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The Mathematical Mechanic by Mark Levi

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10

A BICYCLE WHEEL AND THE GAUSS-BONNET THEOREM

10.1 Introduction

This chapter tells an interesting story on how playing with a bicycle wheel can connect to a fundamental theorem from differential geometry. The internal angles in a planar triangle add up to 180°. This fact can be restated in a more general and yet more basic way: if I walk around a closed curve in the plane, then my nose, treated as a vector, will rotate by 2π (provided that I always look straight ahead).1

Does the same hold for an inhabitant of a curved surface? Figure 10.1 shows a triangular path on the sphere. Two of the sides lie on meridians and one lies the equator. To a resident of the sphere the sides of the triangle appear to be straight lines.2 A plane flying around ...

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