Why is 4-color theorem easier on a torus?

[@mathemaniac]

Originally, I combined the previous video and this one, and found that it was probably too long, and so I split it up. The next video is also related to Euler characteristic - can you guess what it is?

[@1.4142]

Puzzle: how to color to minimize population loss if all blue countries turn into water

[@mrblakeboy1420]

we’ll start by having the colours red, yellow, green, and purple. now we have replaced the ocean with land and actually that’s probably more population loss than just taking away countries

[@sirati9770]

Solution: choose yellow, orange, red, and white as colours

[@xaigamer3129]

Puzzle: how to color to minimize population loss if all orange countries turn into lava

[@hongkonger885]

@@sirati9770the you'd have five colors

[@fluoriteByte]

Puzzle: how to color to minimize population loss if all white countries turned into semen

[@Blingsss]

The 7-color theorem for a torus really shows how fascinating math can get when we explore surfaces beyond the plane! It’s incredible that proving it is simpler than the 4-color theorem. Studying concepts like this alongside tools like SolutionInn has really helped me understand the beauty of topology and geometry.

[@johnchessant3012]

the fact that that formula still "works" in the case g = 0 is one of the freakiest things in all of math

[@NotSomeJustinWithoutAMoustache]

This four-color problem becoming easier to prove/disprove when you move from a flat surface to a torus reminds me of how the goat grazing problem got easier when you increase dimensions to the 3D analog with the tethered bird. It seems like some math problems have a tendency of becoming simpler when you increase dimensions.

[@mathemaniac]

Not all maths problems will become easier when you increase dimensions, but yes, sometimes, the seemingly more difficult problem can in fact be easier to solve.

[@bridgeon7502]

Thanks, I now know I can get a perfect 7 icing distribution on my donuts.

[@rdbury507]

This is something I've been curious about for a while, so thanks for the (relatively) simple explanation. The torus is topologically the same as a parallelogram if the parallelogram is joined to itself at the edges to form an infinitely repeating pattern. It's fairly simple to color a hexagonal tiling of the plane in a repeating pattern so that each hexagon and its six neighbors are all different colors. The pattern is repeating, so it can mapped to a torus and the resulting graph is the complete graph on seven vertices, thus proving that 7 colors are necessary.

[@mathemaniac]

For some reason, I haven't thought of this. In fact, I was only reasonably sure (but not certain) that the necessity of 7 colours was proved by Heawood himself, and I was almost certain that he proved the sufficiency of 7 colours. This reasoning gives me much higher confidence that Heawood definitely could have seen this construction and therefore the 7-colour theorem (and the bound being sharp) was almost certainly already proved by then, i.e. 1890.

[@carly09et]

The idea is better on a mobius strip - this links all platonic solids as duals

[@52flyingbicycles]

The Heawood characteristic applying to a plane is simultaneously awesome and a great reminder of why learning math is a lot more than memorizing results

[@inciaradible7144]

Ah, so that's why our maps aren't projected onto a torus; you need way more colours.

[@thehyperfinestructure6550]

Man please keep up this good work. Your videos really do help in understanding mathematics and what's happening visually!

[@mynamemywish0]

Loved the shoutout at the end. I have been looking for mathy (and ML-y) channels for a while. Found some amazing ones off of SOME-x submissions. Maybe a list of small channels with high quality content would be an amazing resource for the community :)

[@mostly_mental]

Excellent video as always. And thanks for the shoutout!

[@kruksog]

Ive just started the video, so i dont know where this is going to go, but i wanted to respond to the statement "a torus seems more complicated than the plane, yet the 7 color theorem was proved so much earlier than the 4 color theorem." Ive found this phenomenon in math more than a couple times: what may initially seem like additional complexity is actually additional structure you can kind of "grab onto" when proving a statement about that object. This isnt always the case, but id bet its happening here.

[@mathemaniac]

I know why you'd think this, but this is fundamentally different here: those more complicated structures usually constrain the object more, i.e. you are considering a special case, and so you can use more properties of them. This is different in the sense that the torus is not a special case of a plane / sphere.

[@proxye-mail1698]

@mathemaniac As another commenter pointed out, the torus is a quotient space of the plane, so in a certain sense the torus *is* a special case of a plane.

[@Magnelibra]

This was excellent, the math is well above my understanding truly but we are getting closer to understanding our realm, this was great.

[@e-pluszak9419]

I mean it is logically equivalent, but it's waaaay more intuitive to phrase this proof of χ ≤ maximal_degree + 1 in a constructive way: "there's a vertex whose degree is ≤ k - 1, remove it, what is left still has a vertex with degree ≤ k - 1 remove it etc. keep going until you've removed everything and now add them back in reverse order being able to choose a colour at each step as at every step the new vertex has at most k - 1 neighbors, so at most k - 1 impossible colours" rather than do something abstract like "assume there are counterexamples, pick the smallest of them"

[@mathemaniac]

Maybe it's just me, but I find your argument more difficult to follow than picking a minimal example (?)

[@pseudolullus]

Cool topic! It reminds me of topological surface codes in quantum computing, like Kitaev on toruses :D

[@DeathSugar]

can we redraw torus graph in term of knots?

[@SteveMacSticky]

Command Line Interface Torus?

[@-taehyun]

awesome vid

[@convergentseries3508]

I'm actually kind of surprised by how simple the proof is in this case (as long as you're willing to black box some stuff about Euler characteristic, at least)! Good work as always.

[@Kavoshgar-c2i]

Pleas make a video about manifolds

[@carly09et]

And this is why Wi-Fi has 13 channels. Mobile phone cells are mod 7 ...

[@g3452sgp]

Who is talking? Is AI talking? Or real human is talking?

[@TheOneMaddin]

Question: Why is 4-color theorem easier on the torus? Answer: because it is false.

[@WilliamWizer]

I can't figure WHY it's so hard to prove the 4-color theorem. you only need the chrome logo and a bit of common sense.

[@mathemaniac]

You need to prove that ANY graph can be coloured in 4 colours. If I remember correctly, the 4-colour theorem was *thought* to be proved a lot earlier, but people noticed a very subtle error that invalidated the argument.

[@WilliamWizer]

@mathemaniac indeed it's absurdly easy to prove that any 2 dimensional graph can be colored in only 4 colors. you just need to look at the chrome logo. it has only 4 colors and it's impossible to add a fifth that touches all four without breaking their union. a circle area surrounded by an annulus split in 3 areas that each cover 120º of said annulus. any attempt to add a fifth color that touches the existing four will have to include a radius of the annulus forcing two of the connected colors to be separated. no need for more. you can't add a fifth region unless you break the existing unions allowing to reuse colors.