Clocks and Curvature (Geometry and the Imagination)
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Conway, Doyle, Gilman, Thurston; The Geometry Center  
The total curvature of any surface topologically equivalent to the sphere is 4 pi. This can be seen very simply from the definition of the curvature of a region in terms of the angle of rotation when the surface is rolled around on the plane; the only problem is the perennial one of keeping proper track of multiples of pi when measuring the angle of rotation. Since are trying to show that the total curvature is a specific multiple of pi, this problem is crucial. So to begin with let's think carefully about how to reckon these angles correctly...  


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