Theorema Egregium: why all maps are wrong

Head to https://squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac. Website for files download (remember to use the password shown in the video!): https://www.mathemaniac.co.uk/download The Mercator projection is the standard world map, but it famously makes Greenland and Africa the same size, but in reality, Greenland is so much smaller. Gall-Peters projection aims to solve exactly this area mismatch problem, but the shape resulted is horrible, and actually unsuitable for any navigation, unlike Mercator. Can we make a world map that preserves both areas (like Gall-Peters) and angles (like Mercator)? No, and the reason why is Theorema Egregium, the subject of the video. Traditionally, Theorema Egregium was proved with a lot of tedious calculations, and somehow magically, you can compute the curvature with the "first fundamental form", whatever that means. It took until more than a century later than its original discovery that a geometric proof was found, and is presented here. Theorema Egregium, more traditional proof, going through first and second fundamental forms: https://www.dpmms.cam.ac.uk/~gpp24/dg... Tristan Needham's book on visual differential geometry: https://www.amazon.co.uk/Visual-Diffe... Video chapters: 00:00 Introduction 02:40 Chapter 1: Curvature 10:32 Chapter 2: Spherical areas 17:34 Chapter 3.1: Gauss map preserves parallel transport 22:15 Chapter 3.2: Geodesics preserved 27:16 Chapter 3.3: Parallel transport preserved 31:46 Chapter 3.4: Area = holonomy on sphere 36:43 Chapter 4: Tying everything together 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:   / mathemaniacyt   Instagram:   / _mathemaniac_   Twitter:   / mathemaniacyt   Patreon:   / 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!