To South African Flora In 1890, percy John heawood stated that he had discovered an error in Kempe sproofan error so serious that he was unable to repair it. In his paper 3, http://io.uwinnipeg.ca/~ooellerm/guthrie/FourColor.html
Extractions: In the 1850's Francis Guthrie was the first mathematician to formulate the Four Color Problem . He asked whether it is possible to color any map with four or fewer colors so that adjacent regions (those that share a common boundary) are colored differently. At the time when he posed the problem, he was a student at University College in London. He attempted to prove that the counties of any map could be colored in this map with four colors. However, he was not entirely satisfied with his proof, so he mentioned his problem to his brother Frederick, who, in turn, mentioned it to his instructor, the famous Augustus De Morgan (after whom De Morgan's Laws of set theory are named). In a letter dated October 23, 1852, De Morgan mentioned the problem to Sir William Rowan Hamilton (for whom hamiltonian graphs are named). In his response, Hamilton, perhaps displaying his insight into the difficulty of mathematical problems, replied to De Morgan that he did not plan to consider this problem in the near future. Evidently, De Morgan spoke often of this problem with other mathematicians. Indeed, De Morgan is credited with writing an anonymous article in the April 14, 1860, issue of the journal Athenaeum in which he discusses the Four Colour Problem. This is the first known published reference to the problem.
Last Doubts Removed About The Proof Of The Four Color Theorem But he was mistaken, and eleven years later percy John heawood pointed out asignificant error in the argument. heawood was however able to salvage enough http://www.maa.org/devlin/devlin_01_05.html
Extractions: Search MAA Online MAA Home January 2005 At a scientific meeting in France last December, Dr. Georges Gonthier, a mathematician who works at Microsoft Research in Cambridge, England, described how he had used a new computer technology called a mathematical assistant to verify a proof of the famous Four Color Theorem, hopefully putting to rest any doubts about the result that had remained since the first proof of the theorem was announced in 1976. The story of the Four Color Problem begins in October 1852, when Francis Guthrie, a young mathematics graduate from University College London, was coloring in a map showing the counties of England. As he did so it occurred to him that the maximum number of colors required to color any map seemed likely to be four. The coloring has to meet the obvious requirement that no two regions (countries, counties, or whatever) sharing a length of common boundary should be given the same color. Guthrie's question became known as the Four Color Problem, and it grew to be the second most famous unsolved problem in mathematics after Fermat's last theorem.
Extractions: by Robin Wilson Robin Wilson can write. Of course, we've known that for a long time, from his many previous works (sometimes coauthored): thirteen books (by my last count) on graph theory and combinatorics; four volumes on Gilbert and Sullivan; four on the history of mathematics; one on mathematical stamps; and now a book telling the story of the solution of the four color problem. In writing it always helps to have a topic that is by its very nature an entertaining tale with a large and colorful cast of characters. In the author's own words, taken from the preface, we are told that this cast includes: "Lewis Carroll, the Bishop of London, a professor of French literature, an April Fool hoaxer, a botanist who loved heather, a mathematician with a passion for golf, a man who set his watch just once a year, a bridegroom who spent his honeymoon colouring maps, and a Californian traffic cop." Now let's take a look at the problem: Can every map be colored with at most four colors in such a way that neighboring countries are colored differently? The author explains, for his nonmathematical audience, that a proof that four colors suffice must show that all maps can be colored with four colors only. Showing that millions or billions of maps can be colored with four colors will not do.
Gresham College | Lecture Archive surprise when percy heawood of Durham dropped his bombshell in 1890. heawood also tried to generalise the idea of map colouring to other surfaces. http://www.gresham.ac.uk/event.asp?PageId=39&EventId=187
No. 1961: The Four-Color Problem Eleven years later, percy John heawood discovered an error in Kempe s proof.For an incorrect result to be published, much less unchallenged, http://www.uh.edu/engines/epi1961.htm
Extractions: THE FOUR COLOR PROBLEM John H. Lienhard presents guest Andrew Boyd Click here for audio of Episode 1961. Today, guest scientist Andrew Boyd colors maps. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them. I n 1852, Francis Guthrie found himself trying to color a map of the counties of England. It's very helpful to have a map where every bordering county is a different color, and Guthrie wondered how few colors he could use and still do this. Three colors wouldn't work, but he found he could make do with four. So he wondered, would four colors be enough for any map? Little did he realize that his question would lay its hold on generations of mathematicians. The seeming simplicity of the four color problem led countless people to try their hand at it over the years, including some of the world's most renowned mathematicians. Hermann Minkowski once told his students the problem remained unsolved because third-rate mathematicians had worked on it, only to admit much later that, "heaven is angered at my arrogance; my proof is also defective." The first would-be proof of the result was published in the American Journal of Mathematics in 1879, almost thirty years after Guthrie first posed the problem. One Alfred Bray Kempe, received great acclaim for it. He was made a Fellow of the Royal Society in England and was ultimately knighted.
