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 N J Wildberger at UNSW.

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Here are the Insights into Mathematics Playlists:

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Here are the Wild Egg Maths Playlists (some available only to Members!)

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