the four color theorem...
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From: noemata on 13 May 2010 13:26 First, I have to say that I'm not a mathematician. Maybe that's why I don't understand why the four color theorem has been so difficult to prove. The theorem states that no more than four colors are necessary to color the regions of any map to separate them. My understanding goes like this: First you try to draw a counterexample. Then you realize it's impossible. And then you realize why: All the regions have to touch all other regions - three goes fine, the fourth has to surround at least one of them which in turn frees its color for reuse. This problem seems to translate nicely into graph theory where the colors are dots and their contacts are lines between. Given this, it's possible to draw a graph with 1, 2, 3 and 4 dots with lines connecting all-to-all without any line crossing each other. If you try to add a 5. dot, its connecting lines will have to cross an existing line; so, it's not possible to draw a graph with five dots without any line crossing another line. See image: http://noemata.net/ontheball/graph-4-color-theorem.png This seems trivial, so why has the theorem been difficult to prove? Restating the theorem as: It is impossible to draw a two-dimensional map where five or more colors are necessary, it then seems intuitively true via graph theory. Now the problem might still exist, in the word "necessary" - who knows? For me this is proof enough. On the other hand my ideas are often crank... But what's wrong?
From: Hero on 13 May 2010 15:15 noemata wrote: > First, I have to say that I'm not a mathematician. Maybe that's why I > don't understand why the four color theorem has been so difficult to > prove. > The theorem states that no more than four colors are necessary to > color > the regions of any map to separate them. > My understanding goes like this: > First you try to draw a counterexample. Then you realize it's > impossible. And then you realize why: All the regions have to touch > all > other regions - three goes fine, the fourth has to surround at least > one > of them which in turn frees its color for reuse. > This problem > seems > to translate nicely into graph theory where the > colors are dots and their contacts are lines between. Given this, it's > possible to draw a graph with 1, 2, 3 and 4 dots with lines connecting > all-to-all without any line crossing each other. If you try to add a > 5. > dot, its connecting lines will have to cross an existing line; so, > it's > not possible to draw a graph with five dots without any line crossing > another line. See image:http://noemata.net/ontheball/graph-4-color-theorem.png > This seems trivial, so why has the theorem been difficult to prove? > Restating the theorem as: It is impossible to draw a two-dimensional > map > where five or more colors are necessary, it then seems intuitively > true > via graph theory. Now the problem might still exist, in the word > "necessary" - who knows? > For me this is proof enough. On the other hand my ideas are often > crank... But what's wrong? It seems... but You see, someone has a drawing with 273 points arranged in a special manner and in this there are 6 points which connect directly, without crossing. You might not believe and want to see this drawing. No - it's up to You, to show that this drawing does not exist. So You need some fantasy to explain this or hard work in drawing all possible graphs of 273 points - if You want to prove, and not just believe Your intuiton. And to be honest, points are no coloured areas. So if there is a structural identy between a coloured map and line-connected points, this is not total, it's also differing in some respects. You must show as well, that for this problem the identy in structur exists. With friendly greetings Hero
From: spudnik on 13 May 2010 15:26 the "mapping" from polygons (countries) to vertices is a simple, dual relationship, and it has been used on 4CT since the beginning, probably to save coloring costs in printing. > And to be honest, points are no coloured areas. So if there is a > structural identy between a coloured map and line-connected > points, this is not total, it's also differing in some respects. thus: on the wayside, directly proportional means, not inversely proportional, as in the "inverse second-power law" that Hooke derived from Kepler's orbital constraints (and, it has nothing in particular to do with "skwares" .-) thus: the M-set's property of "universality" or self- similarity -- the mini-bugs like the big cardioid -- is strictly an artifact of the floating-point spec (and its many implimentations); however, monsieur M. could only beg the question, over ten years ago, because he never bothered to speak with the engineers at his IBM fellowship. (he had not gotten any further, when he came to campus & spoke, again, couple o'years ago .-) > http://arxiv.org/PS_cache/hep-ph/pdf/0112/0112317v1.pdf > Fractals are usually build with complex numbers, like the Mandelbrodt thus: yeah, but what is the integer, Avagadro's number?... do you know the surfer's value of pi? thus: well, that is where the problem with assigning a particle to a wave, a la de Broglie et al, comes. the assumption, that causes folks to say "particle," is that because a quantum (wave) of light is absorbed by one atom of siver dioxide (say, in the photographic emulsion; or, other detector) --some how-- that it must be that a rock of light hit the electronic orbital (although this is never specified, as to how it could be, and the whole problem of EM is also hard to describe, and is confounded with the absurd notion of the "plane wave"). this is really all of a confusion from Newton's "geometrical optics," that is, the "ray" of light, which is just one "normal" to the wave (or Huyghens wavelet). > You assume the particle exits both slits because you assume the > particle creates the interference pattern in and of itself. thus: about your five "cloture" events, the real problem is that "the Fed" was