Extractions: Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. What is the greatest number of color fields that can be arranged so that each maintains a border with all others? Bernard Frize's Heawood, 1999, a pair of painted sculptures in tire permanent collection of the MAMVP, and Heawood, 2003, the thirteen digital prints that introduce this show of the artist's mostly recent paintings, address this thorny question. The works' namesake, British mathematician Percy John Heawood, labored over this and related problems (which originated in cartography) in the years surrounding the turn of the last century; at one point, exploring three dimensional forms, he determined that no more than eight fields of color can abut one another on the surface of a double torus (a volume shaped, in accidental analogy, exactly like a three-dimensional figure eight). The twin Heawood sculptures are based on this formula.
ArtForum: Sergio Lombardo: Fondazione Mudima (named after their inventor, the mathematician percy John heawood) from 2003 . A heawood map is a subdivision of flat surfaces into sectors that the http://www.findarticles.com/p/articles/mi_m0268/is_5_43/ai_n9510547
Extractions: Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Sergio Lombardo has cut an anomalous, isolated, and eclectic figure on the Italian scene since the late '50s. Best known for his contribution to Italian Pop art in paintings of large, black silhouettes of famous people captured in gesti tipici, or characteristic gestures, Lombardo suddenly pulled away from that movement, around 1965, to tackle aesthetic research of a scientific and psychological natureapplying algorithms to "aesthetic stimuli" to test "empirical valuation." In 1979 he founded the Rivista di Psicologia dell'Arte (Journal of the Psychology of Art). [ILLUSTRATION OMITTED] "Beauty," the artist writes, "does not lie in the fact that we are dealing with an extremely difficult puzzle, but in the fact that the extreme complexity of these structures is a function of aesthetic values that cannot be achieved in uncontrived fashion." He continues: "These paintings cannot be imitated, but only copied, or reproduced.... Since in these maps every formal or chromatic decision has been subjected to experiential proof and responds to very complex laws of optimization, they can only be reproduced just as they are, or they are made worse." All this brings to mind the so-called micro-aesthetics of German philosopher Max Bense (1910-1990), so fashionable in certain abstract art circles in the early '60sa milieu in which aesthetic research focused on finding points of contact with mathematical or at most psychological objectification to establish a clear and scientific basis for concepts of pleasure and beauty.
MMS Online Graph Theory Course Introduction However, in 1890, another British mathematician, percy John heawood, found amistake in Kempe s work. The problem remained unsolved until 1976, when Kenneth http://www.math.lsa.umich.edu/mmss/coursesONLINE/graph/
BSHM: Gazetteer -- D A plaque to percy heawood (18611955) in Durham Castle records his efforts ingetting the Castle restored. heawood was Vice-Chancellor of the University and http://www.dcs.warwick.ac.uk/bshm/zingaz/D.html
Extractions: The British Society for the History of Mathematics HOME About BSHM BSHM Council Join BSHM ... Search Main Gazetteer A B C D ... Z Written by David Singmaster (zingmast@sbu.ac.uk ). Links to relevant external websites are being added occasionally to this gazetteer but the BSHM has no control over the availability or contents of these links. Please inform the BSHM Webster (A.Mann@gre.ac.uk) of any broken links. [When the gazetteer was edited for serial publication in the BSHM Newsletter, references were omitted since the bibliography was too substantial to be included. Publication on the web permits references to be included for material now being added to the website, but they are still absent from material originally prepared for the Newsletter - TM, August 2002] Return to the top. The father of Charles Lutwidge Dodgson (Lewis Carroll, 1832-1898) was vicar at All Saints, Daresbury. Carroll was born at the Old Parsonage, Morphany Lane, Newton-by-Daresbury, and lived there until 1843. The house burned down in 1883. Only the gateposts remain. Another source says the site, on Glebe Farm, has a pillar and a plaque. There is a commemorative window in the church, and a 'Wonderland' weathervane, showing the Mad Hatter, the White Rabbit and Alice, on the local primary school. The Sessions House and adjoining barn are to be restored as an exhibition centre by the Lewis Carroll Birthplace Trust.
