hi sir your counterexample either has a mistake or i'm missing something. you forgot to connect the blue circle on the left to orange below it and blue circle on right to green below it at 16:08. what this ended up doing is that blue circle on the left can simply be changed to orange without any harm (i'm not talking about alternating sequence just the one circle) and similarly the blue circle on the right can be changed to green without any harm.therefore eliminating both blue circles and blue color hence can be used to color that vertex. this can be fixed by connecting blue circle on the left to orange below it and blue circle on the right to green below it.similarly green and orange circles at the bottom must be connected as well.
Thank you for the video that highlights where mathematics did go wrong for once. I saw Kempe's proof in another video, and I had heard it was wrong, but I still needed a day to find a counterexample. Still a rewarding experience.
This video is great, thank you. Do you have a link to the source where you gathered this information on the four colour theorem? I would like to read more about this method.
Come on Math At Andrews, please try to answer my question: Is the max number of nodes with each-to-each connections in 2D (4) related to the minimum number of colors to separate map regions in 2D... Would calculating how many connectors cross for 5+ nodes help.. In 1D it's 2 nodes and 2 colors, in 3D, given infinite sized nodes/regions it's potentially infinite colors/connectors needed? (although finite, node+connector/region size-related formulas are computable).. I just want to know whether I'm totally barking up the wrong tree or on to something. I am a decent programmer, just not a super-advanced mathematician. I understand the official proof. We can share the prize! LOL.
@@Achill101 lets say you have 8 blocks, 4 on top of 4, to make a cube. Could the blocks use only 4 colors, so non of the colors touch a block of the same color?
Come on Math at Andrews, try and answer my question.. Is the max number of each-to-each connected objects without connectors crossing in 2D (4) related to the minimum number of colors needed to separate each region on a 2D map.. Would counting the number of crossed connector for more objects help with the proof... In 1D only 2 colors are needed to separate map regions and only 2 objects can be connected each to each.. In 3D the algos break down to infinity if we have infinite sized objects / regions.. Come on, help me out, we can share the prize if I'm correct!.. It's bugging me again. Put my mind at rest and tell me why I'm barking up the wrong tree, if I am... Cheers for highlighting the problem and solution, got me thinking.
You should have explained why edges can't overlap in the real-world graphs or at least state that it was an exercise for the viewer. Also, you skipped the case when two adjacent colors are the same.
The graph solution is much more complicated than mine... In 2D, the maximum number of nodes that can be connected to each other (each to each) without connectors crossing is 4... I know it's not an actual mathematical proof but it is a graphical abstraction of the same problem. I'd like to know if I'm correct, perhaps you could tell me, I don't know any proper mathematicians.
@@guysheaperd4377 .. What do you mean by research? I have researched the current graph proof and understand it enough to implement it. Mine is much simpler to implement as a graphic formula (each to each * node count). Showing 4 is the max without lines crossing can be done using a drawing and bitmap search routine, and no doubt mathematically, formulaicly. Is it the same problem as the map region color problem, that's the question... II think yes, but I cannot prove it is. Mine is a maximum constant, the map region color assertion is a minimum constant. but I think they are two sides of the same problem... I just can't prove it...
@@PrivateSi sorry it took too long to reply. listen budd. i ain't no harvard grad. i didnt even graduate school. i cant understand this... whatever this is... i hope u prove it bud...gud knowin u . peace
@@guysheaperd4377 .. Cheers. I didn't fully understand the official proof from this vid either, I had to read a few explanations on different web sites and run through the algo a few steps before I got the gist. T'aint easy! It's on my back burner, I have no idea how to show the two constants are related, but they sure 'seem' to be. Have a good 'un.
privateSzi--- you are wrong, it is hard to explain you why you are wrong because you are not a professional mathematician therefore you cannot have a _precise_ feeling where you make an error and where you are 100% correct. having said that --- i will try though to explain you why you are wrong in your jugdement. see continuation below. if you don't understand what I say below, I would not make additional explanation because it will be useless. here is my explanation: technically -- is is much easier to prove or disprove the 4CT theorem when we describe the 4 coloring theorm (= 4CT) in the "dual version" of 4CT (the dual version is when: we claim that we can color vertices by 4 colors for which 4TC holds if there is a vertices coloring by 4 colors only, for which any two vertices which are connected by an edge should have two different colors). the non dual version (=the theorem about "maps"), is somewhat obscure in its description because "map" it s an obscure object (= not clearly defined). therefore you can "prove" with errors in the proof about maps because the obscure description of map hides your errors in your proof.
Kempe’s approach is so intuitive! Thank you for sharing. You’ve helped me summarize the four color theorem for my high schoolers.
your teaching style is amazing thanks a lot! I watched this video several times!
hi sir your counterexample either has a mistake or i'm missing something.
you forgot to connect the blue circle on the left to orange below it and blue circle on right to green below it at 16:08.
what this ended up doing is that blue circle on the left can simply be changed to orange without any harm (i'm not talking about alternating sequence just the one circle) and similarly the blue circle on the right can be changed to green without any harm.therefore eliminating both blue circles and blue color hence can be used to color that vertex.
this can be fixed by connecting blue circle on the left to orange below it and blue circle on the right to green below it.similarly green and orange circles at the bottom must be connected as well.
