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.

## 3 thoughts on “AlgTop15: Rational curvature of a polytope”

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Quite correct, Leibnitz would have gotten a copy, or made a copy, of this letter.

Thanks for making these videos!

You give the impression that Descarte wrote to Leibnitz, but Descarte lived 31 March 1596 — 11 February 1650, and Leibnitz lived July 1, 1646 — November 14, 1716. I guess you meant Leibnitz got a hold of this letter from Descarte.

Stangle, LMAO.