American Scientist Online - Map Quest It was considered correct for 11 years, until percy John heawood, a mathematicianand classical scholar at Durham Colleges, discovered its fatal flaw. http://www.americanscientist.org/template/AssetDetail/assetid/21979
Extractions: Home Current Issue Archives Bookshelf ... Subscribe In This Section Reviewed in This Issue Book Reviews by Issue New Books Received Publishers' Directory ... Virtual Bookshelf Archive Site Search Advanced Search Visitor Login Username Password Help with login Forgot your password? Change your username see list of all reviews from this issue: July-August 2003 MATHEMATICS Daniel S. Silver Four Colors Suffice: How the Map Problem Was Solved . Robin Wilson. xiv + 262 pp. Princeton University Press. First published by Penguin Books in 2002. $24.95 Progress in mathematics is inevitable: Given enough time, even the most fearsome problem surrenders to some new attack. The fall of Fermat's Last Theorem to Andrew Wiles in the mid-1990s, after battles spanning three and a half centuries, is an example. A less well-known instance is Kenneth Appel and Wolfgang Haken's conquest in 1976 of the Four-Color Problem, the subject of this lively and captivating book by mathematician Robin Wilson. "A student of mine asked me today to give him a reason for a fact which I did not know was a fact—and do not yet," wrote the English mathematician Augustus De Morgan in 1852 to his Irish friend and colleague Sir William Rowan Hamilton. The "fact" is that only four colors are required to color any map in such a way that adjacent regions receive different colors. The student was Frederick Guthrie, but it was Frederick's older brother, Francis, who first proposed it. Francis decided that it must be true after coloring a map of the counties in England, and he allowed Frederick to submit the challenge to De Morgan. "What do you say?" continued De Morgan in his letter to Hamilton. "The more I think of it the more evident it seems."
Extractions: Algebra Explorations Astronomy Biology Chemistry ... NEXT >> Mathematicians have puzzled over this question for more than a century. The so-called "four-color map problem" gained attention in the 1850's, when a mathematics student in London named Francis Guthrie asked whether it is possible to color any map using four or fewer colors so that regions sharing a common boundary are colored differently. He mentioned this perplexing question to his brother Frederick Guthrie, who in turn mentioned it to his teacher, the famous mathematician Augustus De Morgan. In the years to follow, De Morgan discussed the problem with several other mathematicians who also became intrigued by this seemingly simple problem. In 1879, a British lawyer and amateur mathematician named Alfred Bray Kempe announced that he had proved that no more than four colors would be needed for any map. Eleven years later, however, a mathematician named Percy John Heawood found an error in Hempe's proof. Heawood then proved that any map could be completed with five colors. Finally, in 1976, two math professors at the University of Illinois, Kenneth Appel and Wolfgang Haken, used a computer to help prove that only four colors are necessary to complete any map. They had solved one of the biggest problems in mathematics!
VEDA MATEMATIKOVÉ V HISTORII percy John heawood Jirí Svrek percy heawoodnavtevoval Gymnázium královny Elisabeth v Ipswichu, kde v roce 1880 získal http://pes.internet.cz/veda/clanky/15266_0_0_0.html
Extractions: Archiv vydání Nadpis Autor Text èlánku Národní licence na elektronickou informaèní subu ProQuest 5000, díky ní mají od loòského roku vysoké koly, Akademie vìd ÈR, veøejné knihovny a dalí instituce monost bezplatného pøístupu ke ètyøem tisícùm odborných èasopisù, byla doèasnì -od 1. záøí do 31. prosince 2001 - rozíøena o dalí tøi významné oborové databáze: AGRICOLA PlusText, ERIC PlusText a MEDLINE with Full Text. Hlavní pøednost tìchto informaèních zdrojù spoèívá v tom, e spojují kvalitní indexování èlánkù pomocí øízených slovníkù (tezaurù) èásteènì s jejich plnými texty. AGRICOLA PlusText pokrývá zemìdìlství, veterinární lékaøství, lesnictví, ekologii, zoologii, botaniku a výivu. Databáze zahrnuje bibliografické záznamy z více ne 830 èasopisù od roku 1970 (z toho je 52 titulù k dispozici v plném znìní s prùmìrnì ètyøletou retrospektivou - napø. Biochemistry, Bioscience, Canadian Journal of Zoology, Journal of Nutrition aj.). ERIC FullText je zamìøena na rùzné aspekty vzdìlávání (vèetnì vývojové psychologie, protidrogové prevence, managementu kol, poradenství, výuky odborných pøedmìtù, celoivotního vzdìlávání) a rovnì na knihovnictví. Úplná retrospektiva bibliografických záznamù z 1060 èasopisù sahá do roku 1966 (zhruba ètvrtina z nich je pøístupná v plnotextové podobì od roku 1994 - napø. Adult Education, Communication and Education, Harvard Educational Policy, Journal of Higher Education aj.). MEDLINE with Full Text obsahuje 11 miliónù záznamù ze 4300 medicínských èasopisù, z nich 230 je pøístupných v plném znìní od roku 1997 (napø. AIDS Care, British Medical Journal, Canadian Medical Association Journal, JAMA, Journal of the American Medical Association, Lancet aj.). Texty jsou doprovázeny reprodukcemi fotografií, schémat a tabulek.
