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?
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
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.
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.
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 :)
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.
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?
Thanks, I now know I can get a perfect 7 icing distribution on my donuts.
Puzzle: how to color to minimize population loss if all blue countries turn into water
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
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.
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.
The idea is better on a mobius strip - this links all platonic solids as duals
Man please keep up this good work. Your videos really do help in understanding mathematics and what's happening visually!
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 :)
Excellent video as always. And thanks for the shoutout!
Cool topic! It reminds me of topological surface codes in quantum computing, like Kitaev on toruses :D
This was excellent, the math is well above my understanding truly but we are getting closer to understanding our realm, this was great.
awesome vid
And this is why Wi-Fi has 13 channels. Mobile phone cells are mod 7 ...
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.