We introduce a new, rational definition of the curvature of a polytope. This removes the usual pi’s that occur in such formulas, giving a more direct connection to the Euler number: total curvature equals Euler number. We use our new normalization of angle called turn-angle, or “tangle” to define the curvature of a polygon P at a vertex A. This number is obtained by studying the opposite cone at the vertex A, whose faces are perpendicular to the edges of P meeting at A. A classical theorem of Harriot on spherical triangles is important. This the 15th lecture in this beginner’s course on Algebraic Topology given by Assoc Prof NJ Wildberger at UNSW.

# AlgTop15: Rational curvature of a polytope

3