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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
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