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

Video Description

My favourites for SoMEpi: https://www.youtube.com/playlist?list=PLDcSwjT2BF_UQ0PfLWRojkpk8u5y-QhUH (1) Fibonacci game: https://www.youtube.com/watch?v=MratE0dzrHc (2) Visual proof of integral (not in the official list for technical reasons, but I came across this while I was doing the voting process): https://www.youtube.com/watch?v=V2gv-BP9SpU (3) Visual proof of combinatorial identity: https://www.youtube.com/watch?v=w1OQD1bCgyI (4) Elementary argument for higher-dimensional spherical volumes: https://www.youtube.com/watch?v=cZgXDOI1dRQ (5) Different way of thinking about Taylor series: https://www.youtube.com/watch?v=fMC4qk3S1Lw Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: 7colourtheoremontorus 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: (1) 4-colour theorem map image: https://mapsontheweb.zoom-maps.com/post/101089185091/the-four-colour-theorem-you-need-only-four (2) I used this picture to reconstruct the 7-colour map: https://demonstrations.wolfram.com/SevenColoringOfATorus/ (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: https://www.jstor.org/stable/27955905 https://www.pnas.org/doi/pdf/10.1073/pnas.60.2.438 (4) The proof itself: [Lecture notes on Part II Graph theory in 2021, using p. 34-35] https://ak2316.user.srcf.net/files/ii-graph-theory/graph-theory.pdf [Lecture notes for same course in 2007, but less detailed, p. 27-28] https://tartarus.org/gareth/maths/notes/ii/Graph_Theory.pdf (5) Link to the GeoGebra file directly: https://www.mathemaniac.co.uk/s/Torus-7-colour-map-for-public.ggb (6) Vihart’s video on the 7-colour theorem, showing it is possible but not proving it: https://www.youtube.com/watch?v=CruQylWSfoU This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Video chapters: 00:00 Introduction 02:38 Degree sum = 2E 04:05 3F ≤ 2E 06:46 The proof - minimal example 09:33 Combining inequalities 13:22 Is the bound sharp? 15:09 My #SoMEpi favourites Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!