7-colour theorem on the torus | Euler characteristic #2

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We only need 4 colours to colour any map on a plane. What if we want to colour a map on a torus? Turns out we need 7 colours, but why? The 7-colour theorem was proved way earlier than the 4-colour one, but why? A torus seems to be more complicated than a plane! Find out by watching this video.

Sources / Further reading:

(3) Heawood’s original paper is this: Heawood, P. J. (1890). "Map colour theorem". Quarterly Journal of Mathematics. 24: 332–338.

However, I can’t find it online, and couldn’t verify myself whether he actually proved the 7-colour theorem in this 1890 paper. The sources below (papers from 1950s to 60s) claimed that Heawood did this in the 1890 paper:

(4) The proof itself:

[Lecture notes on Part II Graph theory in 2021, using p. 34-35]

[Lecture notes for same course in 2007, but less detailed, p. 27-28]

(6) Vihart’s video on the 7-colour theorem, showing it is possible but not proving it:

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[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?

[.]

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

[johnchessant]

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

[bridgeon]

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

[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.

[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.

[rdbury]

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.

[kaidatong]

when I saw the title, I immediately thought, it can't be easy to do the impossible of proving a false statement

[convergentseries]

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.

[inciaradible]

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

[flyingbicycles]

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

[thehyperfinestructure]

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

[Khashayarissi-obyj]

So beautiful. Hoping for more videos.
To step onto the path of knowledge and this is what a beautiful path. To share the joy of knowing with others and this is what a beautiful feeling.
May God bless your lives.

[mostly_mental]

Excellent video as always. And thanks for the shoutout!

[Magnelibra]

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

[mynamemywish]

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 :)

[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.

[cubefromblender]

Including the ocean, it is not possible to do the 4 color theorem on a map, because or saint martin
France, Netherlands, Belgium, Germany, and the Sea all border eachother, requiring 5 colors (france and netherlands border on saint martin)

[TheOneMaddin]

Question: Why is 4-color theorem easier on the torus?

Answer: because it is false.

[DeathSugar]

can we redraw torus graph in term of knots?