Rational Curvature of Polytopes and the Euler Number - Algebraic Topology Lecture 16

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.

Introduction
Harriott's theorem
Ex. A has degree 3
Ex. A has degree 4
Tangles of opposite Cone at vertex A has internal tangles
Theorem. IF A polygon P has vertex A with tangles of facesat A.
Tetrahedron
Ex.2 Cube
Ex.3 Octahedron
Ex.4 Cosahedron
Ex.5 Dodecahedron
Problem 18. Computing directly the curvature at a vertex
Problem 19. Compute directly the curvatures