never properly ratified (and is unconstitutional for that reason, if not directly; it is modeled upon the Federal Reserve System of England). of coursel the 527 cmtes. have essentially taken over the TV advertizing on all national issues & candidates, through an Act that was passed unaanimously in both houses. > "Senate rules don't trump the Constitution" --http://GreaterVoice.org/60 thus: I've been saying, for a while, that if "green" gasoline can be made, and gasoline fuel cells, what is the problem with Fossilized Fuels (TM), which ain't fossilized? ... anyway, see "Green Freedom" in the article, which is not quite what I was refering to! > Thorium has other interesting features. For example, in > oxide form as would probably be used, Thorium has a > higher thermal conductivity than Uranium oxide. That > means the fuel will be cooler for any given power output. > It's got interesting mechanical properties also. > There are a number of new reactor designs being touted. > http://thorium.50webs.com/ thus: Copenhagen's "reifiying" of the mere probabilities of detection, is the biggest problem, whence comes both "perfect vacuum" and "quantum foam" etc. ad vomitorium, as well as the brain-dead "photon" of massless and momentumless and pointy rocks o'light, perfectly aimed at the recieving cone in your eye, like a small pizza pie. thus: all vacuums are good, if they suck hard enough, but there is no absolute vacuum, either on theoretical or Copenhagenskooler fuzzy math grounds. thus: magnetohydrodynamics is probably the way to go, yes; not "perfect vacuum or bearings" -- and, where did the link about YORP, include any thing about the air-pressure?... seems to me, it's assuming Pascal's old, perfected Plenum. twist your mind away from the "illustrated in _Conceptual Physics/for Dummies_" nothingness of the massless & momentumless & pointy "photon" of the Nobel-winning "effect" in an electronic device -- yeah, CCDs -- the Committee's lame attempt to "save the dysappearance" of Newton's corpuscle. also, please don't brag about free God-am energy, til you can demonstrate it in a perpetuum mobile! > It stops because it has bad bearings. These asteroids thus: so, a lightmill is that thing with black & white vanes on a spindle in a relative vacuum? you can't rely on "rocks o'light" to impart momentum to these vanes, only to be absorbed electromagnetically by atoms in them; then, perhaps, the "warm side" will have some aerodynamic/thermal effect on the air in the bulb, compared to the cool one. thus: even if neutrinos don't exist, Michelson and Morely didn't get no results! > Could neutrino availability affect decay rates? thus: every technique has problems. like, you can't grow hemp-for haemorrhoids under a photovoltaic, without a good lightbulb. the real problem is that, if Santa Monica is any indication, the solar-subsidy bandwagon is part of the cargo-cult from Southwest Asia (as is the compact flourescent lightbub, the LED lightbulb etc. ad vomitorium). > Government subsidies, and fat returns on PVs? --Light: A History! http://wlym.com
From: Hero on 13 May 2010 15:49 spudnik wrote: > the "mapping" from polygons (countries) > to vertices is a simple, dual relationship, and > it has been used on 4CT since the beginning, > probably to save coloring costs in printing. That it has been used is not revealing, that it might deliver next time a wrong result. You have to prove. > > > And to be honest, points are no coloured areas. So if there is a > > structural identy between a coloured map and line-connected > > points, this is not total, it's also differing in some respects. > > thus: > on the wayside, .... > thus: > the M-set's property of "universality" or self- .... > thus: > yeah, but what is the integer, Avagadro's number?... ...... > thus: > well, that is where the problem with assigning a particle ,,, > thus: > about your five "cloture" events, the real problem is that ...... > thus: > I've been saying, for a while, that if "green" gasoline can ...... > thus: > Copenhagen's "reifiying" of the mere probabilities ..... > thus: > all vacuums are good, if they suck hard enough, but > thus: > magnetohydrodynamics is probably the way to go, yes; .... > thus: > so, a lightmill is that thing with black & white vanes ...... > thus: > even if neutrinos don't exist, ...... > thus: > every technique has problems. like, ---- > ... etc. ad vomitorium). With friendly greetings Hero
From: master1729 on 13 May 2010 13:25
> First, I have to say that I'm not a mathematician. > Maybe that's why I > don't understand why the four color theorem has been > so difficult to > prove. > The theorem states that no more than four colors are > necessary to > color > the regions of any map to separate them. > My understanding goes like this: > First you try to draw a counterexample. Then you > realize it's > impossible. And then you realize why: All the regions > have to touch > all > other regions - three goes fine, the fourth has to > surround at least > one > of them which in turn frees its color for reuse. > This problem seems to translate nicely into graph > theory where the > colors are dots and their contacts are lines between. > Given this, it's > possible to draw a graph with 1, 2, 3 and 4 dots with > lines connecting > all-to-all without any line crossing each other. If > you try to add a > 5. > dot, its connecting lines will have to cross an > existing line; so, > it's > not possible to draw a graph with five dots without > any line crossing > another line. See image: > http://noemata.net/ontheball/graph-4-color-theorem.png > This seems trivial, so why has the theorem been > difficult to prove? > Restating the theorem as: It is impossible to draw a > two-dimensional > map > where five or more colors are necessary, it then > seems intuitively > true > via graph theory. Now the problem might still exist, > in the word > "necessary" - who knows? > For me this is proof enough. On the other hand my > ideas are often > crank... But what's wrong? i have a proof using infinite descent. |