The 4 Color Map Problem His argument was considered correct until 1890 when percy John heawood discovereda flaw. Work by many people continued and the conjecture was finally http://www.math.utah.edu/~pa/math/4color.html
Extractions: Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah Suppose you have a map. Let's rule out degeneracies where a country has separate parts (like the continental U.S. and Alaska). Suppose you want to color all countries so they are easy to distinguish. In particular you want to color neighboring countries with different colors. How many colors do you need at most? (Two countries are " neighboring" if they share a border segment that consists of more than one point. If sharing one point was enough to be neighbors you could divide a pie into arbitrarily many slices all of which share the center, requiring as many colors as there are slices). In 1852, Francis Guthrie wrote to his brother Frederick saying it seemed that four colors were always sufficient, did Frederick know a proof. Frederick asked his advisor Augustus De Morgan. Morgan did not know either. In 1878 the mathematician Arthur Cayley presented the problem to the London Mathematical Society. Less than a year later Alfred Bray Kempe published a paper purporting to show that the conjecture is true. His argument was considered correct until 1890 when Percy John Heawood discovered a flaw. Work by many people continued and the conjecture was finally proved true in 1976 by Kenneth Appel and Wolfgang Haken.
J Theol Studies -- Index By Author ( 1911, Os-XIII[ 49]) heawood, percy J. PDF HOWORTH, HH PDF. J. JAMES, MR PDF. L. LATIMER JACKSON,H. PDF. S. SOUTER, A. PDF SOUTER, A. PDF SPAGNOLO, http://jts.oxfordjournals.org/content/volos-XIII/issue49/aindex.dtl
J Theol Studies -- Table Of Contents ( 1911, Os-XIII [49]) percy J. heawood {vav} {mem} {alef} AND {mem} {alef} J Theol Studies 1911 osXIII66-73; doi10.1093/jts/os-XIII.49.66 PDF http://jts.oxfordjournals.org/content/volos-XIII/issue49/index.dtl
Extractions: To see an article , click its [Full Text] or [PDF] link. To review many abstracts , check the boxes to the left of the titles you want, and click the 'Get All Checked Abstract(s)' button. To see one abstract at a time , click its [Abstract] link. H. H. HOWORTH THE INFLUENCE OF ST JEROME ON THE CANON OF THE WESTERN CHURCH. III.
Earliest Known Uses Of Some Of The Words Of Mathematics (F) MapColour Theorem is the title of a paper by percy John heawood (1861-1955)which appeared in the Quarterly Journal of Pure and Applied Mathematics in http://members.aol.com/jeff570/f.html
Extractions: Earliest Known Uses of Some of the Words of Mathematics (F) Last revision: July 28, 2005 F DISTRIBUTION. The F distribution was tabulated - and the letter introduced - by G. W. Snedecor Calculation and Interpretation of Analysis of Variance and Covariance (1934). (David, 1995). It is sometimes called Snedecor's F distribution although the letter was chosen to honor Fisher. Fisher's original development of the analysis of variance (see the entry variance.) was not based on F but on a related quantity z. (see the entry z and the z distribution.) The term F distribution is found in Leo A. Aroian, "A study of R. A. Fisher's z distribution and the related F distribution," Ann. Math. Statist. FACTOR (noun). Fibonacci (1202) used factus ex multiplicatione (Smith vol. 2, page 105). Factor appears in English in 1673 in Elements of Algebra by John Kersey: "The Quantities given to be multiplied one by the other are called Factors." FACTOR (verb) appears in English in 1848 in Algebra by J. Ray: "The principal use of factoring, is to shorten the work, and simplify the results of algebraic operations." Factorize (spelled "factorise") is found in 1886 in Algebra by G. Chrystal (OED2).
BBC - Radio 4 - Another 5 Numbers - The Number Four Then in 1890, percy heawood, a lecturer at Durham, discovered a flaw in Kempe stheory. When revised, it suggested that every map could be happily http://www.bbc.co.uk/radio4/science/another51.shtml
Extractions: Simon Singh's journey begins with the number 4, which for over a century has fuelled one of the most elusive problems in mathematics: is it true that any map can be coloured with just 4 colours so that no two neighbouring countries have the same colour? This question has tested some of the most imaginative minds - including Lewis Carroll's - and the eventual solution has aided the design of some of the world's most complex air and road networks. Most people's memories of geography at school will include two things. Firstly, there was learning by rote the groundnut crop yield figures for Senegal in the mid '70s, and secondly, there was colouring-in maps. Many hours were spent shading-in maps of Birmingham, scrounging around for different colour pens to distinguish Smethick from Oldbury. Now, a puzzle which owes more to maths than geography posits the question, how many different colour pens did you really need to pilfer from your mate's pencil-case in order to do Brum, so that no two adjacent suburbs had the same colour. Memories of youthful exuberance would suggest every pen in your pal's possession, but mathematically, the answer is 4.
