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Counting the same thing twice — page 14

One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.
Peaks, pits and passes. Height along one direction on two bumpy surfaces: the torus has 5 peaks, 5 pits and 10 passes, total 0; the sphere has 1, 2 and 1, total 2. Topology

Peaks, pits and passes

Take any lumpy closed surface and any direction of height. Peaks plus pits minus passes is 2 on a sphere and 0 on a torus, whatever the lumps, in every direction. Average one vertex's score over all directions and it becomes the corner's angle defect, computed from nothing but comparisons. So Gauss–Bonnet follows from a count of which neighbours lie higher.

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