Why All Maps Are Wrong - Theorema Egregium

Explore the fascinating world of cartography and mathematics in this 40-minute video that delves into the Theorema Egregium and its implications for map projections. Discover why all maps are inherently flawed and learn about the challenges in creating accurate representations of Earth's surface. Examine the limitations of popular projections like Mercator and Gall-Peters, and understand why it's impossible to create a map that preserves both area and angles simultaneously. Follow along as the video presents a geometric proof of the Theorema Egregium, breaking down complex concepts into digestible chapters covering curvature, spherical areas, Gauss maps, geodesics, and parallel transport. Gain insights into the mathematical principles underlying cartography and expand your understanding of differential geometry.

Introduction
Chapter 1: Curvature
Chapter 2: Spherical areas
Chapter 3.1: Gauss map preserves parallel transport
Chapter 3.2: Geodesics preserved
Chapter 3.3: Parallel transport preserved
Chapter 3.4: Area = holonomy on sphere
Chapter 4: Tying everything together