Thank you so much.This helped me to prepare for the seminar named ' Kempe Chains'
I was happy to find an explanation of the 5 color theorem in clear english!
Thank you so much
You have made us to fall in love with graph theory
Thank you for the video that highlights where mathematics did go wrong for once. I saw Kempe's proof in another video, and I had heard it was wrong, but I still needed a day to find a counterexample. Still a rewarding experience.
It took mathematicians 10 years to notice it was wrong and find a counterexample, so kudos to you for doing so in a day!
@@MathatAndrews - I had the advantage to know that there had to be a fault, at least all the mathematicians told me :-)
For the counterpoint example, what if you just changed the top pink color to a blue? Or is it specifically against Kempe
Great video. You deserve much more views and likes.
👏👏 thankyou sir... understood well ❤
Very well explained. Thank you.
This video is great, thank you. Do you have a link to the source where you gathered this information on the four colour theorem? I would like to read more about this method.
Come on Math At Andrews, please try to answer my question: Is the max number of nodes with each-to-each connections in 2D (4) related to the minimum number of colors to separate map regions in 2D... Would calculating how many connectors cross for 5+ nodes help.. In 1D it's 2 nodes and 2 colors, in 3D, given infinite sized nodes/regions it's potentially infinite colors/connectors needed? (although finite, node+connector/region size-related formulas are computable).. I just want to know whether I'm totally barking up the wrong tree or on to something. I am a decent programmer, just not a super-advanced mathematician. I understand the official proof. We can share the prize! LOL.
Very nice. Except the outro is not respectful to the people who finished it, by only crediting 'supercomputers' or something like that.
初見です
いつも拝見さしていただいております！！！
but isnt it flawed to assume that things can only connect based ona 2D plane? what if you didnt consider 2D or 3D planes? then what would happen???
I don't understand your question and what a 3D plane should be. The four color problem was posed for maps, which are 2-d representations.
@@Achill101 what about a 3D map?
@@christopherwalsh3101 - the 4-color theorem holds for a 2-d map.
. . . What is 3-d "map"? How does it look like?
@@Achill101 lets say you have 8 blocks, 4 on top of 4, to make a cube. Could the blocks use only 4 colors, so non of the colors touch a block of the same color?
@@Achill101 reality is in 3D. therefore, math should always describe a 3D world, not hypothetical 2D universes.
Come on Math at Andrews, try and answer my question.. Is the max number of each-to-each connected objects without connectors crossing in 2D (4) related to the minimum number of colors needed to separate each region on a 2D map.. Would counting the number of crossed connector for more objects help with the proof... In 1D only 2 colors are needed to separate map regions and only 2 objects can be connected each to each.. In 3D the algos break down to infinity if we have infinite sized objects / regions.. Come on, help me out, we can share the prize if I'm correct!.. It's bugging me again. Put my mind at rest and tell me why I'm barking up the wrong tree, if I am... Cheers for highlighting the problem and solution, got me thinking.
You should have explained why edges can't overlap in the real-world graphs or at least state that it was an exercise for the viewer. Also, you skipped the case when two adjacent colors are the same.
The graph solution is much more complicated than mine... In 2D, the maximum number of nodes that can be connected to each other (each to each) without connectors crossing is 4... I know it's not an actual mathematical proof but it is a graphical abstraction of the same problem. I'd like to know if I'm correct, perhaps you could tell me, I don't know any proper mathematicians.
hey, i tried and got one just like yours....interested in doing some research together?
@@guysheaperd4377 .. What do you mean by research? I have researched the current graph proof and understand it enough to implement it. Mine is much simpler to implement as a graphic formula (each to each * node count). Showing 4 is the max without lines crossing can be done using a drawing and bitmap search routine, and no doubt mathematically, formulaicly. Is it the same problem as the map region color problem, that's the question... II think yes, but I cannot prove it is. Mine is a maximum constant, the map region color assertion is a minimum constant. but I think they are two sides of the same problem... I just can't prove it...
@@PrivateSi sorry it took too long to reply. listen budd. i ain't no harvard grad. i didnt even graduate school. i cant understand this... whatever this is... i hope u prove it bud...gud knowin u . peace
@@guysheaperd4377 .. Cheers. I didn't fully understand the official proof from this vid either, I had to read a few explanations on different web sites and run through the algo a few steps before I got the gist. T'aint easy! It's on my back burner, I have no idea how to show the two constants are related, but they sure 'seem' to be. Have a good 'un.
privateSzi---
you are wrong, it is hard to explain you why you are wrong because you are not a professional mathematician therefore you cannot have a _precise_ feeling where you make an error and where you are 100% correct.
having said that --- i will try though to explain you why you are wrong in your jugdement. see continuation below.
if you don't understand what I say below, I would not make additional explanation because it will be useless.
here is my explanation:
technically -- is is much easier to prove or disprove the 4CT theorem when we describe the 4
coloring theorm (= 4CT) in the "dual version" of 4CT (the dual version is when: we claim that we can color vertices by 4 colors for which 4TC holds if there is a vertices coloring by 4 colors only, for which any two vertices which are connected by an edge should have two different colors). the non dual version (=the theorem about "maps"), is somewhat obscure in its description because "map" it s an obscure object (= not clearly defined). therefore you can "prove" with errors in the proof about maps because the obscure description of map hides your errors in your proof.
高評価押しました！！