Explore Gauss's perspective on curvature and the Theorema Egregium in this comprehensive differential geometry lecture. Delve into the Gauss-Rodrigues map and its relation to curvature, examining the coefficients of first and second fundamental forms. Investigate a paraboloid example to compare with previous curvature discussions. Discover a discrete analog of curvature for polyhedra, tracing back to Descartes, and learn how it provides an easy justification for Gauss's Theorema Egregium. Examine the Gauss-Bonnet theorem in this context, understanding how the total curvature of a closed surface with sphere topology relates to its Euler characteristic. Gain insights into both smooth and discrete differential geometry, exploring concepts such as normal level surfaces, curve fitting, and polyhedron vertices through detailed computations and explanations.