In this video lecture, Prof. N.J. Wildberger talks about rational curvature of polytopes and the Euler number.

We show that the total curvature of a polyhedron is equal to its Euler number. This only works with the rational formulation of curvature, using an analog of the turn angle suitable for the 2 dimensional sphere.

1. Introduction to Algebraic Topology
2. One-dimensional Objects
3. Homeomorphism and the Group Structure on a Circle
4. Two-dimensional Surfaces: The Sphere
5. More on the Sphere
6. Two-dimensional Objects: The Torus and Genus
7. Non-orientable Surfaces: The Mobius Band
8. The Klein Bottle and Projective Plane
9. Polyhedra and Euler's Formula
10. Applications of Euler's Formula and Graphs
11. More on Graphs and Euler's Formula
12. Rational Curvature, Winding and Turning
13. Duality for Polygons and the Fundamental Theorem of Algebra
14. More Applications of Winding Numbers
15. The Ham Sandwich Theorem and the Continuum
16. Rational Curvature of a Polytope
17. Rational Curvature of Polytopes and the Euler Number
18. Classification of Combinatorial Surfaces (Part I)
19. Classification of Combinatorial Surfaces (Part II)
20. An Algebraic ZIP Proof
21. The Geometry of Surfaces
22. The Two-holed Torus and 3-Crosscaps Surface
23. The Two-holed Torus and 3-Crosscaps Surface
24. Knots and Surfaces (Part I)
25. Knots and Surfaces (Part II)
26. The Fundamental Group
27. More on the Fundamental Group

In this course, Prof. N.J. Wildberger gives 26 video lectures on Algebraic Topology.  This is a beginner's course in Algebraic Topology given by Assoc. Prof. N J Wildberger of the School of Mathematics and Statistics, UNSW. It features a visual approach to the subject that stresses the importance of familiarity with specific examples. It also introduces 'rational curvature', a simple but important innovation. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.