Explore the relationship between the total curvature of a polyhedron and its Euler number in this 35-minute lecture on Algebraic Topology. Delve into Harriot's theorem on spherical polygon areas and examine the rational formulation of curvature using an analog of the turn angle for the 2-dimensional sphere. Analyze various polyhedra, including tetrahedrons, cubes, octahedrons, icosahedrons, and dodecahedrons, to understand their curvature properties. Work through practical examples and problems to directly compute curvatures at vertices, gaining a deeper understanding of this fundamental concept in geometry and topology.